fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
classes | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
mul_indicator_empty (f : α → M) : mul_indicator (∅ : set α) f = λa, 1 :=
mul_indicator_eq_one.2 $ disjoint_empty _ | mul_indicator_empty (f : α → M) : mul_indicator (∅ : set α) f = λa, 1 | mul_indicator_eq_one.2 $ disjoint_empty _ | lemma | set.mul_indicator_empty | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 135 | 136 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_empty' (f : α → M) : mul_indicator (∅ : set α) f = 1 :=
mul_indicator_empty f | mul_indicator_empty' (f : α → M) : mul_indicator (∅ : set α) f = 1 | mul_indicator_empty f | lemma | set.mul_indicator_empty' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 138 | 139 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_one (s : set α) :
mul_indicator s (λx, (1:M)) = λx, (1:M) :=
mul_indicator_eq_one.2 $ by simp only [mul_support_one, empty_disjoint] | mul_indicator_one (s : set α) :
mul_indicator s (λx, (1:M)) = λx, (1:M) | mul_indicator_eq_one.2 $ by simp only [mul_support_one, empty_disjoint] | lemma | set.mul_indicator_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 143 | 145 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_one' {s : set α} : s.mul_indicator (1 : α → M) = 1 :=
mul_indicator_one M s | mul_indicator_one' {s : set α} : s.mul_indicator (1 : α → M) = 1 | mul_indicator_one M s | lemma | set.mul_indicator_one' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 147 | 148 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_mul_indicator (s t : set α) (f : α → M) :
mul_indicator s (mul_indicator t f) = mul_indicator (s ∩ t) f :=
funext $ λx, by { simp only [mul_indicator], split_ifs, repeat {simp * at * {contextual := tt}} } | mul_indicator_mul_indicator (s t : set α) (f : α → M) :
mul_indicator s (mul_indicator t f) = mul_indicator (s ∩ t) f | funext $ λx, by { simp only [mul_indicator], split_ifs, repeat {simp * at * {contextual := tt}} } | lemma | set.mul_indicator_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 152 | 154 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_inter_mul_support (s : set α) (f : α → M) :
mul_indicator (s ∩ mul_support f) f = mul_indicator s f :=
by rw [← mul_indicator_mul_indicator, mul_indicator_mul_support] | mul_indicator_inter_mul_support (s : set α) (f : α → M) :
mul_indicator (s ∩ mul_support f) f = mul_indicator s f | by rw [← mul_indicator_mul_indicator, mul_indicator_mul_support] | lemma | set.mul_indicator_inter_mul_support | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 156 | 158 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mul_indicator (h : M → β) (f : α → M) {s : set α} {x : α}
[decidable_pred (∈ s)] :
h (s.mul_indicator f x) = s.piecewise (h ∘ f) (const α (h 1)) x :=
by letI := classical.dec_pred (∈ s); convert s.apply_piecewise f (const α 1) (λ _, h) | comp_mul_indicator (h : M → β) (f : α → M) {s : set α} {x : α}
[decidable_pred (∈ s)] :
h (s.mul_indicator f x) = s.piecewise (h ∘ f) (const α (h 1)) x | by letI := classical.dec_pred (∈ s); convert s.apply_piecewise f (const α 1) (λ _, h) | lemma | set.comp_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | null | 160 | 163 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_comp_right {s : set α} (f : β → α) {g : α → M} {x : β} :
mul_indicator (f ⁻¹' s) (g ∘ f) x = mul_indicator s g (f x) :=
by { simp only [mul_indicator], split_ifs; refl } | mul_indicator_comp_right {s : set α} (f : β → α) {g : α → M} {x : β} :
mul_indicator (f ⁻¹' s) (g ∘ f) x = mul_indicator s g (f x) | by { simp only [mul_indicator], split_ifs; refl } | lemma | set.mul_indicator_comp_right | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 165 | 167 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_image {s : set α} {f : β → M} {g : α → β} (hg : injective g)
{x : α} : mul_indicator (g '' s) f (g x) = mul_indicator s (f ∘ g) x :=
by rw [← mul_indicator_comp_right, preimage_image_eq _ hg] | mul_indicator_image {s : set α} {f : β → M} {g : α → β} (hg : injective g)
{x : α} : mul_indicator (g '' s) f (g x) = mul_indicator s (f ∘ g) x | by rw [← mul_indicator_comp_right, preimage_image_eq _ hg] | lemma | set.mul_indicator_image | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 169 | 171 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_comp_of_one {g : M → N} (hg : g 1 = 1) :
mul_indicator s (g ∘ f) = g ∘ (mul_indicator s f) :=
begin
funext,
simp only [mul_indicator],
split_ifs; simp [*]
end | mul_indicator_comp_of_one {g : M → N} (hg : g 1 = 1) :
mul_indicator s (g ∘ f) = g ∘ (mul_indicator s f) | begin
funext,
simp only [mul_indicator],
split_ifs; simp [*]
end | lemma | set.mul_indicator_comp_of_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 173 | 179 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mul_indicator_const (c : M) (f : M → N) (hf : f 1 = 1) :
(λ x, f (s.mul_indicator (λ x, c) x)) = s.mul_indicator (λ x, f c) :=
(mul_indicator_comp_of_one hf).symm | comp_mul_indicator_const (c : M) (f : M → N) (hf : f 1 = 1) :
(λ x, f (s.mul_indicator (λ x, c) x)) = s.mul_indicator (λ x, f c) | (mul_indicator_comp_of_one hf).symm | lemma | set.comp_mul_indicator_const | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 181 | 183 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_preimage (s : set α) (f : α → M) (B : set M) :
(mul_indicator s f)⁻¹' B = s.ite (f ⁻¹' B) (1 ⁻¹' B) :=
by letI := classical.dec_pred (∈ s); exact piecewise_preimage s f 1 B | mul_indicator_preimage (s : set α) (f : α → M) (B : set M) :
(mul_indicator s f)⁻¹' B = s.ite (f ⁻¹' B) (1 ⁻¹' B) | by letI := classical.dec_pred (∈ s); exact piecewise_preimage s f 1 B | lemma | set.mul_indicator_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"classical.dec_pred"
] | null | 185 | 187 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_one_preimage (s : set M) :
t.mul_indicator 1 ⁻¹' s ∈ ({set.univ, ∅} : set (set α)) :=
begin
classical,
rw [mul_indicator_one', preimage_one],
split_ifs; simp
end | mul_indicator_one_preimage (s : set M) :
t.mul_indicator 1 ⁻¹' s ∈ ({set.univ, ∅} : set (set α)) | begin
classical,
rw [mul_indicator_one', preimage_one],
split_ifs; simp
end | lemma | set.mul_indicator_one_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 189 | 195 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_const_preimage_eq_union (U : set α) (s : set M) (a : M)
[decidable (a ∈ s)] [decidable ((1 : M) ∈ s)] :
U.mul_indicator (λ x, a) ⁻¹' s = (if a ∈ s then U else ∅) ∪ (if (1 : M) ∈ s then Uᶜ else ∅) :=
begin
rw [mul_indicator_preimage, preimage_one, preimage_const],
split_ifs; simp [← compl_eq_univ_d... | mul_indicator_const_preimage_eq_union (U : set α) (s : set M) (a : M)
[decidable (a ∈ s)] [decidable ((1 : M) ∈ s)] :
