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
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|---|---|---|---|---|---|---|---|---|---|---|
is_associated_prime.map_of_injective
(h : is_associated_prime I M) (hf : function.injective f) :
is_associated_prime I M' | begin
obtain ⟨x, rfl⟩ := h.2,
refine ⟨h.1, ⟨f x, _⟩⟩,
ext r,
rw [submodule.mem_annihilator_span_singleton, submodule.mem_annihilator_span_singleton,
← map_smul, ← f.map_zero, hf.eq_iff],
end | lemma | is_associated_prime.map_of_injective | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"is_associated_prime",
"submodule.mem_annihilator_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv.is_associated_prime_iff (l : M ≃ₗ[R] M') :
is_associated_prime I M ↔ is_associated_prime I M' | ⟨λ h, h.map_of_injective l l.injective, λ h, h.map_of_injective l.symm l.symm.injective⟩ | lemma | linear_equiv.is_associated_prime_iff | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"is_associated_prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_is_associated_prime_of_subsingleton [subsingleton M] : ¬ is_associated_prime I M | begin
rintro ⟨hI, x, hx⟩,
apply hI.ne_top,
rwa [subsingleton.elim x 0, submodule.span_singleton_eq_bot.mpr rfl,
submodule.annihilator_bot] at hx
end | lemma | not_is_associated_prime_of_subsingleton | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"is_associated_prime",
"submodule.annihilator_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_le_is_associated_prime_of_is_noetherian_ring [H : is_noetherian_ring R]
(x : M) (hx : x ≠ 0) :
∃ P : ideal R, is_associated_prime P M ∧ (R ∙ x).annihilator ≤ P | begin
have : (R ∙ x).annihilator ≠ ⊤,
{ rwa [ne.def, ideal.eq_top_iff_one, submodule.mem_annihilator_span_singleton, one_smul] },
obtain ⟨P, ⟨l, h₁, y, rfl⟩, h₃⟩ := set_has_maximal_iff_noetherian.mpr H
({ P | (R ∙ x).annihilator ≤ P ∧ P ≠ ⊤ ∧ ∃ y : M, P = (R ∙ y).annihilator })
⟨(R ∙ x).annihilator, rfl.l... | lemma | exists_le_is_associated_prime_of_is_noetherian_ring | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"ideal",
"ideal.eq_top_iff_one",
"is_associated_prime",
"is_noetherian_ring",
"one_smul",
"or_iff_not_imp_left",
"smul_smul",
"smul_zero",
"submodule.annihilator_eq_top_iff",
"submodule.mem_annihilator_span_singleton",
"submodule.span",
"submodule.span_singleton_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated_primes.subset_of_injective (hf : function.injective f) :
associated_primes R M ⊆ associated_primes R M' | λ I h, h.map_of_injective f hf | lemma | associated_primes.subset_of_injective | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"associated_primes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linear_equiv.associated_primes.eq (l : M ≃ₗ[R] M') :
associated_primes R M = associated_primes R M' | le_antisymm (associated_primes.subset_of_injective l l.injective)
(associated_primes.subset_of_injective l.symm l.symm.injective) | lemma | linear_equiv.associated_primes.eq | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"associated_primes",
"associated_primes.subset_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated_primes.eq_empty_of_subsingleton [subsingleton M] : associated_primes R M = ∅ | begin
ext, simp only [set.mem_empty_iff_false, iff_false], apply not_is_associated_prime_of_subsingleton
end | lemma | associated_primes.eq_empty_of_subsingleton | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"associated_primes",
"not_is_associated_prime_of_subsingleton",
"set.mem_empty_iff_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated_primes.nonempty [is_noetherian_ring R] [nontrivial M] :
(associated_primes R M).nonempty | begin
obtain ⟨x, hx⟩ := exists_ne (0 : M),
obtain ⟨P, hP, _⟩ := exists_le_is_associated_prime_of_is_noetherian_ring R x hx,
