statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
monoid_with_zero.to_mul_action_with_zero : mul_action_with_zero R R | { ..mul_zero_class.to_smul_with_zero R,
..monoid.to_mul_action R } | instance | monoid_with_zero.to_mul_action_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"monoid.to_mul_action",
"mul_action_with_zero",
"mul_zero_class.to_smul_with_zero"
] | See also `semiring.to_module` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero.to_opposite_mul_action_with_zero : mul_action_with_zero Rᵐᵒᵖ R | { ..mul_zero_class.to_opposite_smul_with_zero R,
..monoid.to_opposite_mul_action R } | instance | monoid_with_zero.to_opposite_mul_action_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"monoid.to_opposite_mul_action",
"mul_action_with_zero",
"mul_zero_class.to_opposite_smul_with_zero"
] | Like `monoid_with_zero.to_mul_action_with_zero`, but multiplies on the right. See also
`semiring.to_opposite_module` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_action_with_zero.subsingleton
[mul_action_with_zero R M] [subsingleton R] : subsingleton M | ⟨λ x y, by rw [←one_smul R x, ←one_smul R y, subsingleton.elim (1 : R) 0, zero_smul, zero_smul]⟩ | lemma | mul_action_with_zero.subsingleton | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_action_with_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_with_zero.nontrivial
[mul_action_with_zero R M] [nontrivial M] : nontrivial R | (subsingleton_or_nontrivial R).resolve_left $ λ hR, not_subsingleton M $
by exactI mul_action_with_zero.subsingleton R M | lemma | mul_action_with_zero.nontrivial | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_action_with_zero",
"mul_action_with_zero.subsingleton",
"nontrivial",
"not_subsingleton",
"subsingleton_or_nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.injective.mul_action_with_zero
(f : zero_hom M' M) (hf : function.injective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
mul_action_with_zero R M' | { ..hf.mul_action f smul, ..hf.smul_with_zero f smul } | def | function.injective.mul_action_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_action_with_zero",
"zero_hom"
] | Pullback a `mul_action_with_zero` structure along an injective zero-preserving homomorphism.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.surjective.mul_action_with_zero
(f : zero_hom M M') (hf : function.surjective f) (smul : ∀ (a : R) b, f (a • b) = a • f b) :
mul_action_with_zero R M' | { ..hf.mul_action f smul, ..hf.smul_with_zero f smul } | def | function.surjective.mul_action_with_zero | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_action_with_zero",
"zero_hom"
] | Pushforward a `mul_action_with_zero` structure along a surjective zero-preserving homomorphism.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_action_with_zero.comp_hom (f : R' →*₀ R) : mul_action_with_zero R' M | { smul := (•) ∘ f,
mul_smul := λ r s m, by simp [mul_smul],
one_smul := λ m, by simp,
.. smul_with_zero.comp_hom M f.to_zero_hom} | def | mul_action_with_zero.comp_hom | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"mul_action_with_zero",
"one_smul",
"smul_with_zero.comp_hom"
] | Compose a `mul_action_with_zero` with a `monoid_with_zero_hom`, with action `f r' • m` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_inv₀ [smul_comm_class α β β] [is_scalar_tower α β β] (c : α) (x : β) :
(c • x)⁻¹ = c⁻¹ • x⁻¹ | begin
obtain rfl | hc := eq_or_ne c 0,
{ simp only [inv_zero, zero_smul] },
obtain rfl | hx := eq_or_ne x 0,
{ simp only [inv_zero, smul_zero] },
{ refine inv_eq_of_mul_eq_one_left _,
rw [smul_mul_smul, inv_mul_cancel hc, inv_mul_cancel hx, one_smul] }
end | lemma | smul_inv₀ | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"eq_or_ne",
"inv_eq_of_mul_eq_one_left",
"inv_mul_cancel",
"inv_zero",
"is_scalar_tower",
"one_smul",
"smul_comm_class",
"smul_mul_smul",
"smul_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_monoid_with_zero_hom {α β : Type*} [monoid_with_zero α] [mul_zero_one_class β]
[mul_action_with_zero α β] [is_scalar_tower α β β] [smul_comm_class α β β] :
α × β →*₀ β | { map_zero' := smul_zero _,