U.mul_indicator (λ x, a) ⁻¹' s = (if a ∈ s then U else ∅) ∪ (if (1 : M) ∈ s then Uᶜ else ∅) | begin
rw [mul_indicator_preimage, preimage_one, preimage_const],
split_ifs; simp [← compl_eq_univ_diff]
end | lemma | set.mul_indicator_const_preimage_eq_union | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 197 | 203 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_const_preimage (U : set α) (s : set M) (a : M) :
U.mul_indicator (λ x, a) ⁻¹' s ∈ ({set.univ, U, Uᶜ, ∅} : set (set α)) :=
begin
classical,
rw [mul_indicator_const_preimage_eq_union],
split_ifs; simp
end | mul_indicator_const_preimage (U : set α) (s : set M) (a : M) :
U.mul_indicator (λ x, a) ⁻¹' s ∈ ({set.univ, U, Uᶜ, ∅} : set (set α)) | begin
classical,
rw [mul_indicator_const_preimage_eq_union],
split_ifs; simp
end | lemma | set.mul_indicator_const_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 205 | 211 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_one_preimage [has_zero M] (U : set α) (s : set M) :
U.indicator 1 ⁻¹' s ∈ ({set.univ, U, Uᶜ, ∅} : set (set α)) :=
indicator_const_preimage _ _ 1 | indicator_one_preimage [has_zero M] (U : set α) (s : set M) :
U.indicator 1 ⁻¹' s ∈ ({set.univ, U, Uᶜ, ∅} : set (set α)) | indicator_const_preimage _ _ 1 | lemma | set.indicator_one_preimage | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 213 | 215 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_preimage_of_not_mem (s : set α) (f : α → M)
{t : set M} (ht : (1:M) ∉ t) :
(mul_indicator s f)⁻¹' t = f ⁻¹' t ∩ s :=
by simp [mul_indicator_preimage, pi.one_def, set.preimage_const_of_not_mem ht] | mul_indicator_preimage_of_not_mem (s : set α) (f : α → M)
{t : set M} (ht : (1:M) ∉ t) :
(mul_indicator s f)⁻¹' t = f ⁻¹' t ∩ s | by simp [mul_indicator_preimage, pi.one_def, set.preimage_const_of_not_mem ht] | lemma | set.mul_indicator_preimage_of_not_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"pi.one_def",
"set.preimage_const_of_not_mem"
] | null | 217 | 220 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_range_mul_indicator {r : M} {s : set α} {f : α → M} :
r ∈ range (mul_indicator s f) ↔ (r = 1 ∧ s ≠ univ) ∨ (r ∈ f '' s) :=
by simp [mul_indicator, ite_eq_iff, exists_or_distrib, eq_univ_iff_forall, and_comm, or_comm,
@eq_comm _ r 1] | mem_range_mul_indicator {r : M} {s : set α} {f : α → M} :
r ∈ range (mul_indicator s f) ↔ (r = 1 ∧ s ≠ univ) ∨ (r ∈ f '' s) | by simp [mul_indicator, ite_eq_iff, exists_or_distrib, eq_univ_iff_forall, and_comm, or_comm,
@eq_comm _ r 1] | lemma | set.mem_range_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"exists_or_distrib",
"ite_eq_iff"
] | null | 222 | 225 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_rel_mul_indicator {r : M → M → Prop} (h1 : r 1 1)
(ha : a ∈ s → r (f a) (g a)) :
r (mul_indicator s f a) (mul_indicator s g a) :=
by { simp only [mul_indicator], split_ifs with has has, exacts [ha has, h1] } | mul_indicator_rel_mul_indicator {r : M → M → Prop} (h1 : r 1 1)
(ha : a ∈ s → r (f a) (g a)) :
r (mul_indicator s f a) (mul_indicator s g a) | by { simp only [mul_indicator], split_ifs with has has, exacts [ha has, h1] } | lemma | set.mul_indicator_rel_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 227 | 230 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_union_mul_inter_apply (f : α → M) (s t : set α) (a : α) :
mul_indicator (s ∪ t) f a * mul_indicator (s ∩ t) f a =
mul_indicator s f a * mul_indicator t f a :=
by by_cases hs : a ∈ s; by_cases ht : a ∈ t; simp * | mul_indicator_union_mul_inter_apply (f : α → M) (s t : set α) (a : α) :
mul_indicator (s ∪ t) f a * mul_indicator (s ∩ t) f a =
mul_indicator s f a * mul_indicator t f a | by by_cases hs : a ∈ s; by_cases ht : a ∈ t; simp * | lemma | set.mul_indicator_union_mul_inter_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 237 | 240 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_union_mul_inter (f : α → M) (s t : set α) :
mul_indicator (s ∪ t) f * mul_indicator (s ∩ t) f = mul_indicator s f * mul_indicator t f :=
funext $ mul_indicator_union_mul_inter_apply f s t | mul_indicator_union_mul_inter (f : α → M) (s t : set α) :
mul_indicator (s ∪ t) f * mul_indicator (s ∩ t) f = mul_indicator s f * mul_indicator t f | funext $ mul_indicator_union_mul_inter_apply f s t | lemma | set.mul_indicator_union_mul_inter | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 242 | 244 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_union_of_not_mem_inter (h : a ∉ s ∩ t) (f : α → M) :
mul_indicator (s ∪ t) f a = mul_indicator s f a * mul_indicator t f a :=
by rw [← mul_indicator_union_mul_inter_apply f s t, mul_indicator_of_not_mem h, mul_one] | mul_indicator_union_of_not_mem_inter (h : a ∉ s ∩ t) (f : α → M) :
mul_indicator (s ∪ t) f a = mul_indicator s f a * mul_indicator t f a | by rw [← mul_indicator_union_mul_inter_apply f s t, mul_indicator_of_not_mem h, mul_one] | lemma | set.mul_indicator_union_of_not_mem_inter | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one"
] | null | 246 | 248 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_union_of_disjoint (h : disjoint s t) (f : α → M) :
mul_indicator (s ∪ t) f = λa, mul_indicator s f a * mul_indicator t f a :=
funext $ λa, mul_indicator_union_of_not_mem_inter (λ ha, h.le_bot ha) _ | mul_indicator_union_of_disjoint (h : disjoint s t) (f : α → M) :
mul_indicator (s ∪ t) f = λa, mul_indicator s f a * mul_indicator t f a | funext $ λa, mul_indicator_union_of_not_mem_inter (λ ha, h.le_bot ha) _ | lemma | set.mul_indicator_union_of_disjoint | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint"
] | null | 250 | 252 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_mul (s : set α) (f g : α → M) :
mul_indicator s (λa, f a * g a) = λa, mul_indicator s f a * mul_indicator s g a :=
by { funext, simp only [mul_indicator], split_ifs, { refl }, rw mul_one } | mul_indicator_mul (s : set α) (f g : α → M) :
mul_indicator s (λa, f a * g a) = λa, mul_indicator s f a * mul_indicator s g a | by { funext, simp only [mul_indicator], split_ifs, { refl }, rw mul_one } | lemma | set.mul_indicator_mul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one"
] | null | 254 | 256 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_mul' (s : set α) (f g : α → M) :
mul_indicator s (f * g) = mul_indicator s f * mul_indicator s g :=
mul_indicator_mul s f g | mul_indicator_mul' (s : set α) (f g : α → M) :
mul_indicator s (f * g) = mul_indicator s f * mul_indicator s g | mul_indicator_mul s f g | lemma | set.mul_indicator_mul' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 258 | 260 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_compl_mul_self_apply (s : set α) (f : α → M) (a : α) :
mul_indicator sᶜ f a * mul_indicator s f a = f a :=