exact ⟨P, hP⟩,
end | lemma | associated_primes.nonempty | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"associated_primes",
"exists_le_is_associated_prime_of_is_noetherian_ring",
"exists_ne",
"is_noetherian_ring",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_associated_prime.annihilator_le (h : is_associated_prime I M) :
(⊤ : submodule R M).annihilator ≤ I | begin
obtain ⟨hI, x, rfl⟩ := h,
exact submodule.annihilator_mono le_top,
end | lemma | is_associated_prime.annihilator_le | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"is_associated_prime",
"le_top",
"submodule",
"submodule.annihilator_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_associated_prime.eq_radical (hI : I.is_primary) (h : is_associated_prime J (R ⧸ I)) :
J = I.radical | begin
obtain ⟨hJ, x, e⟩ := h,
have : x ≠ 0,
{ rintro rfl, apply hJ.1,
rwa [submodule.span_singleton_eq_bot.mpr rfl, submodule.annihilator_bot] at e },
obtain ⟨x, rfl⟩ := ideal.quotient.mkₐ_surjective R _ x,
replace e : ∀ {y}, y ∈ J ↔ x * y ∈ I,
{ intro y, rw [e, submodule.mem_annihilator_span_singleton,... | lemma | is_associated_prime.eq_radical | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"ideal.quotient.mk_eq_mk",
"ideal.quotient.mkₐ_eq_mk",
"ideal.quotient.mkₐ_surjective",
"is_associated_prime",
"mul_comm",
"smul_eq_mul",
"submodule.annihilator_bot",
"submodule.mem_annihilator_span_singleton",
"submodule.quotient.mk_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
associated_primes.eq_singleton_of_is_primary [is_noetherian_ring R] (hI : I.is_primary) :
associated_primes R (R ⧸ I) = {I.radical} | begin
ext J,
rw [set.mem_singleton_iff],
refine ⟨is_associated_prime.eq_radical hI, _⟩,
rintro rfl,
haveI : nontrivial (R ⧸ I) := ⟨⟨(I^.quotient.mk : _) 1, (I^.quotient.mk : _) 0, _⟩⟩,
obtain ⟨a, ha⟩ := associated_primes.nonempty R (R ⧸ I),
exact ha.eq_radical hI ▸ ha,
rw [ne.def, ideal.quotient.eq, sub... | lemma | associated_primes.eq_singleton_of_is_primary | ring_theory.ideal | src/ring_theory/ideal/associated_prime.lean | [
"linear_algebra.span",
"ring_theory.ideal.operations",
"ring_theory.ideal.quotient_operations",
"ring_theory.noetherian"
] | [
"associated_primes",
"associated_primes.nonempty",
"ideal.eq_top_iff_one",
"ideal.quotient.eq",
"is_noetherian_ring",
"nontrivial",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ideal (R : Type u) [semiring R] | submodule R R | def | ideal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"semiring",
"submodule"
] | A (left) ideal in a semiring `R` is an additive submonoid `s` such that
`a * b ∈ s` whenever `b ∈ s`. If `R` is a ring, then `s` is an additive subgroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_mem : (0 : α) ∈ I | I.zero_mem | lemma | ideal.zero_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mem : a ∈ I → b ∈ I → a + b ∈ I | I.add_mem | lemma | ideal.add_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_left : b ∈ I → a * b ∈ I | I.smul_mem a | lemma | ideal.mul_mem_left | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {I J : ideal α} (h : ∀ x, x ∈ I ↔ x ∈ J) : I = J | submodule.ext h | lemma | ideal.ext | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"submodule.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_mem (I : ideal α) {ι : Type*} {t : finset ι} {f : ι → α} :
(∀c∈t, f c ∈ I) → (∑ i in t, f i) ∈ I | submodule.sum_mem I | lemma | ideal.sum_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"finset",
"ideal",
"submodule.sum_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_of_unit_mem
(x y : α) (hx : x ∈ I) (h : y * x = 1) : I = ⊤ | eq_top_iff.2 $ λ z _, calc
z = z * (y * x) : by simp [h]
... = (z * y) * x : eq.symm $ mul_assoc z y x