.. smul_monoid_hom } | def | smul_monoid_with_zero_hom | algebra | src/algebra/smul_with_zero.lean | [
"algebra.group_power.basic",
"algebra.ring.opposite",
"group_theory.group_action.opposite",
"group_theory.group_action.prod"
] | [
"is_scalar_tower",
"monoid_with_zero",
"mul_action_with_zero",
"mul_zero_one_class",
"smul_comm_class",
"smul_monoid_hom",
"smul_zero"
] | Scalar multiplication as a monoid homomorphism with zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
squarefree [monoid R] (r : R) : Prop | ∀ x : R, x * x ∣ r → is_unit x | def | squarefree | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"is_unit",
"monoid"
] | An element of a monoid is squarefree if the only squares that
divide it are the squares of units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit.squarefree [comm_monoid R] {x : R} (h : is_unit x) :
squarefree x | λ y hdvd, is_unit_of_mul_is_unit_left (is_unit_of_dvd_unit hdvd h) | lemma | is_unit.squarefree | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"comm_monoid",
"is_unit",
"is_unit_of_dvd_unit",
"is_unit_of_mul_is_unit_left",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree_one [comm_monoid R] : squarefree (1 : R) | is_unit_one.squarefree | lemma | squarefree_one | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"comm_monoid",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_squarefree_zero [monoid_with_zero R] [nontrivial R] : ¬ squarefree (0 : R) | begin
erw [not_forall],
exact ⟨0, by simp⟩,
end | lemma | not_squarefree_zero | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"monoid_with_zero",
"nontrivial",
"not_forall",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree.ne_zero [monoid_with_zero R] [nontrivial R] {m : R}
(hm : squarefree (m : R)) : m ≠ 0 | begin
rintro rfl,
exact not_squarefree_zero hm,
end | lemma | squarefree.ne_zero | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"monoid_with_zero",
"nontrivial",
"not_squarefree_zero",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible.squarefree [comm_monoid R] {x : R} (h : irreducible x) :
squarefree x | begin
rintros y ⟨z, hz⟩,
rw mul_assoc at hz,
rcases h.is_unit_or_is_unit hz with hu | hu,
{ exact hu },
{ apply is_unit_of_mul_is_unit_left hu },
end | lemma | irreducible.squarefree | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"comm_monoid",
"irreducible",
"is_unit_of_mul_is_unit_left",
"mul_assoc",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime.squarefree [cancel_comm_monoid_with_zero R] {x : R} (h : prime x) :
squarefree x | h.irreducible.squarefree | lemma | prime.squarefree | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"cancel_comm_monoid_with_zero",
"prime",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree.of_mul_left [comm_monoid R] {m n : R} (hmn : squarefree (m * n)) : squarefree m | (λ p hp, hmn p (dvd_mul_of_dvd_left hp n)) | lemma | squarefree.of_mul_left | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"comm_monoid",
"dvd_mul_of_dvd_left",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree.of_mul_right [comm_monoid R] {m n : R} (hmn : squarefree (m * n)) : squarefree n | (λ p hp, hmn p (dvd_mul_of_dvd_right hp m)) | lemma | squarefree.of_mul_right | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"comm_monoid",
"dvd_mul_of_dvd_right",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree.squarefree_of_dvd [comm_monoid R]
{x y : R} (hdvd : x ∣ y) (hsq : squarefree y) :
squarefree x | λ a h, hsq _ (h.trans hdvd) | lemma | squarefree.squarefree_of_dvd | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"comm_monoid",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree.gcd_right (a : α) {b : α} (hb : squarefree b) :
squarefree (gcd a b) | hb.squarefree_of_dvd (gcd_dvd_right _ _) | lemma | squarefree.gcd_right | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree.gcd_left {a : α} (b : α) (ha : squarefree a) :
squarefree (gcd a b) | ha.squarefree_of_dvd (gcd_dvd_left _ _) | lemma | squarefree.gcd_left | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree_iff_multiplicity_le_one (r : R) :
squarefree r ↔ ∀ x : R, multiplicity x r ≤ 1 ∨ is_unit x | begin
refine forall_congr (λ a, _),