classical.by_cases (λ ha : a ∈ s, by simp [ha]) (λ ha, by simp [ha]) | mul_indicator_compl_mul_self_apply (s : set α) (f : α → M) (a : α) :
mul_indicator sᶜ f a * mul_indicator s f a = f a | classical.by_cases (λ ha : a ∈ s, by simp [ha]) (λ ha, by simp [ha]) | lemma | set.mul_indicator_compl_mul_self_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 262 | 264 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_compl_mul_self (s : set α) (f : α → M) :
mul_indicator sᶜ f * mul_indicator s f = f :=
funext $ mul_indicator_compl_mul_self_apply s f | mul_indicator_compl_mul_self (s : set α) (f : α → M) :
mul_indicator sᶜ f * mul_indicator s f = f | funext $ mul_indicator_compl_mul_self_apply s f | lemma | set.mul_indicator_compl_mul_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 266 | 268 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_self_mul_compl_apply (s : set α) (f : α → M) (a : α) :
mul_indicator s f a * mul_indicator sᶜ f a = f a :=
classical.by_cases (λ ha : a ∈ s, by simp [ha]) (λ ha, by simp [ha]) | mul_indicator_self_mul_compl_apply (s : set α) (f : α → M) (a : α) :
mul_indicator s f a * mul_indicator sᶜ f a = f a | classical.by_cases (λ ha : a ∈ s, by simp [ha]) (λ ha, by simp [ha]) | lemma | set.mul_indicator_self_mul_compl_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 270 | 272 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_self_mul_compl (s : set α) (f : α → M) :
mul_indicator s f * mul_indicator sᶜ f = f :=
funext $ mul_indicator_self_mul_compl_apply s f | mul_indicator_self_mul_compl (s : set α) (f : α → M) :
mul_indicator s f * mul_indicator sᶜ f = f | funext $ mul_indicator_self_mul_compl_apply s f | lemma | set.mul_indicator_self_mul_compl | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 274 | 276 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_mul_eq_left {f g : α → M}
(h : disjoint (mul_support f) (mul_support g)) :
(mul_support f).mul_indicator (f * g) = f :=
begin
refine (mul_indicator_congr $ λ x hx, _).trans mul_indicator_mul_support,
have : g x = 1, from nmem_mul_support.1 (disjoint_left.1 h hx),
rw [pi.mul_apply, this, mul_one]... | mul_indicator_mul_eq_left {f g : α → M}
(h : disjoint (mul_support f) (mul_support g)) :
(mul_support f).mul_indicator (f * g) = f | begin
refine (mul_indicator_congr $ λ x hx, _).trans mul_indicator_mul_support,
have : g x = 1, from nmem_mul_support.1 (disjoint_left.1 h hx),
rw [pi.mul_apply, this, mul_one]
end | lemma | set.mul_indicator_mul_eq_left | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"mul_one",
"pi.mul_apply"
] | null | 278 | 285 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_mul_eq_right {f g : α → M}
(h : disjoint (mul_support f) (mul_support g)) :
(mul_support g).mul_indicator (f * g) = g :=
begin
refine (mul_indicator_congr $ λ x hx, _).trans mul_indicator_mul_support,
have : f x = 1, from nmem_mul_support.1 (disjoint_right.1 h hx),
rw [pi.mul_apply, this, one_mu... | mul_indicator_mul_eq_right {f g : α → M}
(h : disjoint (mul_support f) (mul_support g)) :
(mul_support g).mul_indicator (f * g) = g | begin
refine (mul_indicator_congr $ λ x hx, _).trans mul_indicator_mul_support,
have : f x = 1, from nmem_mul_support.1 (disjoint_right.1 h hx),
rw [pi.mul_apply, this, one_mul]
end | lemma | set.mul_indicator_mul_eq_right | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"one_mul",
"pi.mul_apply"
] | null | 287 | 294 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_mul_compl_eq_piecewise
[decidable_pred (∈ s)] (f g : α → M) :
s.mul_indicator f * sᶜ.mul_indicator g = s.piecewise f g :=
begin
ext x,
by_cases h : x ∈ s,
{ rw [piecewise_eq_of_mem _ _ _ h, pi.mul_apply, set.mul_indicator_of_mem h,
set.mul_indicator_of_not_mem (set.not_mem_compl_iff.2 h), ... | mul_indicator_mul_compl_eq_piecewise
[decidable_pred (∈ s)] (f g : α → M) :
s.mul_indicator f * sᶜ.mul_indicator g = s.piecewise f g | begin
ext x,
by_cases h : x ∈ s,
{ rw [piecewise_eq_of_mem _ _ _ h, pi.mul_apply, set.mul_indicator_of_mem h,
set.mul_indicator_of_not_mem (set.not_mem_compl_iff.2 h), mul_one] },
{ rw [piecewise_eq_of_not_mem _ _ _ h, pi.mul_apply, set.mul_indicator_of_not_mem h,
set.mul_indicator_of_mem (set.mem_c... | lemma | set.mul_indicator_mul_compl_eq_piecewise | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one",
"one_mul",
"pi.mul_apply",
"set.mem_compl",
"set.mul_indicator_of_mem",
"set.mul_indicator_of_not_mem"
] | null | 296 | 306 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_hom {α} (M) [mul_one_class M] (s : set α) : (α → M) →* (α → M) :=
{ to_fun := mul_indicator s,
map_one' := mul_indicator_one M s,
map_mul' := mul_indicator_mul s } | mul_indicator_hom {α} (M) [mul_one_class M] (s : set α) : (α → M) →* (α → M) | { to_fun := mul_indicator s,
map_one' := mul_indicator_one M s,
map_mul' := mul_indicator_mul s } | def | set.mul_indicator_hom | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one_class"
] | `set.mul_indicator` as a `monoid_hom`. | 310 | 314 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_smul_apply (s : set α) (r : α → M) (f : α → A) (x : α) :
indicator s (λ x, r x • f x) x = r x • indicator s f x :=
by { dunfold indicator, split_ifs, exacts [rfl, (smul_zero (r x)).symm] } | indicator_smul_apply (s : set α) (r : α → M) (f : α → A) (x : α) :
indicator s (λ x, r x • f x) x = r x • indicator s f x | by { dunfold indicator, split_ifs, exacts [rfl, (smul_zero (r x)).symm] } | lemma | set.indicator_smul_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"smul_zero"
] | null | 322 | 324 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_smul (s : set α) (r : α → M) (f : α → A) :
indicator s (λ (x : α), r x • f x) = λ (x : α), r x • indicator s f x :=
funext $ indicator_smul_apply s r f | indicator_smul (s : set α) (r : α → M) (f : α → A) :
indicator s (λ (x : α), r x • f x) = λ (x : α), r x • indicator s f x | funext $ indicator_smul_apply s r f | lemma | set.indicator_smul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 326 | 328 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_const_smul_apply (s : set α) (r : M) (f : α → A) (x : α) :
indicator s (λ x, r • f x) x = r • indicator s f x :=
indicator_smul_apply s (λ x, r) f x | indicator_const_smul_apply (s : set α) (r : M) (f : α → A) (x : α) :
indicator s (λ x, r • f x) x = r • indicator s f x | indicator_smul_apply s (λ x, r) f x | lemma | set.indicator_const_smul_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 330 | 332 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_const_smul (s : set α) (r : M) (f : α → A) :
indicator s (λ (x : α), r • f x) = λ (x : α), r • indicator s f x :=
funext $ indicator_const_smul_apply s r f | indicator_const_smul (s : set α) (r : M) (f : α → A) :