... ∈ I : I.mul_mem_left _ hx | theorem | ideal.eq_top_of_unit_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_of_is_unit_mem {x} (hx : x ∈ I) (h : is_unit x) : I = ⊤ | let ⟨y, hy⟩ := h.exists_left_inv in eq_top_of_unit_mem I x y hx hy | theorem | ideal.eq_top_of_is_unit_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_top_iff_one : I = ⊤ ↔ (1:α) ∈ I | ⟨by rintro rfl; trivial,
λ h, eq_top_of_unit_mem _ _ 1 h (by simp)⟩ | theorem | ideal.eq_top_iff_one | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_top_iff_one : I ≠ ⊤ ↔ (1:α) ∉ I | not_congr I.eq_top_iff_one | theorem | ideal.ne_top_iff_one | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unit_mul_mem_iff_mem {x y : α} (hy : is_unit y) : y * x ∈ I ↔ x ∈ I | begin
refine ⟨λ h, _, λ h, I.mul_mem_left y h⟩,
obtain ⟨y', hy'⟩ := hy.exists_left_inv,
have := I.mul_mem_left y' h,
rwa [← mul_assoc, hy', one_mul] at this,
end | theorem | ideal.unit_mul_mem_iff_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit",
"mul_assoc",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span (s : set α) : ideal α | submodule.span α s | def | ideal.span | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"submodule.span"
] | The ideal generated by a subset of a ring | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule_span_eq {s : set α} :
submodule.span α s = ideal.span s | rfl | lemma | ideal.submodule_span_eq | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal.span",
"submodule.span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_empty : span (∅ : set α) = ⊥ | submodule.span_empty | lemma | ideal.span_empty | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_univ : span (set.univ : set α) = ⊤ | submodule.span_univ | lemma | ideal.span_univ | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_union (s t : set α) : span (s ∪ t) = span s ⊔ span t | submodule.span_union _ _ | lemma | ideal.span_union | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_Union {ι} (s : ι → set α) : span (⋃ i, s i) = ⨆ i, span (s i) | submodule.span_Union _ | lemma | ideal.span_Union | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span {s : set α} (x) : x ∈ span s ↔ ∀ p : ideal α, s ⊆ p → x ∈ p | mem_Inter₂ | lemma | ideal.mem_span | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_span {s : set α} : s ⊆ span s | submodule.subset_span | lemma | ideal.subset_span | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.subset_span"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_le {s : set α} {I} : span s ≤ I ↔ s ⊆ I | submodule.span_le | lemma | ideal.span_le | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_mono {s t : set α} : s ⊆ t → span s ≤ span t | submodule.span_mono | lemma | ideal.span_mono | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_eq : span (I : set α) = I | submodule.span_eq _ | lemma | ideal.span_eq | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_one : span ({1} : set α) = ⊤ | (eq_top_iff_one _).2 $ subset_span $ mem_singleton _ | lemma | ideal.span_singleton_one | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_insert {s : set α} {x y} :
x ∈ span (insert y s) ↔ ∃ a (z ∈ span s), x = a * y + z | submodule.mem_span_insert | lemma | ideal.mem_span_insert | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.mem_span_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton' {x y : α} :
x ∈ span ({y} : set α) ↔ ∃ a, a * y = x | submodule.mem_span_singleton | lemma | ideal.mem_span_singleton' | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.mem_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_le_iff_mem {x : α} : span {x} ≤ I ↔ x ∈ I | submodule.span_singleton_le_iff_mem _ _ | lemma | ideal.span_singleton_le_iff_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_singleton_le_iff_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_left_unit {a : α} (h2 : is_unit a) (x : α) :