rw [← sq, pow_dvd_iff_le_multiplicity, or_iff_not_imp_left, not_le, imp_congr _ iff.rfl],
simpa using part_enat.add_one_le_iff_lt (part_enat.coe_ne_top 1)
end | lemma | multiplicity.squarefree_iff_multiplicity_le_one | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"is_unit",
"multiplicity",
"or_iff_not_imp_left",
"part_enat.add_one_le_iff_lt",
"part_enat.coe_ne_top",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_prime_left {a b : R} (ha : prime a) (hb : b ≠ 0) :
multiplicity.finite a b | begin
classical,
revert hb,
refine wf_dvd_monoid.induction_on_irreducible b (by contradiction) (λ u hu hu', _)
(λ b p hb hp ih hpb, _),
{ rw [multiplicity.finite_iff_dom, multiplicity.is_unit_right ha.not_unit hu],
exact part_enat.dom_coe 0, },
{ refine multiplicity.finite_mul ha
(multiplicity.f... | lemma | multiplicity.finite_prime_left | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"ih",
"multiplicity",
"multiplicity.finite",
"multiplicity.finite_iff_dom",
"multiplicity.finite_mul",
"multiplicity.is_unit_right",
"multiplicity.squarefree_iff_multiplicity_le_one",
"part_enat.dom_coe",
"part_enat.dom_of_le_coe",
"prime",
"wf_dvd_monoid.induction_on_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :
(∀ x : R, irreducible x → ¬ x * x ∣ r) ↔ ((r = 0 ∧ ∀ x : R, ¬irreducible x) ∨ squarefree r) | begin
symmetry,
split,
{ rintro (⟨rfl, h⟩ | h),
{ simpa using h },
intros x hx t,
exact hx.not_unit (h x t) },
intro h,
rcases eq_or_ne r 0 with rfl | hr,
{ exact or.inl (by simpa using h) },
right,
intros x hx,
by_contra i,
have : x ≠ 0,
{ rintro rfl,
apply hr,
simpa only [zer... | lemma | irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"by_contra",
"eq_or_ne",
"irreducible",
"mul_dvd_mul",
"mul_zero",
"squarefree",
"wf_dvd_monoid.exists_irreducible_factor",
"zero_dvd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree_iff_irreducible_sq_not_dvd_of_ne_zero {r : R} (hr : r ≠ 0) :
squarefree r ↔ ∀ x : R, irreducible x → ¬ x * x ∣ r | by simpa [hr] using (irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree r).symm | lemma | squarefree_iff_irreducible_sq_not_dvd_of_ne_zero | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"irreducible",
"irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible
{r : R} (hr : ∃ (x : R), irreducible x) :
squarefree r ↔ ∀ x : R, irreducible x → ¬ x * x ∣ r | begin
rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ←not_exists],
simp only [hr, not_true, false_or, and_false],
end | lemma | squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"irreducible",
"irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical.squarefree {x : R} (h0 : x ≠ 0) (h : is_radical x) : squarefree x | begin
rintro z ⟨w, rfl⟩,
specialize h 2 (z * w) ⟨w, by simp_rw [pow_two, mul_left_comm, ← mul_assoc]⟩,
rwa [← one_mul (z * w), mul_assoc, mul_dvd_mul_iff_right, ← is_unit_iff_dvd_one] at h,
rw [mul_assoc, mul_ne_zero_iff] at h0, exact h0.2,
end | theorem | is_radical.squarefree | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"is_radical",
"is_unit_iff_dvd_one",
"mul_assoc",
"mul_dvd_mul_iff_right",
"mul_left_comm",
"mul_ne_zero_iff",
"one_mul",
"pow_two",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree.is_radical {x : R} (hx : squarefree x) : is_radical x | (is_radical_iff_pow_one_lt 2 one_lt_two).2 $ λ y hy, and.right $ (dvd_gcd_iff x x y).1
begin
by_cases gcd x y = 0, { rw h, apply dvd_zero },
replace hy := ((dvd_gcd_iff x x _).2 ⟨dvd_rfl, hy⟩).trans gcd_pow_right_dvd_pow_gcd,
obtain ⟨z, hz⟩ := gcd_dvd_left x y,
nth_rewrite 0 hz at hy ⊢,
rw [pow_two, mul_dvd_m... | theorem | squarefree.is_radical | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"dvd_gcd_iff",
"dvd_rfl",
"dvd_zero",
"gcd_pow_right_dvd_pow_gcd",
"is_radical",
"is_radical_iff_pow_one_lt",
"mul_dvd_mul_iff_left",
"mul_right_comm",
"one_lt_two",
"pow_two",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical_iff_squarefree_or_zero {x : R} : is_radical x ↔ squarefree x ∨ x = 0 | ⟨λ hx, (em $ x = 0).elim or.inr (λ h, or.inl $ hx.squarefree h),
or.rec squarefree.is_radical $ by { rintro rfl, rw zero_is_radical_iff, apply_instance }⟩ | theorem | is_radical_iff_squarefree_or_zero | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"em",