indicator s (λ (x : α), r • f x) = λ (x : α), r • indicator s f x | funext $ indicator_const_smul_apply s r f | lemma | set.indicator_const_smul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 334 | 336 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_inv' (s : set α) (f : α → G) :
mul_indicator s (f⁻¹) = (mul_indicator s f)⁻¹ :=
(mul_indicator_hom G s).map_inv f | mul_indicator_inv' (s : set α) (f : α → G) :
mul_indicator s (f⁻¹) = (mul_indicator s f)⁻¹ | (mul_indicator_hom G s).map_inv f | lemma | set.mul_indicator_inv' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"map_inv"
] | null | 343 | 345 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_inv (s : set α) (f : α → G) :
mul_indicator s (λa, (f a)⁻¹) = λa, (mul_indicator s f a)⁻¹ :=
mul_indicator_inv' s f | mul_indicator_inv (s : set α) (f : α → G) :
mul_indicator s (λa, (f a)⁻¹) = λa, (mul_indicator s f a)⁻¹ | mul_indicator_inv' s f | lemma | set.mul_indicator_inv | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 347 | 349 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_div (s : set α) (f g : α → G) :
mul_indicator s (λ a, f a / g a) =
λ a, mul_indicator s f a / mul_indicator s g a :=
(mul_indicator_hom G s).map_div f g | mul_indicator_div (s : set α) (f g : α → G) :
mul_indicator s (λ a, f a / g a) =
λ a, mul_indicator s f a / mul_indicator s g a | (mul_indicator_hom G s).map_div f g | lemma | set.mul_indicator_div | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"map_div"
] | null | 351 | 354 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_div' (s : set α) (f g : α → G) :
mul_indicator s (f / g) = mul_indicator s f / mul_indicator s g :=
mul_indicator_div s f g | mul_indicator_div' (s : set α) (f g : α → G) :
mul_indicator s (f / g) = mul_indicator s f / mul_indicator s g | mul_indicator_div s f g | lemma | set.mul_indicator_div' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 356 | 358 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_compl (s : set α) (f : α → G) :
mul_indicator sᶜ f = f * (mul_indicator s f)⁻¹ :=
eq_mul_inv_of_mul_eq $ s.mul_indicator_compl_mul_self f | mul_indicator_compl (s : set α) (f : α → G) :
mul_indicator sᶜ f = f * (mul_indicator s f)⁻¹ | eq_mul_inv_of_mul_eq $ s.mul_indicator_compl_mul_self f | lemma | set.mul_indicator_compl | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"eq_mul_inv_of_mul_eq"
] | null | 360 | 362 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_compl {G} [add_group G] (s : set α) (f : α → G) :
indicator sᶜ f = f - indicator s f :=
by rw [sub_eq_add_neg, indicator_compl'] | indicator_compl {G} [add_group G] (s : set α) (f : α → G) :
indicator sᶜ f = f - indicator s f | by rw [sub_eq_add_neg, indicator_compl'] | lemma | set.indicator_compl | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"add_group"
] | null | 364 | 366 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_diff (h : s ⊆ t) (f : α → G) :
mul_indicator (t \ s) f = mul_indicator t f * (mul_indicator s f)⁻¹ :=
eq_mul_inv_of_mul_eq $ by { rw [pi.mul_def, ←mul_indicator_union_of_disjoint, diff_union_self,
union_eq_self_of_subset_right h], exact disjoint_sdiff_self_left } | mul_indicator_diff (h : s ⊆ t) (f : α → G) :
mul_indicator (t \ s) f = mul_indicator t f * (mul_indicator s f)⁻¹ | eq_mul_inv_of_mul_eq $ by { rw [pi.mul_def, ←mul_indicator_union_of_disjoint, diff_union_self,
union_eq_self_of_subset_right h], exact disjoint_sdiff_self_left } | lemma | set.mul_indicator_diff | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint_sdiff_self_left",
"eq_mul_inv_of_mul_eq",
"pi.mul_def"
] | null | 368 | 371 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_diff {G : Type*} [add_group G] {s t : set α} (h : s ⊆ t) (f : α → G) :
indicator (t \ s) f = indicator t f - indicator s f :=
by rw [indicator_diff' h, sub_eq_add_neg] | indicator_diff {G : Type*} [add_group G] {s t : set α} (h : s ⊆ t) (f : α → G) :
indicator (t \ s) f = indicator t f - indicator s f | by rw [indicator_diff' h, sub_eq_add_neg] | lemma | set.indicator_diff | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"add_group"
] | null | 373 | 375 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mul_indicator_subset_of_eq_one [has_one N] (f : α → N)
(g : α → N → M) {s t : finset α} (h : s ⊆ t) (hg : ∀ a, g a 1 = 1) :
∏ i in s, g i (f i) = ∏ i in t, g i (mul_indicator ↑s f i) :=
begin
rw ← finset.prod_subset h _,
{ apply finset.prod_congr rfl,
intros i hi,
congr,
symmetry,
exact mul... | prod_mul_indicator_subset_of_eq_one [has_one N] (f : α → N)
(g : α → N → M) {s t : finset α} (h : s ⊆ t) (hg : ∀ a, g a 1 = 1) :
∏ i in s, g i (f i) = ∏ i in t, g i (mul_indicator ↑s f i) | begin
rw ← finset.prod_subset h _,
{ apply finset.prod_congr rfl,
intros i hi,
congr,
symmetry,
exact mul_indicator_of_mem hi _ },
{ refine λ i hi hn, _,
convert hg i,
exact mul_indicator_of_not_mem hn _ }
end | lemma | set.prod_mul_indicator_subset_of_eq_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"finset",
"finset.prod_congr",
"finset.prod_subset"
] | Consider a product of `g i (f i)` over a `finset`. Suppose `g` is a
function such as `pow`, which maps a second argument of `1` to
`1`. Then if `f` is replaced by the corresponding multiplicative indicator
function, the `finset` may be replaced by a possibly larger `finset`
without changing the value of the sum. | 388 | 401 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_mul_indicator_subset (f : α → M) {s t : finset α} (h : s ⊆ t) :
∏ i in s, f i = ∏ i in t, mul_indicator ↑s f i :=
prod_mul_indicator_subset_of_eq_one _ (λ a b, b) h (λ _, rfl) | prod_mul_indicator_subset (f : α → M) {s t : finset α} (h : s ⊆ t) :
∏ i in s, f i = ∏ i in t, mul_indicator ↑s f i | prod_mul_indicator_subset_of_eq_one _ (λ a b, b) h (λ _, rfl) | lemma | set.prod_mul_indicator_subset | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"finset"
] | Taking the product of an indicator function over a possibly larger `finset` is the same as
taking the original function over the original `finset`. | 414 | 418 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.finset.prod_mul_indicator_eq_prod_filter
(s : finset ι) (f : ι → α → M) (t : ι → set α) (g : ι → α) [decidable_pred (λ i, g i ∈ t i)]:
∏ i in s, mul_indicator (t i) (f i) (g i) = ∏ i in s.filter (λ i, g i ∈ t i), f i (g i) :=
begin
refine (finset.prod_filter_mul_prod_filter_not s (λ i, g i ∈ t i) _).symm.t... | _root_.finset.prod_mul_indicator_eq_prod_filter
(s : finset ι) (f : ι → α → M) (t : ι → set α) (g : ι → α) [decidable_pred (λ i, g i ∈ t i)]:
∏ i in s, mul_indicator (t i) (f i) (g i) = ∏ i in s.filter (λ i, g i ∈ t i), f i (g i) | begin
refine (finset.prod_filter_mul_prod_filter_not s (λ i, g i ∈ t i) _).symm.trans _,
refine eq.trans _ (mul_one _),
exact congr_arg2 (*)
(finset.prod_congr rfl $ λ x hx, mul_indicator_of_mem (finset.mem_filter.1 hx).2 _)
