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UnluckyOrangutan
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tactic-haveDraft8

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obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hm.symm hu)).1
refine_1
mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v → (((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1) → (((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1) → ((if True then u else 1) = if m = m then v else 1) → (((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1) → k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k l m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1 hm : (if True then u else 1) = if m = l then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1 hkm : k ≠ m hmn : m ≠ n ⊢ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n
rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk
refine_1
((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk
refine_1
(if True then u else 1) = if m = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk
refine_1
((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk
refine_1
(if True then u else 1) = if k = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
(if True then u else 1) = if m = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
(if True then u else 1) * 1 = (if k = m then u else 1) * if k = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
(if True then u else 1) = if m = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
if_neg hkm,
refine_1
(if True then u else 1) * 1 = 1 * if k = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = (if k = m then u else 1) * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
one_mul,
refine_1
((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = 1 * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
one_mul,
refine_1
(if True then u else 1) = if m = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = 1 * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
one_mul,
refine_1
((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = 1 * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
one_mul,
refine_1
(if True then u else 1) * 1 = if k = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = 1 * if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
mul_one
refine_1
((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
mul_one
refine_1
(if True then u else 1) = if m = m then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
mul_one
refine_1
((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
mul_one
refine_1
(if True then u else 1) = if k = n then v else 1
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) * 1 = if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hk.symm hu)).1
refine_1
m ≠ k → mulSingle k u * mulSingle m v = mulSingle m u * mulSingle k v → ((if True then u else 1) = if k = k then v else 1) → (((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = k then v else 1) → (((if k = m then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = m then v else 1) → k = m ∧ m = k ∨ u = v ∧ k = k ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = k
I : Type u inst✝¹ : DecidableEq I M : Type u_4 inst✝ : CommMonoid M k m n : I u v : M hu : u ≠ 1 hv : v ≠ 1 hkm : k ≠ m hmn : m ≠ n h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v hk : (if True then u else 1) = if k = n then v else 1 hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1 hm : (if True then u else 1) = if m = m then v else 1 hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1 ⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n
ext x
h
(mulSingle i x✝ ∘ (⇑σ : n → m)) x = mulSingle ((σ.symm : m → n) i) x✝ x
α : Type u_2 m : Type u_4 n : Type u_5 inst✝² : DecidableEq n inst✝¹ : DecidableEq m inst✝ : One α σ : n ≃ m i : m x : α ⊢ mulSingle i x ∘ (⇑σ : n → m) = mulSingle ((σ.symm : m → n) i) x
← curry_mulSingle ⟨a, b⟩,
curry_mulSingle
uncurry (curry (Pi.mulSingle ⟨a, b⟩ x)) = Pi.mulSingle ⟨a, b⟩ x
α : Type u_4 β : α → Type u_5 γ : (a : α) → β a → Type u_6 inst✝² : DecidableEq α inst✝¹ : (a : α) → DecidableEq (β a) inst✝ : (a : α) → (b : β a) → One (γ a b) a : α b : β a x : γ a b ⊢ uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle ⟨a, b⟩ x