"is_radical",
"squarefree",
"squarefree.is_radical",
"zero_is_radical_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical_iff_squarefree_of_ne_zero {x : R} (h : x ≠ 0) : is_radical x ↔ squarefree x | ⟨is_radical.squarefree h, squarefree.is_radical⟩ | theorem | is_radical_iff_squarefree_of_ne_zero | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"is_radical",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree_iff_nodup_normalized_factors [normalization_monoid R] [decidable_eq R] {x : R}
(x0 : x ≠ 0) : squarefree x ↔ multiset.nodup (normalized_factors x) | begin
have drel : decidable_rel (has_dvd.dvd : R → R → Prop),
{ classical,
apply_instance, },
haveI := drel,
rw [multiplicity.squarefree_iff_multiplicity_le_one, multiset.nodup_iff_count_le_one],
haveI := nontrivial_of_ne x 0 x0,
split; intros h a,
{ by_cases hmem : a ∈ normalized_factors x,
{ hav... | lemma | unique_factorization_monoid.squarefree_iff_nodup_normalized_factors | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"multiplicity.multiplicity_le_multiplicity_of_dvd_left",
"multiplicity.squarefree_iff_multiplicity_le_one",
"multiset.count_eq_zero_of_not_mem",
"multiset.nodup",
"multiset.nodup_iff_count_le_one",
"nontrivial_of_ne",
"normalization_monoid",
"normalize",
"or_iff_not_imp_right",
"squarefree",
"wf... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_pow_iff_dvd_of_squarefree {x y : R} {n : ℕ} (hsq : squarefree x) (h0 : n ≠ 0) :
x ∣ y ^ n ↔ x ∣ y | begin
classical,
haveI := unique_factorization_monoid.to_gcd_monoid R,
exact ⟨hsq.is_radical n y, λ h, h.pow h0⟩,
end | lemma | unique_factorization_monoid.dvd_pow_iff_dvd_of_squarefree | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"squarefree",
"unique_factorization_monoid.to_gcd_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree_nat_abs {n : ℤ} : squarefree n.nat_abs ↔ squarefree n | by simp_rw [squarefree, nat_abs_surjective.forall, ←nat_abs_mul, nat_abs_dvd_iff_dvd,
is_unit_iff_nat_abs_eq, nat.is_unit_iff] | lemma | int.squarefree_nat_abs | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"nat.is_unit_iff",
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
squarefree_coe_nat {n : ℕ} : squarefree (n : ℤ) ↔ squarefree n | by rw [←squarefree_nat_abs, nat_abs_of_nat] | lemma | int.squarefree_coe_nat | algebra | src/algebra/squarefree.lean | [
"ring_theory.unique_factorization_domain"
] | [
"squarefree"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support [has_zero A] (f : α → A) : set α | {x | f x ≠ 0} | def | function.support | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | `support` of a function is the set of points `x` such that `f x ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_support (f : α → M) : set α | {x | f x ≠ 1} | def | function.mul_support | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | `mul_support` of a function is the set of points `x` such that `f x ≠ 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_support_eq_preimage (f : α → M) : mul_support f = f ⁻¹' {1}ᶜ | rfl | lemma | function.mul_support_eq_preimage | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nmem_mul_support {f : α → M} {x : α} :
x ∉ mul_support f ↔ f x = 1 | not_not | lemma | function.nmem_mul_support | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compl_mul_support {f : α → M} :
(mul_support f)ᶜ = {x | f x = 1} | ext $ λ x, nmem_mul_support | lemma | function.compl_mul_support | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_mul_support {f : α → M} {x : α} :
x ∈ mul_support f ↔ f x ≠ 1 | iff.rfl | lemma | function.mem_mul_support | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_subset_iff {f : α → M} {s : set α} :
mul_support f ⊆ s ↔ ∀ x, f x ≠ 1 → x ∈ s | iff.rfl | lemma | function.mul_support_subset_iff | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_subset_iff' {f : α → M} {s : set α} :
mul_support f ⊆ s ↔ ∀ x ∉ s, f x = 1 | forall_congr $ λ x, not_imp_comm | lemma | function.mul_support_subset_iff' | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"not_imp_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_eq_iff {f : α → M} {s : set α} :