(finset.prod_eq_one $ λ x hx, mul_indicator_of_not_mem (finset.mem_filter.1 hx).2... | lemma | finset.prod_mul_indicator_eq_prod_filter | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"congr_arg2",
"finset",
"finset.prod_congr",
"finset.prod_eq_one",
"finset.prod_filter_mul_prod_filter_not",
"mul_one"
] | null | 420 | 429 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_finset_prod (I : finset ι) (s : set α) (f : ι → α → M) :
mul_indicator s (∏ i in I, f i) = ∏ i in I, mul_indicator s (f i) :=
(mul_indicator_hom M s).map_prod _ _ | mul_indicator_finset_prod (I : finset ι) (s : set α) (f : ι → α → M) :
mul_indicator s (∏ i in I, f i) = ∏ i in I, mul_indicator s (f i) | (mul_indicator_hom M s).map_prod _ _ | lemma | set.mul_indicator_finset_prod | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"finset",
"map_prod"
] | null | 431 | 433 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_finset_bUnion {ι} (I : finset ι)
(s : ι → set α) {f : α → M} : (∀ (i ∈ I) (j ∈ I), i ≠ j → disjoint (s i) (s j)) →
mul_indicator (⋃ i ∈ I, s i) f = λ a, ∏ i in I, mul_indicator (s i) f a :=
begin
classical,
refine finset.induction_on I _ _,
{ intro h, funext, simp },
assume a I haI ih hI,
fu... | mul_indicator_finset_bUnion {ι} (I : finset ι)
(s : ι → set α) {f : α → M} : (∀ (i ∈ I) (j ∈ I), i ≠ j → disjoint (s i) (s j)) →
mul_indicator (⋃ i ∈ I, s i) f = λ a, ∏ i in I, mul_indicator (s i) f a | begin
classical,
refine finset.induction_on I _ _,
{ intro h, funext, simp },
assume a I haI ih hI,
funext,
rw [finset.prod_insert haI, finset.set_bUnion_insert, mul_indicator_union_of_not_mem_inter, ih _],
{ assume i hi j hj hij,
exact hI i (finset.mem_insert_of_mem hi) j (finset.mem_insert_of_mem hj... | lemma | set.mul_indicator_finset_bUnion | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"exists_prop",
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.prod_insert",
"finset.set_bUnion_insert",
"ih",
"ne_of_mem_of_not_mem",
"not_and",
"not_exists"
] | null | 435 | 451 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_finset_bUnion_apply {ι} (I : finset ι)
(s : ι → set α) {f : α → M} (h : ∀ (i ∈ I) (j ∈ I), i ≠ j → disjoint (s i) (s j)) (x : α) :
mul_indicator (⋃ i ∈ I, s i) f x = ∏ i in I, mul_indicator (s i) f x :=
by rw set.mul_indicator_finset_bUnion I s h | mul_indicator_finset_bUnion_apply {ι} (I : finset ι)
(s : ι → set α) {f : α → M} (h : ∀ (i ∈ I) (j ∈ I), i ≠ j → disjoint (s i) (s j)) (x : α) :
mul_indicator (⋃ i ∈ I, s i) f x = ∏ i in I, mul_indicator (s i) f x | by rw set.mul_indicator_finset_bUnion I s h | lemma | set.mul_indicator_finset_bUnion_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"disjoint",
"finset",
"set.mul_indicator_finset_bUnion"
] | null | 453 | 456 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_mul (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) = λa, indicator s f a * indicator s g a :=
by { funext, simp only [indicator], split_ifs, { refl }, rw mul_zero } | indicator_mul (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) = λa, indicator s f a * indicator s g a | by { funext, simp only [indicator], split_ifs, { refl }, rw mul_zero } | lemma | set.indicator_mul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_zero"
] | null | 464 | 466 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_mul_left (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) a = indicator s f a * g a :=
by { simp only [indicator], split_ifs, { refl }, rw [zero_mul] } | indicator_mul_left (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) a = indicator s f a * g a | by { simp only [indicator], split_ifs, { refl }, rw [zero_mul] } | lemma | set.indicator_mul_left | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"zero_mul"
] | null | 468 | 470 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_mul_right (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) a = f a * indicator s g a :=
by { simp only [indicator], split_ifs, { refl }, rw [mul_zero] } | indicator_mul_right (s : set α) (f g : α → M) :
indicator s (λa, f a * g a) a = f a * indicator s g a | by { simp only [indicator], split_ifs, { refl }, rw [mul_zero] } | lemma | set.indicator_mul_right | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_zero"
] | null | 472 | 474 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inter_indicator_mul {t1 t2 : set α} (f g : α → M) (x : α) :
(t1 ∩ t2).indicator (λ x, f x * g x) x = t1.indicator f x * t2.indicator g x :=
by { rw [← set.indicator_indicator], simp [indicator] } | inter_indicator_mul {t1 t2 : set α} (f g : α → M) (x : α) :
(t1 ∩ t2).indicator (λ x, f x * g x) x = t1.indicator f x * t2.indicator g x | by { rw [← set.indicator_indicator], simp [indicator] } | lemma | set.inter_indicator_mul | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 476 | 478 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inter_indicator_one {s t : set α} :
(s ∩ t).indicator (1 : _ → M) = s.indicator 1 * t.indicator 1 :=
funext (λ _, by simpa only [← inter_indicator_mul, pi.mul_apply, pi.one_apply, one_mul]) | inter_indicator_one {s t : set α} :
(s ∩ t).indicator (1 : _ → M) = s.indicator 1 * t.indicator 1 | funext (λ _, by simpa only [← inter_indicator_mul, pi.mul_apply, pi.one_apply, one_mul]) | lemma | set.inter_indicator_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"one_mul",
"pi.mul_apply",
"pi.one_apply"
] | null | 486 | 488 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_prod_one {s : set α} {t : set β} {x : α} {y : β} :
(s ×ˢ t).indicator (1 : _ → M) (x, y) = s.indicator 1 x * t.indicator 1 y :=
by { classical, simp [indicator_apply, ←ite_and] } | indicator_prod_one {s : set α} {t : set β} {x : α} {y : β} :
(s ×ˢ t).indicator (1 : _ → M) (x, y) = s.indicator 1 x * t.indicator 1 y | by { classical, simp [indicator_apply, ←ite_and] } | lemma | set.indicator_prod_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 490 | 492 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_eq_zero_iff_not_mem {U : set α} {x : α} :
indicator U 1 x = (0 : M) ↔ x ∉ U :=
by { classical, simp [indicator_apply, imp_false] } | indicator_eq_zero_iff_not_mem {U : set α} {x : α} :
indicator U 1 x = (0 : M) ↔ x ∉ U | by { classical, simp [indicator_apply, imp_false] } | lemma | set.indicator_eq_zero_iff_not_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"imp_false"
] | null | 496 | 498 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_eq_one_iff_mem {U : set α} {x : α} :
indicator U 1 x = (1 : M) ↔ x ∈ U :=
by { classical, simp [indicator_apply, imp_false] } | indicator_eq_one_iff_mem {U : set α} {x : α} :
indicator U 1 x = (1 : M) ↔ x ∈ U | by { classical, simp [indicator_apply, imp_false] } | lemma | set.indicator_eq_one_iff_mem | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"imp_false"