uncurry_curry
uncurry_curry
Pi.mulSingle ⟨a, b⟩ x = Pi.mulSingle ⟨a, b⟩ x
α : Type u_4 β : α → Type u_5 γ : (a : α) → β a → Type u_6 inst✝² : DecidableEq α inst✝¹ : (a : α) → DecidableEq (β a) inst✝ : (a : α) → (b : β a) → One (γ a b) a : α b : β a x : γ a b ⊢ uncurry (curry (Pi.mulSingle ⟨a, b⟩ x)) = Pi.mulSingle ⟨a, b⟩ x
obtain h|h := hy.lt_or_lt
intro
y < 1
α : Type u inst✝³ : CommGroup α inst✝² : LinearOrder α inst✝¹ : IsOrderedMonoid α inst✝ : Nontrivial α y : α hy : y ≠ 1 ⊢ ∃ a, 1 < a
obtain h|h := hy.lt_or_lt
intro
1 < y
α : Type u inst✝³ : CommGroup α inst✝² : LinearOrder α inst✝¹ : IsOrderedMonoid α inst✝ : Nontrivial α y : α hy : y ≠ 1 h : y < 1 ⊢ ∃ a, 1 < a
induction ha using closure_induction with | mem r hr => exact AddSubmonoid.mem_closure.mpr fun y hy => hy (mul_mem hr hb) | one => simp only [zero_mul, zero_mem _] | mul r s _ _ hr hs => simpa only [add_mul] using add_mem hr hs
mem
r ∈ (↑S : Set R) → r * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) hb : b ∈ S ⊢ a * b ∈ closure (↑S : Set R)
induction ha using closure_induction with | mem r hr => exact AddSubmonoid.mem_closure.mpr fun y hy => hy (mul_mem hr hb) | one => simp only [zero_mul, zero_mem _] | mul r s _ _ hr hs => simpa only [add_mul] using add_mem hr hs
one
0 * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) hb : b ∈ S mem : _fvar.884 ∈ (↑S : Set R) → _fvar.884 * b ∈ closure (↑S : Set R) ⊢ a * b ∈ closure (↑S : Set R)
induction ha using closure_induction with | mem r hr => exact AddSubmonoid.mem_closure.mpr fun y hy => hy (mul_mem hr hb) | one => simp only [zero_mul, zero_mem _] | mul r s _ _ hr hs => simpa only [add_mul] using add_mem hr hs
mul
r ∈ closure (↑S : Set R) → s ∈ closure (↑S : Set R) → r * b ∈ closure (↑S : Set R) → s * b ∈ closure (↑S : Set R) → (r + s) * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) hb : b ∈ S mem : _fvar.884 ∈ (↑S : Set R) → _fvar.884 * b ∈ closure (↑S : Set R) one : 0 * b ∈ closure (↑S : Set R) ⊢ a * b ∈ closure (↑S : Set R)
| mem r hr =>
one
0 * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R hb : b ∈ S r : R hr : r ∈ (↑S : Set R) ⊢ r * b ∈ closure (↑S : Set R)
| mem r hr =>
mul
r ∈ closure (↑S : Set R) → s ∈ closure (↑S : Set R) → r * b ∈ closure (↑S : Set R) → s * b ∈ closure (↑S : Set R) → (r + s) * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R hb : b ∈ S r : R hr : r ∈ (↑S : Set R) one : 0 * b ∈ closure (↑S : Set R) ⊢ r * b ∈ closure (↑S : Set R)
| one =>
mem
r ∈ (↑S : Set R) → r * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R hb : b ∈ S ⊢ 0 * b ∈ closure (↑S : Set R)
| one =>
mul
r ∈ closure (↑S : Set R) → s ∈ closure (↑S : Set R) → r * b ∈ closure (↑S : Set R) → s * b ∈ closure (↑S : Set R) → (r + s) * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R hb : b ∈ S mem : _fvar.884 ∈ (↑S : Set R) → _fvar.884 * b ∈ closure (↑S : Set R) ⊢ 0 * b ∈ closure (↑S : Set R)
| mul r s _ _ hr hs =>
mem
r ∈ (↑S : Set R) → r * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R hb : b ∈ S r s : R hx✝ : r ∈ closure (↑S : Set R) hy✝ : s ∈ closure (↑S : Set R) hr : r * b ∈ closure (↑S : Set R) hs : s * b ∈ closure (↑S : Set R) ⊢ (r + s) * b ∈ closure (↑S : Set R)
| mul r s _ _ hr hs =>
one
0 * b ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R hb : b ∈ S r s : R hx✝ : r ∈ closure (↑S : Set R) hy✝ : s ∈ closure (↑S : Set R) hr : r * b ∈ closure (↑S : Set R) hs : s * b ∈ closure (↑S : Set R) mem : _fvar.884 ∈ (↑S : Set R) → _fvar.884 * b ∈ closure (↑S : Set R) ⊢ (r + s) * b ∈ closure (↑S : Set R)
induction hb using closure_induction with | mem r hr => exact MulMemClass.mul_right_mem_add_closure ha hr | one => simp only [mul_zero, zero_mem _] | mul r s _ _ hr hs => simpa only [mul_add] using add_mem hr hs
mem
r ∈ (↑S : Set R) → a * r ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) hb : b ∈ closure (↑S : Set R) ⊢ a * b ∈ closure (↑S : Set R)
induction hb using closure_induction with | mem r hr => exact MulMemClass.mul_right_mem_add_closure ha hr | one => simp only [mul_zero, zero_mem _] | mul r s _ _ hr hs => simpa only [mul_add] using add_mem hr hs
one
a * 0 ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) hb : b ∈ closure (↑S : Set R) mem : _fvar.2592 ∈ (↑S : Set R) → a * _fvar.2592 ∈ closure (↑S : Set R) ⊢ a * b ∈ closure (↑S : Set R)