mul_support f = s ↔ ((∀ x, x ∈ s → f x ≠ 1) ∧ (∀ x, x ∉ s → f x = 1)) | by simp only [set.ext_iff, mem_mul_support, ne.def, imp_not_comm, ← forall_and_distrib,
← iff_def, ← xor_iff_not_iff', ← xor_iff_iff_not] | lemma | function.mul_support_eq_iff | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"forall_and_distrib",
"iff_def",
"imp_not_comm",
"set.ext_iff",
"xor_iff_iff_not",
"xor_iff_not_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_disjoint_iff {f : α → M} {s : set α} :
disjoint (mul_support f) s ↔ eq_on f 1 s | by simp_rw [←subset_compl_iff_disjoint_right, mul_support_subset_iff', not_mem_compl_iff, eq_on,
pi.one_apply] | lemma | function.mul_support_disjoint_iff | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"disjoint",
"pi.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_mul_support_iff {f : α → M} {s : set α} :
disjoint s (mul_support f) ↔ eq_on f 1 s | by rw [disjoint.comm, mul_support_disjoint_iff] | lemma | function.disjoint_mul_support_iff | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"disjoint",
"disjoint.comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_eq_empty_iff {f : α → M} :
mul_support f = ∅ ↔ f = 1 | by { simp_rw [← subset_empty_iff, mul_support_subset_iff', funext_iff], simp } | lemma | function.mul_support_eq_empty_iff | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_nonempty_iff {f : α → M} :
(mul_support f).nonempty ↔ f ≠ 1 | by rw [nonempty_iff_ne_empty, ne.def, mul_support_eq_empty_iff] | lemma | function.mul_support_nonempty_iff | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_subset_insert_image_mul_support (f : α → M) :
range f ⊆ insert 1 (f '' mul_support f) | by simpa only [range_subset_iff, mem_insert_iff, or_iff_not_imp_left]
using λ x (hx : x ∈ mul_support f), mem_image_of_mem f hx | lemma | function.range_subset_insert_image_mul_support | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"or_iff_not_imp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_one' : mul_support (1 : α → M) = ∅ | mul_support_eq_empty_iff.2 rfl | lemma | function.mul_support_one' | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_one : mul_support (λ x : α, (1 : M)) = ∅ | mul_support_one' | lemma | function.mul_support_one | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_const {c : M} (hc : c ≠ 1) :
mul_support (λ x : α, c) = set.univ | by { ext x, simp [hc] } | lemma | function.mul_support_const | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_binop_subset (op : M → N → P) (op1 : op 1 1 = 1)
(f : α → M) (g : α → N) :
mul_support (λ x, op (f x) (g x)) ⊆ mul_support f ∪ mul_support g | λ x hx, not_or_of_imp (λ hf hg, hx $ by simp only [hf, hg, op1]) | lemma | function.mul_support_binop_subset | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"not_or_of_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_sup [semilattice_sup M] (f g : α → M) :
mul_support (λ x, f x ⊔ g x) ⊆ mul_support f ∪ mul_support g | mul_support_binop_subset (⊔) sup_idem f g | lemma | function.mul_support_sup | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"semilattice_sup",
"sup_idem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_inf [semilattice_inf M] (f g : α → M) :
mul_support (λ x, f x ⊓ g x) ⊆ mul_support f ∪ mul_support g | mul_support_binop_subset (⊓) inf_idem f g | lemma | function.mul_support_inf | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"inf_idem",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_max [linear_order M] (f g : α → M) :
mul_support (λ x, max (f x) (g x)) ⊆ mul_support f ∪ mul_support g | mul_support_sup f g | lemma | function.mul_support_max | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_min [linear_order M] (f g : α → M) :
mul_support (λ x, min (f x) (g x)) ⊆ mul_support f ∪ mul_support g | mul_support_inf f g | lemma | function.mul_support_min | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_supr [conditionally_complete_lattice M] [nonempty ι]
(f : ι → α → M) :
mul_support (λ x, ⨆ i, f i x) ⊆ ⋃ i, mul_support (f i) | begin
rw mul_support_subset_iff',
simp only [mem_Union, not_exists, nmem_mul_support],
intros x hx,
simp only [hx, csupr_const]