span ({a * x} : set α) = span {x} | begin
apply le_antisymm; rw [span_singleton_le_iff_mem, mem_span_singleton'],
exacts [⟨a, rfl⟩, ⟨_, h2.unit.inv_mul_cancel_left x⟩],
end | lemma | ideal.span_singleton_mul_left_unit | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_insert (x) (s : set α) : span (insert x s) = span ({x} : set α) ⊔ span s | submodule.span_insert x s | lemma | ideal.span_insert | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_eq_bot {s : set α} : span s = ⊥ ↔ ∀ x ∈ s, (x:α) = 0 | submodule.span_eq_bot | lemma | ideal.span_eq_bot | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_eq_bot {x} : span ({x} : set α) = ⊥ ↔ x = 0 | submodule.span_singleton_eq_bot | lemma | ideal.span_singleton_eq_bot | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.span_singleton_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_ne_top {α : Type*} [comm_semiring α] {x : α} (hx : ¬ is_unit x) :
ideal.span ({x} : set α) ≠ ⊤ | (ideal.ne_top_iff_one _).mpr $ λ h1, let ⟨y, hy⟩ := ideal.mem_span_singleton'.mp h1 in
hx ⟨⟨x, y, mul_comm y x ▸ hy, hy⟩, rfl⟩ | lemma | ideal.span_singleton_ne_top | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_semiring",
"ideal.ne_top_iff_one",
"ideal.span",
"is_unit",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_zero : span (0 : set α) = ⊥ | by rw [←set.singleton_zero, span_singleton_eq_bot] | lemma | ideal.span_zero | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_one : span (1 : set α) = ⊤ | by rw [←set.singleton_one, span_singleton_one] | lemma | ideal.span_one | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_eq_top_iff_finite (s : set α) :
span s = ⊤ ↔ ∃ s' : finset α, ↑s' ⊆ s ∧ span (s' : set α) = ⊤ | begin
simp_rw eq_top_iff_one,
exact ⟨submodule.mem_span_finite_of_mem_span, λ ⟨s', h₁, h₂⟩, span_mono h₁ h₂⟩
end | lemma | ideal.span_eq_top_iff_finite | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton_sup {S : Type*} [comm_semiring S] {x y : S} {I : ideal S} :
x ∈ ideal.span {y} ⊔ I ↔ ∃ (a : S) (b ∈ I), a * y + b = x | begin
rw submodule.mem_sup,
split,
{ rintro ⟨ya, hya, b, hb, rfl⟩,
obtain ⟨a, rfl⟩ := mem_span_singleton'.mp hya,
exact ⟨a, b, hb, rfl⟩ },
{ rintro ⟨a, b, hb, rfl⟩,
exact ⟨a * y, ideal.mem_span_singleton'.mpr ⟨a, rfl⟩, b, hb, rfl⟩ }
end | lemma | ideal.mem_span_singleton_sup | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_semiring",
"ideal",
"ideal.span",
"submodule.mem_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_rel (r : α → α → Prop) : ideal α | submodule.span α { x | ∃ (a b) (h : r a b), x + b = a } | def | ideal.of_rel | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"submodule.span"
] | The ideal generated by an arbitrary binary relation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime (I : ideal α) : Prop | (ne_top' : I ≠ ⊤)
(mem_or_mem' : ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I) | class | ideal.is_prime | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | An ideal `P` of a ring `R` is prime if `P ≠ R` and `xy ∈ P → x ∈ P ∨ y ∈ P` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime_iff {I : ideal α} :
is_prime I ↔ I ≠ ⊤ ∧ ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I | ⟨λ h, ⟨h.1, λ _ _, h.2⟩, λ h, ⟨h.1, λ _ _, h.2⟩⟩ | theorem | ideal.is_prime_iff | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.ne_top {I : ideal α} (hI : I.is_prime) : I ≠ ⊤ | hI.1 | theorem | ideal.is_prime.ne_top | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.mem_or_mem {I : ideal α} (hI : I.is_prime) {x y : α} :
x * y ∈ I → x ∈ I ∨ y ∈ I | hI.2 | theorem | ideal.is_prime.mem_or_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.mem_or_mem_of_mul_eq_zero {I : ideal α} (hI : I.is_prime)