] | null | 500 | 502 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_one_inj {U V : set α} (h : indicator U (1 : α → M) = indicator V 1) : U = V :=
by { ext, simp_rw [← indicator_eq_one_iff_mem M, h] } | indicator_one_inj {U V : set α} (h : indicator U (1 : α → M) = indicator V 1) : U = V | by { ext, simp_rw [← indicator_eq_one_iff_mem M, h] } | lemma | set.indicator_one_inj | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 504 | 505 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply_le' (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y) :
mul_indicator s f a ≤ y :=
begin
by_cases ha : a ∈ s,
{ simpa [ha] using hfg ha },
{ simpa [ha] using hg ha },
end | mul_indicator_apply_le' (hfg : a ∈ s → f a ≤ y) (hg : a ∉ s → 1 ≤ y) :
mul_indicator s f a ≤ y | begin
by_cases ha : a ∈ s,
{ simpa [ha] using hfg ha },
{ simpa [ha] using hg ha },
end | lemma | set.mul_indicator_apply_le' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 516 | 522 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_le' (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a ∉ s, 1 ≤ g a) :
mul_indicator s f ≤ g :=
λ a, mul_indicator_apply_le' (hfg _) (hg _) | mul_indicator_le' (hfg : ∀ a ∈ s, f a ≤ g a) (hg : ∀ a ∉ s, 1 ≤ g a) :
mul_indicator s f ≤ g | λ a, mul_indicator_apply_le' (hfg _) (hg _) | lemma | set.mul_indicator_le' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 524 | 526 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_indicator_apply {y} (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1) :
y ≤ mul_indicator s g a :=
@mul_indicator_apply_le' α Mᵒᵈ ‹_› _ _ _ _ _ hfg hf | le_mul_indicator_apply {y} (hfg : a ∈ s → y ≤ g a) (hf : a ∉ s → y ≤ 1) :
y ≤ mul_indicator s g a | @mul_indicator_apply_le' α Mᵒᵈ ‹_› _ _ _ _ _ hfg hf | lemma | set.le_mul_indicator_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 528 | 530 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_mul_indicator (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1) :
f ≤ mul_indicator s g :=
λ a, le_mul_indicator_apply (hfg _) (hf _) | le_mul_indicator (hfg : ∀ a ∈ s, f a ≤ g a) (hf : ∀ a ∉ s, f a ≤ 1) :
f ≤ mul_indicator s g | λ a, le_mul_indicator_apply (hfg _) (hf _) | lemma | set.le_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 532 | 534 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_le_mul_indicator_apply (h : a ∈ s → 1 ≤ f a) : 1 ≤ mul_indicator s f a :=
le_mul_indicator_apply h (λ _, le_rfl) | one_le_mul_indicator_apply (h : a ∈ s → 1 ≤ f a) : 1 ≤ mul_indicator s f a | le_mul_indicator_apply h (λ _, le_rfl) | lemma | set.one_le_mul_indicator_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"le_rfl"
] | null | 540 | 542 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_le_mul_indicator (h : ∀ a ∈ s, 1 ≤ f a) (a : α) : 1 ≤ mul_indicator s f a :=
one_le_mul_indicator_apply (h a) | one_le_mul_indicator (h : ∀ a ∈ s, 1 ≤ f a) (a : α) : 1 ≤ mul_indicator s f a | one_le_mul_indicator_apply (h a) | lemma | set.one_le_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 544 | 546 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply_le_one (h : a ∈ s → f a ≤ 1) : mul_indicator s f a ≤ 1 :=
mul_indicator_apply_le' h (λ _, le_rfl) | mul_indicator_apply_le_one (h : a ∈ s → f a ≤ 1) : mul_indicator s f a ≤ 1 | mul_indicator_apply_le' h (λ _, le_rfl) | lemma | set.mul_indicator_apply_le_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"le_rfl"
] | null | 548 | 549 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_le_one (h : ∀ a ∈ s, f a ≤ 1) (a : α) :
mul_indicator s f a ≤ 1 :=
mul_indicator_apply_le_one (h a) | mul_indicator_le_one (h : ∀ a ∈ s, f a ≤ 1) (a : α) :
mul_indicator s f a ≤ 1 | mul_indicator_apply_le_one (h a) | lemma | set.mul_indicator_le_one | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 551 | 553 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_le_mul_indicator (h : f a ≤ g a) :
mul_indicator s f a ≤ mul_indicator s g a :=
mul_indicator_rel_mul_indicator le_rfl (λ _, h) | mul_indicator_le_mul_indicator (h : f a ≤ g a) :
mul_indicator s f a ≤ mul_indicator s g a | mul_indicator_rel_mul_indicator le_rfl (λ _, h) | lemma | set.mul_indicator_le_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"le_rfl"
] | null | 555 | 557 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_le_mul_indicator_of_subset (h : s ⊆ t) (hf : ∀ a, 1 ≤ f a)
(a : α) :
mul_indicator s f a ≤ mul_indicator t f a :=
mul_indicator_apply_le' (λ ha, le_mul_indicator_apply (λ _, le_rfl) (λ hat, (hat $ h ha).elim))
(λ ha, one_le_mul_indicator_apply (λ _, hf _)) | mul_indicator_le_mul_indicator_of_subset (h : s ⊆ t) (hf : ∀ a, 1 ≤ f a)
(a : α) :
mul_indicator s f a ≤ mul_indicator t f a | mul_indicator_apply_le' (λ ha, le_mul_indicator_apply (λ _, le_rfl) (λ hat, (hat $ h ha).elim))
(λ ha, one_le_mul_indicator_apply (λ _, hf _)) | lemma | set.mul_indicator_le_mul_indicator_of_subset | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"le_rfl"
] | null | 561 | 565 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_le_self' (hf : ∀ x ∉ s, 1 ≤ f x) : mul_indicator s f ≤ f :=
mul_indicator_le' (λ _ _, le_rfl) hf | mul_indicator_le_self' (hf : ∀ x ∉ s, 1 ≤ f x) : mul_indicator s f ≤ f | mul_indicator_le' (λ _ _, le_rfl) hf | lemma | set.mul_indicator_le_self' | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"le_rfl"
] | null | 567 | 568 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_Union_apply {ι M} [complete_lattice M] [has_one M]
(h1 : (⊥:M) = 1) (s : ι → set α) (f : α → M) (x : α) :
mul_indicator (⋃ i, s i) f x = ⨆ i, mul_indicator (s i) f x :=
begin
by_cases hx : x ∈ ⋃ i, s i,
{ rw [mul_indicator_of_mem hx],
rw [mem_Union] at hx,
refine le_antisymm _ (supr_le $ λ... | mul_indicator_Union_apply {ι M} [complete_lattice M] [has_one M]
(h1 : (⊥:M) = 1) (s : ι → set α) (f : α → M) (x : α) :
mul_indicator (⋃ i, s i) f x = ⨆ i, mul_indicator (s i) f x | begin
by_cases hx : x ∈ ⋃ i, s i,
{ rw [mul_indicator_of_mem hx],
rw [mem_Union] at hx,
refine le_antisymm _ (supr_le $ λ i, mul_indicator_le_self' (λ x hx, h1 ▸ bot_le) x),
rcases hx with ⟨i, hi⟩,
exact le_supr_of_le i (ge_of_eq $ mul_indicator_of_mem hi _) },
{ rw [mul_indicator_of_not_mem hx],
... | lemma | set.mul_indicator_Union_apply | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"bot_le",
"complete_lattice",
"ge_of_eq",
"le_supr_of_le",
"not_exists",
"supr_le"
] | null | 570 | 583 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_le_self (s : set α) (f : α → M) :
mul_indicator s f ≤ f :=
mul_indicator_le_self' $ λ _ _, one_le _ | mul_indicator_le_self (s : set α) (f : α → M) :
mul_indicator s f ≤ f | mul_indicator_le_self' $ λ _ _, one_le _ | lemma | set.mul_indicator_le_self | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"one_le"