end | lemma | function.mul_support_supr | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"conditionally_complete_lattice",
"csupr_const",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_infi [conditionally_complete_lattice M] [nonempty ι]
(f : ι → α → M) :
mul_support (λ x, ⨅ i, f i x) ⊆ ⋃ i, mul_support (f i) | @mul_support_supr _ Mᵒᵈ ι ⟨(1:M)⟩ _ _ f | lemma | function.mul_support_infi | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"conditionally_complete_lattice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_comp_subset {g : M → N} (hg : g 1 = 1) (f : α → M) :
mul_support (g ∘ f) ⊆ mul_support f | λ x, mt $ λ h, by simp only [(∘), *] | lemma | function.mul_support_comp_subset | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_subset_comp {g : M → N} (hg : ∀ {x}, g x = 1 → x = 1)
(f : α → M) :
mul_support f ⊆ mul_support (g ∘ f) | λ x, mt hg | lemma | function.mul_support_subset_comp | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_comp_eq (g : M → N) (hg : ∀ {x}, g x = 1 ↔ x = 1)
(f : α → M) :
mul_support (g ∘ f) = mul_support f | set.ext $ λ x, not_congr hg | lemma | function.mul_support_comp_eq | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_comp_eq_preimage (g : β → M) (f : α → β) :
mul_support (g ∘ f) = f ⁻¹' mul_support g | rfl | lemma | function.mul_support_comp_eq_preimage | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_prod_mk (f : α → M) (g : α → N) :
mul_support (λ x, (f x, g x)) = mul_support f ∪ mul_support g | set.ext $ λ x, by simp only [mul_support, not_and_distrib, mem_union, mem_set_of_eq,
prod.mk_eq_one, ne.def] | lemma | function.mul_support_prod_mk | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"not_and_distrib",
"prod.mk_eq_one",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_prod_mk' (f : α → M × N) :
mul_support f = mul_support (λ x, (f x).1) ∪ mul_support (λ x, (f x).2) | by simp only [← mul_support_prod_mk, prod.mk.eta] | lemma | function.mul_support_prod_mk' | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_along_fiber_subset (f : α × β → M) (a : α) :
mul_support (λ b, f (a, b)) ⊆ (mul_support f).image prod.snd | by tidy | lemma | function.mul_support_along_fiber_subset | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_along_fiber_finite_of_finite
(f : α × β → M) (a : α) (h : (mul_support f).finite) :
(mul_support (λ b, f (a, b))).finite | (h.image prod.snd).subset (mul_support_along_fiber_subset f a) | lemma | function.mul_support_along_fiber_finite_of_finite | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul [mul_one_class M] (f g : α → M) :
mul_support (λ x, f x * g x) ⊆ mul_support f ∪ mul_support g | mul_support_binop_subset (*) (one_mul _) f g | lemma | function.mul_support_mul | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"mul_one_class",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_pow [monoid M] (f : α → M) (n : ℕ) :
mul_support (λ x, f x ^ n) ⊆ mul_support f | begin
induction n with n hfn,
{ simpa only [pow_zero, mul_support_one] using empty_subset _ },
{ simpa only [pow_succ] using (mul_support_mul f _).trans (union_subset subset.rfl hfn) }
end | lemma | function.mul_support_pow | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"monoid",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_inv : mul_support (λ x, (f x)⁻¹) = mul_support f | ext $ λ _, inv_ne_one | lemma | function.mul_support_inv | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"inv_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_inv' : mul_support f⁻¹ = mul_support f | mul_support_inv f | lemma | function.mul_support_inv' | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul_inv :
mul_support (λ x, f x * (g x)⁻¹) ⊆ mul_support f ∪ mul_support g | mul_support_binop_subset (λ a b, a * b⁻¹) (by simp) f g | lemma | function.mul_support_mul_inv | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_div :