{x y : α} (h : x * y = 0) : x ∈ I ∨ y ∈ I | hI.mem_or_mem (h.symm ▸ I.zero_mem) | theorem | ideal.is_prime.mem_or_mem_of_mul_eq_zero | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.mem_of_pow_mem {I : ideal α} (hI : I.is_prime)
{r : α} (n : ℕ) (H : r^n ∈ I) : r ∈ I | begin
induction n with n ih,
{ rw pow_zero at H, exact (mt (eq_top_iff_one _).2 hI.1).elim H },
{ rw pow_succ at H, exact or.cases_on (hI.mem_or_mem H) id ih }
end | theorem | ideal.is_prime.mem_of_pow_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"ih",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_is_prime_iff {I : ideal α} : ¬ I.is_prime ↔ I = ⊤ ∨ ∃ (x ∉ I) (y ∉ I), x * y ∈ I | begin
simp_rw [ideal.is_prime_iff, not_and_distrib, ne.def, not_not, not_forall, not_or_distrib],
exact or_congr iff.rfl
⟨λ ⟨x, y, hxy, hx, hy⟩, ⟨x, hx, y, hy, hxy⟩, λ ⟨x, hx, y, hy, hxy⟩, ⟨x, y, hxy, hx, hy⟩⟩
end | lemma | ideal.not_is_prime_iff | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"ideal.is_prime_iff",
"not_and_distrib",
"not_forall",
"not_not",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_ne_one_of_proper {I : ideal α} (h : I ≠ ⊤) : (0:α) ≠ 1 | λ hz, I.ne_top_iff_one.1 h $ hz ▸ I.zero_mem | theorem | ideal.zero_ne_one_of_proper | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_prime {R : Type*} [ring R] [is_domain R] : (⊥ : ideal R).is_prime | ⟨λ h, one_ne_zero (by rwa [ideal.eq_top_iff_one, submodule.mem_bot] at h),
λ x y h, mul_eq_zero.mp (by simpa only [submodule.mem_bot] using h)⟩ | lemma | ideal.bot_prime | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"ideal.eq_top_iff_one",
"is_domain",
"one_ne_zero",
"ring",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal (I : ideal α) : Prop | (out : is_coatom I) | class | ideal.is_maximal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"is_coatom"
] | An ideal is maximal if it is maximal in the collection of proper ideals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_maximal_def {I : ideal α} : I.is_maximal ↔ is_coatom I | ⟨λ h, h.1, λ h, ⟨h⟩⟩ | theorem | ideal.is_maximal_def | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"is_coatom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal.ne_top {I : ideal α} (h : I.is_maximal) : I ≠ ⊤ | (is_maximal_def.1 h).1 | theorem | ideal.is_maximal.ne_top | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal_iff {I : ideal α} : I.is_maximal ↔
(1:α) ∉ I ∧ ∀ (J : ideal α) x, I ≤ J → x ∉ I → x ∈ J → (1:α) ∈ J | is_maximal_def.trans $ and_congr I.ne_top_iff_one $ forall_congr $ λ J,
by rw [lt_iff_le_not_le]; exact
⟨λ H x h hx₁ hx₂, J.eq_top_iff_one.1 $
H ⟨h, not_subset.2 ⟨_, hx₂, hx₁⟩⟩,
λ H ⟨h₁, h₂⟩, let ⟨x, xJ, xI⟩ := not_subset.1 h₂ in
J.eq_top_iff_one.2 $ H x h₁ xI xJ⟩ | theorem | ideal.is_maximal_iff | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal.eq_of_le {I J : ideal α}
(hI : I.is_maximal) (hJ : J ≠ ⊤) (IJ : I ≤ J) : I = J | eq_iff_le_not_lt.2 ⟨IJ, λ h, hJ (hI.1.2 _ h)⟩ | theorem | ideal.is_maximal.eq_of_le | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal.coprime_of_ne {M M' : ideal α} (hM : M.is_maximal) (hM' : M'.is_maximal)
(hne : M ≠ M') : M ⊔ M' = ⊤ | begin
contrapose! hne with h,
exact hM.eq_of_le hM'.ne_top (le_sup_left.trans_eq (hM'.eq_of_le h le_sup_right).symm)
end | lemma | ideal.is_maximal.coprime_of_ne | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_le_maximal (I : ideal α) (hI : I ≠ ⊤) :
∃ M : ideal α, M.is_maximal ∧ I ≤ M | let ⟨m, hm⟩ := (eq_top_or_exists_le_coatom I).resolve_left hI in ⟨m, ⟨⟨hm.1⟩, hm.2⟩⟩ | theorem | ideal.exists_le_maximal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | **Krull's theorem**: if `I` is an ideal that is not the whole ring, then it is included in some