] | null | 591 | 593 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_apply_le {a : α} {s : set α} {f g : α → M}
(hfg : a ∈ s → f a ≤ g a) :
mul_indicator s f a ≤ g a :=
mul_indicator_apply_le' hfg $ λ _, one_le _ | mul_indicator_apply_le {a : α} {s : set α} {f g : α → M}
(hfg : a ∈ s → f a ≤ g a) :
mul_indicator s f a ≤ g a | mul_indicator_apply_le' hfg $ λ _, one_le _ | lemma | set.mul_indicator_apply_le | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"one_le"
] | null | 595 | 598 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_indicator_le {s : set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) :
mul_indicator s f ≤ g :=
mul_indicator_le' hfg $ λ _ _, one_le _ | mul_indicator_le {s : set α} {f g : α → M} (hfg : ∀ a ∈ s, f a ≤ g a) :
mul_indicator s f ≤ g | mul_indicator_le' hfg $ λ _ _, one_le _ | lemma | set.mul_indicator_le | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"one_le"
] | null | 600 | 602 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_le_indicator_nonneg {β} [linear_order β] [has_zero β] (s : set α) (f : α → β) :
s.indicator f ≤ {x | 0 ≤ f x}.indicator f :=
begin
intro x,
classical,
simp_rw indicator_apply,
split_ifs,
{ exact le_rfl, },
{ exact (not_le.mp h_1).le, },
{ exact h_1, },
{ exact le_rfl, },
end | indicator_le_indicator_nonneg {β} [linear_order β] [has_zero β] (s : set α) (f : α → β) :
s.indicator f ≤ {x | 0 ≤ f x}.indicator f | begin
intro x,
classical,
simp_rw indicator_apply,
split_ifs,
{ exact le_rfl, },
{ exact (not_le.mp h_1).le, },
{ exact h_1, },
{ exact le_rfl, },
end | lemma | set.indicator_le_indicator_nonneg | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"le_rfl"
] | null | 606 | 617 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
indicator_nonpos_le_indicator {β} [linear_order β] [has_zero β] (s : set α) (f : α → β) :
{x | f x ≤ 0}.indicator f ≤ s.indicator f :=
@indicator_le_indicator_nonneg α βᵒᵈ _ _ s f | indicator_nonpos_le_indicator {β} [linear_order β] [has_zero β] (s : set α) (f : α → β) :
{x | f x ≤ 0}.indicator f ≤ s.indicator f | @indicator_le_indicator_nonneg α βᵒᵈ _ _ s f | lemma | set.indicator_nonpos_le_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [] | null | 619 | 621 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.map_mul_indicator
{M N : Type*} [mul_one_class M] [mul_one_class N] (f : M →* N)
(s : set α) (g : α → M) (x : α) :
f (s.mul_indicator g x) = s.mul_indicator (f ∘ g) x :=
congr_fun (set.mul_indicator_comp_of_one f.map_one).symm x | monoid_hom.map_mul_indicator
{M N : Type*} [mul_one_class M] [mul_one_class N] (f : M →* N)
(s : set α) (g : α → M) (x : α) :
f (s.mul_indicator g x) = s.mul_indicator (f ∘ g) x | congr_fun (set.mul_indicator_comp_of_one f.map_one).symm x | lemma | monoid_hom.map_mul_indicator | algebra | src/algebra/indicator_function.lean | [
"algebra.support"
] | [
"mul_one_class",
"set.mul_indicator_comp_of_one"
] | null | 625 | 629 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible [has_mul α] [has_one α] (a : α) : Type u :=
(inv_of : α) (inv_of_mul_self : inv_of * a = 1) (mul_inv_of_self : a * inv_of = 1) | invertible [has_mul α] [has_one α] (a : α) : Type u | (inv_of : α) (inv_of_mul_self : inv_of * a = 1) (mul_inv_of_self : a * inv_of = 1) | class | invertible | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"inv_of_mul_self",
"mul_inv_of_self"
] | `invertible a` gives a two-sided multiplicative inverse of `a`. | 68 | 69 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_of_mul_self [has_mul α] [has_one α] (a : α) [invertible a] : ⅟a * a = 1 :=
invertible.inv_of_mul_self | inv_of_mul_self [has_mul α] [has_one α] (a : α) [invertible a] : ⅟a * a = 1 | invertible.inv_of_mul_self | lemma | inv_of_mul_self | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible"
] | null | 74 | 76 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_of_self [has_mul α] [has_one α] (a : α) [invertible a] : a * ⅟a = 1 :=
invertible.mul_inv_of_self | mul_inv_of_self [has_mul α] [has_one α] (a : α) [invertible a] : a * ⅟a = 1 | invertible.mul_inv_of_self | lemma | mul_inv_of_self | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible"
] | null | 78 | 80 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_of_mul_self_assoc [monoid α] (a b : α) [invertible a] : ⅟a * (a * b) = b :=
by rw [←mul_assoc, inv_of_mul_self, one_mul] | inv_of_mul_self_assoc [monoid α] (a b : α) [invertible a] : ⅟a * (a * b) = b | by rw [←mul_assoc, inv_of_mul_self, one_mul] | lemma | inv_of_mul_self_assoc | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"inv_of_mul_self",
"invertible",
"monoid",
"one_mul"
] | null | 82 | 84 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_of_self_assoc [monoid α] (a b : α) [invertible a] : a * (⅟a * b) = b :=
by rw [←mul_assoc, mul_inv_of_self, one_mul] | mul_inv_of_self_assoc [monoid α] (a b : α) [invertible a] : a * (⅟a * b) = b | by rw [←mul_assoc, mul_inv_of_self, one_mul] | lemma | mul_inv_of_self_assoc | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"monoid",
"mul_inv_of_self",
"one_mul"
] | null | 86 | 88 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_inv_of_mul_self_cancel [monoid α] (a b : α) [invertible b] : a * ⅟b * b = a :=
by simp [mul_assoc] | mul_inv_of_mul_self_cancel [monoid α] (a b : α) [invertible b] : a * ⅟b * b = a | by simp [mul_assoc] | lemma | mul_inv_of_mul_self_cancel | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"monoid",
"mul_assoc"
] | null | 90 | 92 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mul_inv_of_self_cancel [monoid α] (a b : α) [invertible b] : a * b * ⅟b = a :=
by simp [mul_assoc] | mul_mul_inv_of_self_cancel [monoid α] (a b : α) [invertible b] : a * b * ⅟b = a | by simp [mul_assoc] | lemma | mul_mul_inv_of_self_cancel | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"monoid",
"mul_assoc"
] | null | 94 | 96 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_of_eq_right_inv [monoid α] {a b : α} [invertible a] (hac : a * b = 1) : ⅟a = b :=
left_inv_eq_right_inv (inv_of_mul_self _) hac | inv_of_eq_right_inv [monoid α] {a b : α} [invertible a] (hac : a * b = 1) : ⅟a = b | left_inv_eq_right_inv (inv_of_mul_self _) hac | lemma | inv_of_eq_right_inv | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"inv_of_mul_self",
"invertible",
"left_inv_eq_right_inv",
"monoid"
] | null | 98 | 99 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_of_eq_left_inv [monoid α] {a b : α} [invertible a] (hac : b * a = 1) : ⅟a = b :=
(left_inv_eq_right_inv hac (mul_inv_of_self _)).symm | inv_of_eq_left_inv [monoid α] {a b : α} [invertible a] (hac : b * a = 1) : ⅟a = b | (left_inv_eq_right_inv hac (mul_inv_of_self _)).symm | lemma | inv_of_eq_left_inv | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"left_inv_eq_right_inv",