mul_support (λ x, f x / g x) ⊆ mul_support f ∪ mul_support g | mul_support_binop_subset (/) one_div_one f g | lemma | function.mul_support_div | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"one_div_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_one : support (1 : α → R) = univ | support_const one_ne_zero | lemma | function.support_one | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_zero : mul_support (0 : α → R) = univ | mul_support_const zero_ne_one | lemma | function.mul_support_zero | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_nat_cast (hn : n ≠ 0) : support (n : α → R) = univ | support_const $ nat.cast_ne_zero.2 hn | lemma | function.support_nat_cast | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_nat_cast (hn : n ≠ 1) : mul_support (n : α → R) = univ | mul_support_const $ nat.cast_ne_one.2 hn | lemma | function.mul_support_nat_cast | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_int_cast (hn : n ≠ 0) : support (n : α → R) = univ | support_const $ int.cast_ne_zero.2 hn | lemma | function.support_int_cast | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_int_cast (hn : n ≠ 1) : mul_support (n : α → R) = univ | mul_support_const $ int.cast_ne_one.2 hn | lemma | function.mul_support_int_cast | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_smul [has_zero R] [has_zero M] [smul_with_zero R M] [no_zero_smul_divisors R M]
(f : α → R) (g : α → M) :
support (f • g) = support f ∩ support g | ext $ λ x, smul_ne_zero_iff | lemma | function.support_smul | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"no_zero_smul_divisors",
"smul_ne_zero_iff",
"smul_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_mul [mul_zero_class R] [no_zero_divisors R] (f g : α → R) :
support (λ x, f x * g x) = support f ∩ support g | support_smul f g | lemma | function.support_mul | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"mul_zero_class",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_mul_subset_left [mul_zero_class R] (f g : α → R) :
support (λ x, f x * g x) ⊆ support f | λ x hfg hf, hfg $ by simp only [hf, zero_mul] | lemma | function.support_mul_subset_left | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"mul_zero_class",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_mul_subset_right [mul_zero_class R] (f g : α → R) :
support (λ x, f x * g x) ⊆ support g | λ x hfg hg, hfg $ by simp only [hg, mul_zero] | lemma | function.support_mul_subset_right | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"mul_zero",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_smul_subset_right [add_monoid A] [monoid B] [distrib_mul_action B A]
(b : B) (f : α → A) :
support (b • f) ⊆ support f | λ x hbf hf, hbf $ by rw [pi.smul_apply, hf, smul_zero] | lemma | function.support_smul_subset_right | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_monoid",
"distrib_mul_action",
"monoid",
"pi.smul_apply",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_smul_subset_left [has_zero M] [has_zero β] [smul_with_zero M β]
(f : α → M) (g : α → β) :
support (f • g) ⊆ support f | λ x hfg hf, hfg $ by rw [pi.smul_apply', hf, zero_smul] | lemma | function.support_smul_subset_left | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"pi.smul_apply'",
"smul_with_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_const_smul_of_ne_zero [semiring R] [add_comm_monoid M] [module R M]
[no_zero_smul_divisors R M] (c : R) (g : α → M) (hc : c ≠ 0) :
support (c • g) = support g | ext $ λ x, by simp only [hc, mem_support, pi.smul_apply, ne.def, smul_eq_zero, false_or] | lemma | function.support_const_smul_of_ne_zero | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_comm_monoid",
"module",
"no_zero_smul_divisors",
"pi.smul_apply",
"semiring",
"smul_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_inv [group_with_zero G₀] (f : α → G₀) :
support (λ x, (f x)⁻¹) = support f | set.ext $ λ x, not_congr inv_eq_zero | lemma | function.support_inv | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"group_with_zero",
"inv_eq_zero",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_div [group_with_zero G₀] (f g : α → G₀) :
support (λ x, f x / g x) = support f ∩ support g | by simp [div_eq_mul_inv] | lemma | function.support_div | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"div_eq_mul_inv",
"group_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_prod [comm_monoid M] (s : finset α) (f : α → β → M) :
mul_support (λ x, ∏ i in s, f i x) ⊆ ⋃ i ∈ s, mul_support (f i) | begin