maximal ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_maximal [nontrivial α] : ∃ M : ideal α, M.is_maximal | let ⟨I, ⟨hI, _⟩⟩ := exists_le_maximal (⊥ : ideal α) bot_ne_top in ⟨I, hI⟩ | theorem | ideal.exists_maximal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"bot_ne_top",
"ideal",
"nontrivial"
] | Krull's theorem: a nontrivial ring has a maximal ideal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maximal_of_no_maximal {R : Type u} [semiring R] {P : ideal R}
(hmax : ∀ m : ideal R, P < m → ¬is_maximal m) (J : ideal R) (hPJ : P < J) : J = ⊤ | begin
by_contradiction hnonmax,
rcases exists_le_maximal J hnonmax with ⟨M, hM1, hM2⟩,
exact hmax M (lt_of_lt_of_le hPJ hM2) hM1,
end | lemma | ideal.maximal_of_no_maximal | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"by_contradiction",
"ideal",
"semiring"
] | If P is not properly contained in any maximal ideal then it is not properly contained
in any proper ideal | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_pair_comm {x y : α} : (span {x, y} : ideal α) = span {y, x} | by simp only [span_insert, sup_comm] | lemma | ideal.span_pair_comm | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"sup_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_pair {x y z : α} :
z ∈ span ({x, y} : set α) ↔ ∃ a b, a * x + b * y = z | submodule.mem_span_pair | theorem | ideal.mem_span_pair | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"submodule.mem_span_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_pair_add_mul_left {R : Type u} [comm_ring R] {x y : R} (z : R) :
(span {x + y * z, y} : ideal R) = span {x, y} | begin
ext,
rw [mem_span_pair, mem_span_pair],
exact ⟨λ ⟨a, b, h⟩, ⟨a, b + a * z, by { rw [← h], ring1 }⟩,
λ ⟨a, b, h⟩, ⟨a, b - a * z, by { rw [← h], ring1 }⟩⟩
end | lemma | ideal.span_pair_add_mul_left | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_ring",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_pair_add_mul_right {R : Type u} [comm_ring R] {x y : R} (z : R) :
(span {x, y + x * z} : ideal R) = span {x, y} | by rw [span_pair_comm, span_pair_add_mul_left, span_pair_comm] | lemma | ideal.span_pair_add_mul_right | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_ring",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal.exists_inv {I : ideal α}
(hI : I.is_maximal) {x} (hx : x ∉ I) : ∃ y, ∃ i ∈ I, y * x + i = 1 | begin
cases is_maximal_iff.1 hI with H₁ H₂,
rcases mem_span_insert.1 (H₂ (span (insert x I)) x
(set.subset.trans (subset_insert _ _) subset_span)
hx (subset_span (mem_insert _ _))) with ⟨y, z, hz, hy⟩,
refine ⟨y, z, _, hy.symm⟩,
rwa ← span_eq I,
end | theorem | ideal.is_maximal.exists_inv | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"set.subset.trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sup_left {S T : ideal R} : ∀ {x : R}, x ∈ S → x ∈ S ⊔ T | show S ≤ S ⊔ T, from le_sup_left | lemma | ideal.mem_sup_left | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_sup_right {S T : ideal R} : ∀ {x : R}, x ∈ T → x ∈ S ⊔ T | show T ≤ S ⊔ T, from le_sup_right | lemma | ideal.mem_sup_right | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"le_sup_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_supr_of_mem {ι : Sort*} {S : ι → ideal R} (i : ι) :
∀ {x : R}, x ∈ S i → x ∈ supr S | show S i ≤ supr S, from le_supr _ _ | lemma | ideal.mem_supr_of_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"le_supr",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Sup_of_mem {S : set (ideal R)} {s : ideal R}
(hs : s ∈ S) : ∀ {x : R}, x ∈ s → x ∈ Sup S | show s ≤ Sup S, from le_Sup hs | lemma | ideal.mem_Sup_of_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"le_Sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {s : set (ideal R)} {x : R} :