induction hb using closure_induction with | mem r hr => exact MulMemClass.mul_right_mem_add_closure ha hr | one => simp only [mul_zero, zero_mem _] | mul r s _ _ hr hs => simpa only [mul_add] using add_mem hr hs
mul
r ∈ closure (↑S : Set R) → s ∈ closure (↑S : Set R) → a * r ∈ closure (↑S : Set R) → a * s ∈ closure (↑S : Set R) → a * (r + s) ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) hb : b ∈ closure (↑S : Set R) mem : _fvar.2592 ∈ (↑S : Set R) → a * _fvar.2592 ∈ closure (↑S : Set R) one : a * 0 ∈ closure (↑S : Set R) ⊢ a * b ∈ closure (↑S : Set R)
| mem r hr =>
one
a * 0 ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) r : R hr : r ∈ (↑S : Set R) ⊢ a * r ∈ closure (↑S : Set R)
| mem r hr =>
mul
r ∈ closure (↑S : Set R) → s ∈ closure (↑S : Set R) → a * r ∈ closure (↑S : Set R) → a * s ∈ closure (↑S : Set R) → a * (r + s) ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) r : R hr : r ∈ (↑S : Set R) one : a * 0 ∈ closure (↑S : Set R) ⊢ a * r ∈ closure (↑S : Set R)
| one =>
mem
r ∈ (↑S : Set R) → a * r ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) ⊢ a * 0 ∈ closure (↑S : Set R)
| one =>
mul
r ∈ closure (↑S : Set R) → s ∈ closure (↑S : Set R) → a * r ∈ closure (↑S : Set R) → a * s ∈ closure (↑S : Set R) → a * (r + s) ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) mem : _fvar.2592 ∈ (↑S : Set R) → a * _fvar.2592 ∈ closure (↑S : Set R) ⊢ a * 0 ∈ closure (↑S : Set R)
| mul r s _ _ hr hs =>
mem
r ∈ (↑S : Set R) → a * r ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) r s : R hx✝ : r ∈ closure (↑S : Set R) hy✝ : s ∈ closure (↑S : Set R) hr : a * r ∈ closure (↑S : Set R) hs : a * s ∈ closure (↑S : Set R) ⊢ a * (r + s) ∈ closure (↑S : Set R)
| mul r s _ _ hr hs =>
one
a * 0 ∈ closure (↑S : Set R)
M : Type u_1 R : Type u_2 inst✝² : NonUnitalNonAssocSemiring R inst✝¹ : SetLike M R inst✝ : MulMemClass M R S : M a b : R ha : a ∈ closure (↑S : Set R) r s : R hx✝ : r ∈ closure (↑S : Set R) hy✝ : s ∈ closure (↑S : Set R) hr : a * r ∈ closure (↑S : Set R) hs : a * s ∈ closure (↑S : Set R) mem : _fvar.2592 ∈ (↑S : Set R) → a * _fvar.2592 ∈ closure (↑S : Set R) ⊢ a * (r + s) ∈ closure (↑S : Set R)
simp only [ShortComplex.exact_iff_isZero_homology] at hS ⊢
simp
(∀ (i : ι), IsZero (S.map (eval C c i)).homology) → IsZero S.homology
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) hS : ∀ (i : ι), (S.map (eval C c i)).Exact ⊢ S.Exact
rw [IsZero.iff_id_eq_zero]
rw
𝟙 S.homology = 0
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) hS : ∀ (i : ι), IsZero (S.map (eval C c i)).homology ⊢ IsZero S.homology
IsZero.iff_id_eq_zero
this
𝟙 S.homology = 0
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) hS : ∀ (i : ι), IsZero (S.map (eval C c i)).homology ⊢ IsZero S.homology
ext i
h
(𝟙 S.homology).f i = Hom.f 0 i
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) hS : ∀ (i : ι), IsZero (S.map (eval C c i)).homology ⊢ 𝟙 S.homology = 0
constructor
mp
S.Exact → ∀ (i : ι), (S.map (eval C c i)).Exact
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) ⊢ S.Exact ↔ ∀ (i : ι), (S.map (eval C c i)).Exact
constructor
mpr
(∀ (i : ι), (S.map (eval C c i)).Exact) → S.Exact
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) mp : S.Exact → ∀ (i : ι), (S.map (eval C c i)).Exact ⊢ S.Exact ↔ ∀ (i : ι), (S.map (eval C c i)).Exact
constructor
mp
S.ShortExact → ∀ (i : ι), (S.map (eval C c i)).ShortExact
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) ⊢ S.ShortExact ↔ ∀ (i : ι), (S.map (eval C c i)).ShortExact
constructor
mpr
(∀ (i : ι), (S.map (eval C c i)).ShortExact) → S.ShortExact
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) mp : S.ShortExact → ∀ (i : ι), (S.map (eval C c i)).ShortExact ⊢ S.ShortExact ↔ ∀ (i : ι), (S.map (eval C c i)).ShortExact
have := hS.mono_f
mp
Mono S.f
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) hS : S.ShortExact i : ι ⊢ (S.map (eval C c i)).ShortExact
have := hS.epi_g
mp
Epi S.g
C : Type u_1 ι : Type u_2 c : ComplexShape ι inst✝¹ : Category.{u_3, u_1} C inst✝ : Abelian C S : ShortComplex (HomologicalComplex C c) hS : S.ShortExact i : ι this : Mono S.f ⊢ (S.map (eval C c i)).ShortExact