rw mul_support_subset_iff',
simp only [mem_Union, not_exists, nmem_mul_support],
exact λ x, finset.prod_eq_one
end | lemma | function.mul_support_prod | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"comm_monoid",
"finset",
"finset.prod_eq_one",
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_prod_subset [comm_monoid_with_zero A] (s : finset α) (f : α → β → A) :
support (λ x, ∏ i in s, f i x) ⊆ ⋂ i ∈ s, support (f i) | λ x hx, mem_Inter₂.2 $ λ i hi H, hx $ finset.prod_eq_zero hi H | lemma | function.support_prod_subset | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"comm_monoid_with_zero",
"finset",
"finset.prod_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_prod [comm_monoid_with_zero A] [no_zero_divisors A] [nontrivial A]
(s : finset α) (f : α → β → A) :
support (λ x, ∏ i in s, f i x) = ⋂ i ∈ s, support (f i) | set.ext $ λ x, by
simp only [support, ne.def, finset.prod_eq_zero_iff, mem_set_of_eq, set.mem_Inter, not_exists] | lemma | function.support_prod | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"comm_monoid_with_zero",
"finset",
"finset.prod_eq_zero_iff",
"no_zero_divisors",
"nontrivial",
"not_exists",
"set.ext",
"set.mem_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_one_add [has_one R] [add_left_cancel_monoid R] (f : α → R) :
mul_support (λ x, 1 + f x) = support f | set.ext $ λ x, not_congr add_right_eq_self | lemma | function.mul_support_one_add | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_left_cancel_monoid",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_one_add' [has_one R] [add_left_cancel_monoid R] (f : α → R) :
mul_support (1 + f) = support f | mul_support_one_add f | lemma | function.mul_support_one_add' | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_left_cancel_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_add_one [has_one R] [add_right_cancel_monoid R] (f : α → R) :
mul_support (λ x, f x + 1) = support f | set.ext $ λ x, not_congr add_left_eq_self | lemma | function.mul_support_add_one | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_right_cancel_monoid",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_add_one' [has_one R] [add_right_cancel_monoid R] (f : α → R) :
mul_support (f + 1) = support f | mul_support_add_one f | lemma | function.mul_support_add_one' | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_right_cancel_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_one_sub' [has_one R] [add_group R] (f : α → R) :
mul_support (1 - f) = support f | by rw [sub_eq_add_neg, mul_support_one_add', support_neg'] | lemma | function.mul_support_one_sub' | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_one_sub [has_one R] [add_group R] (f : α → R) :
mul_support (λ x, 1 - f x) = support f | mul_support_one_sub' f | lemma | function.mul_support_one_sub | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_inter_mul_support_eq {s : set β} {g : β → α} :
(g '' s ∩ mul_support f) = g '' (s ∩ mul_support (f ∘ g)) | by rw [mul_support_comp_eq_preimage f g, image_inter_preimage] | lemma | set.image_inter_mul_support_eq | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul_single_subset : mul_support (mul_single a b) ⊆ {a} | λ x hx, by_contra $ λ hx', hx $ mul_single_eq_of_ne hx' _ | lemma | pi.mul_support_mul_single_subset | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [
"by_contra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul_single_one : mul_support (mul_single a (1 : B)) = ∅ | by simp | lemma | pi.mul_support_mul_single_one | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_support_mul_single_of_ne (h : b ≠ 1) :
mul_support (mul_single a b) = {a} | mul_support_mul_single_subset.antisymm $
λ x (hx : x = a), by rwa [mem_mul_support, hx, mul_single_eq_same] | lemma | pi.mul_support_mul_single_of_ne | algebra | src/algebra/support.lean | [
"order.conditionally_complete_lattice.basic",
"data.set.finite",
"algebra.big_operators.basic",
"algebra.group.prod",
"algebra.group.pi",
"algebra.module.basic",
"group_theory.group_action.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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