"monoid",
"mul_inv_of_self"
] | null | 101 | 102 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible_unique {α : Type u} [monoid α] (a b : α) [invertible a] [invertible b]
(h : a = b) :
⅟a = ⅟b :=
by { apply inv_of_eq_right_inv, rw [h, mul_inv_of_self], } | invertible_unique {α : Type u} [monoid α] (a b : α) [invertible a] [invertible b]
(h : a = b) :
⅟a = ⅟b | by { apply inv_of_eq_right_inv, rw [h, mul_inv_of_self], } | lemma | invertible_unique | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"inv_of_eq_right_inv",
"invertible",
"monoid",
"mul_inv_of_self"
] | null | 104 | 107 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[monoid α] (a : α) : subsingleton (invertible a) :=
⟨ λ ⟨b, hba, hab⟩ ⟨c, hca, hac⟩, by { congr, exact left_inv_eq_right_inv hba hac } ⟩ | [monoid α] (a : α) : subsingleton (invertible a) | ⟨ λ ⟨b, hba, hab⟩ ⟨c, hca, hac⟩, by { congr, exact left_inv_eq_right_inv hba hac } ⟩ | instance | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"left_inv_eq_right_inv",
"monoid"
] | null | 109 | 110 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invertible.copy' [mul_one_class α] {r : α} (hr : invertible r) (s : α) (si : α)
(hs : s = r) (hsi : si = ⅟r) :
invertible s :=
{ inv_of := si,
inv_of_mul_self := by rw [hs, hsi, inv_of_mul_self],
mul_inv_of_self := by rw [hs, hsi, mul_inv_of_self] } | invertible.copy' [mul_one_class α] {r : α} (hr : invertible r) (s : α) (si : α)
(hs : s = r) (hsi : si = ⅟r) :
invertible s | { inv_of := si,
inv_of_mul_self := by rw [hs, hsi, inv_of_mul_self],
mul_inv_of_self := by rw [hs, hsi, mul_inv_of_self] } | def | invertible.copy' | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"inv_of_mul_self",
"invertible",
"mul_inv_of_self",
"mul_one_class"
] | If `r` is invertible and `s = r` and `si = ⅟r`, then `s` is invertible with `⅟s = si`. | 113 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible.copy [mul_one_class α] {r : α} (hr : invertible r) (s : α) (hs : s = r) :
invertible s :=
hr.copy' _ _ hs rfl | invertible.copy [mul_one_class α] {r : α} (hr : invertible r) (s : α) (hs : s = r) :
invertible s | hr.copy' _ _ hs rfl | def | invertible.copy | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"mul_one_class"
] | If `r` is invertible and `s = r`, then `s` is invertible. | 121 | 124 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unit_of_invertible [monoid α] (a : α) [invertible a] : αˣ :=
{ val := a,
inv := ⅟a,
val_inv := by simp,
inv_val := by simp, } | unit_of_invertible [monoid α] (a : α) [invertible a] : αˣ | { val := a,
inv := ⅟a,
val_inv := by simp,
inv_val := by simp, } | def | unit_of_invertible | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"monoid"
] | An `invertible` element is a unit. | 127 | 132 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_of_invertible [monoid α] (a : α) [invertible a] : is_unit a :=
⟨unit_of_invertible a, rfl⟩ | is_unit_of_invertible [monoid α] (a : α) [invertible a] : is_unit a | ⟨unit_of_invertible a, rfl⟩ | lemma | is_unit_of_invertible | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"is_unit",
"monoid"
] | null | 134 | 135 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.invertible [monoid α] (u : αˣ) : invertible (u : α) :=
{ inv_of := ↑(u⁻¹), inv_of_mul_self := u.inv_mul, mul_inv_of_self := u.mul_inv } | units.invertible [monoid α] (u : αˣ) : invertible (u : α) | { inv_of := ↑(u⁻¹), inv_of_mul_self := u.inv_mul, mul_inv_of_self := u.mul_inv } | def | units.invertible | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"inv_of_mul_self",
"invertible",
"monoid",
"mul_inv_of_self"
] | Units are invertible in their associated monoid. | 138 | 139 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_of_units [monoid α] (u : αˣ) [invertible (u : α)] : ⅟(u : α) = ↑(u⁻¹) :=
inv_of_eq_right_inv u.mul_inv | inv_of_units [monoid α] (u : αˣ) [invertible (u : α)] : ⅟(u : α) = ↑(u⁻¹) | inv_of_eq_right_inv u.mul_inv | lemma | inv_of_units | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"inv_of_eq_right_inv",
"invertible",
"monoid"
] | null | 141 | 142 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit.nonempty_invertible [monoid α] {a : α} (h : is_unit a) : nonempty (invertible a) :=
let ⟨x, hx⟩ := h in ⟨x.invertible.copy _ hx.symm⟩ | is_unit.nonempty_invertible [monoid α] {a : α} (h : is_unit a) : nonempty (invertible a) | let ⟨x, hx⟩ := h in ⟨x.invertible.copy _ hx.symm⟩ | lemma | is_unit.nonempty_invertible | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"is_unit",
"monoid"
] | null | 144 | 145 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit.invertible [monoid α] {a : α} (h : is_unit a) : invertible a :=
classical.choice h.nonempty_invertible | is_unit.invertible [monoid α] {a : α} (h : is_unit a) : invertible a | classical.choice h.nonempty_invertible | def | is_unit.invertible | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"is_unit",
"monoid"
] | Convert `is_unit` to `invertible` using `classical.choice`.
Prefer `casesI h.nonempty_invertible` over `letI := h.invertible` if you want to avoid choice. | 150 | 151 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nonempty_invertible_iff_is_unit [monoid α] (a : α) :
nonempty (invertible a) ↔ is_unit a :=
⟨nonempty.rec $ @is_unit_of_invertible _ _ _, is_unit.nonempty_invertible⟩ | nonempty_invertible_iff_is_unit [monoid α] (a : α) :
nonempty (invertible a) ↔ is_unit a | ⟨nonempty.rec $ @is_unit_of_invertible _ _ _, is_unit.nonempty_invertible⟩ | lemma | nonempty_invertible_iff_is_unit | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"invertible",
"is_unit",
"is_unit_of_invertible",
"monoid"
] | null | 153 | 155 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
invertible_of_group [group α] (a : α) : invertible a :=
⟨a⁻¹, inv_mul_self a, mul_inv_self a⟩ | invertible_of_group [group α] (a : α) : invertible a | ⟨a⁻¹, inv_mul_self a, mul_inv_self a⟩ | def | invertible_of_group | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"group",
"inv_mul_self",
"invertible",
"mul_inv_self"
] | Each element of a group is invertible. | 158 | 159 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_of_eq_group_inv [group α] (a : α) [invertible a] : ⅟a = a⁻¹ :=
inv_of_eq_right_inv (mul_inv_self a) | inv_of_eq_group_inv [group α] (a : α) [invertible a] : ⅟a = a⁻¹ | inv_of_eq_right_inv (mul_inv_self a) | lemma | inv_of_eq_group_inv | algebra | src/algebra/invertible.lean | [
"algebra.group.units",
"algebra.group_with_zero.units.lemmas",
"algebra.ring.defs"
] | [
"group",
"inv_of_eq_right_inv",
"invertible",
"mul_inv_self"
] | null | 161 | 162 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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