x ∈ Inf s ↔ ∀ ⦃I⦄, I ∈ s → x ∈ I | ⟨λ hx I his, hx I ⟨I, infi_pos his⟩, λ H I ⟨J, hij⟩, hij ▸ λ S ⟨hj, hS⟩, hS ▸ H hj⟩ | theorem | ideal.mem_Inf | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"infi_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {I J : ideal R} {x : R} : x ∈ I ⊓ J ↔ x ∈ I ∧ x ∈ J | iff.rfl | lemma | ideal.mem_inf | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi {ι : Sort*} {I : ι → ideal R} {x : R} : x ∈ infi I ↔ ∀ i, x ∈ I i | submodule.mem_infi _ | lemma | ideal.mem_infi | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"infi",
"submodule.mem_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_bot {x : R} : x ∈ (⊥ : ideal R) ↔ x = 0 | submodule.mem_bot _ | lemma | ideal.mem_bot | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"submodule.mem_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi : ideal (ι → α) | { carrier := { x | ∀ i, x i ∈ I },
zero_mem' := λ i, I.zero_mem,
add_mem' := λ a b ha hb i, I.add_mem (ha i) (hb i),
smul_mem' := λ a b hb i, I.mul_mem_left (a i) (hb i) } | def | ideal.pi | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | `I^n` as an ideal of `R^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_pi (x : ι → α) : x ∈ I.pi ι ↔ ∀ i, x i ∈ I | iff.rfl | lemma | ideal.mem_pi | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_is_prime_of_is_chain {s : set (ideal α)} (hs : s.nonempty) (hs' : is_chain (≤) s)
(H : ∀ p ∈ s, ideal.is_prime p) :
(Inf s).is_prime | ⟨λ e, let ⟨x, hx⟩ := hs in (H x hx).ne_top (eq_top_iff.mpr (e.symm.trans_le (Inf_le hx))),
λ x y e, or_iff_not_imp_left.mpr $ λ hx, begin
rw ideal.mem_Inf at hx ⊢ e,
push_neg at hx,
obtain ⟨I, hI, hI'⟩ := hx,
intros J hJ,
cases hs'.total hI hJ,
{ exact h (((H I hI).mem_or_mem (e hI)).resolve_l... | lemma | ideal.Inf_is_prime_of_is_chain | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"Inf_le",
"ideal",
"ideal.is_prime",
"ideal.mem_Inf",
"is_chain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_unit_mem_iff_mem {x y : α} (hy : is_unit y) : x * y ∈ I ↔ x ∈ I | mul_comm y x ▸ unit_mul_mem_iff_mem I hy | theorem | ideal.mul_unit_mem_iff_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton {x y : α} : x ∈ span ({y} : set α) ↔ y ∣ x | mem_span_singleton'.trans $ exists_congr $ λ _, by rw [eq_comm, mul_comm] | lemma | ideal.mem_span_singleton | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton_self (x : α) : x ∈ span ({x} : set α) | mem_span_singleton.mpr dvd_rfl | lemma | ideal.mem_span_singleton_self | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"dvd_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_le_span_singleton {x y : α} :
span ({x} : set α) ≤ span ({y} : set α) ↔ y ∣ x | span_le.trans $ singleton_subset_iff.trans mem_span_singleton | lemma | ideal.span_singleton_le_span_singleton | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_eq_span_singleton {α : Type u} [comm_ring α] [is_domain α] {x y : α} :
span ({x} : set α) = span ({y} : set α) ↔ associated x y | begin
rw [←dvd_dvd_iff_associated, le_antisymm_iff, and_comm],
apply and_congr;
rw span_singleton_le_span_singleton,
end | lemma | ideal.span_singleton_eq_span_singleton | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"associated",
"comm_ring",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_right_unit {a : α} (h2 : is_unit a) (x : α) :
span ({x * a} : set α) = span {x} | by rw [mul_comm, span_singleton_mul_left_unit h2] | lemma | ideal.span_singleton_mul_right_unit | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"is_unit",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_eq_top {x} : span ({x} : set α) = ⊤ ↔ is_unit x | by rw [is_unit_iff_dvd_one, ← span_singleton_le_span_singleton, span_singleton_one,
eq_top_iff] | lemma | ideal.span_singleton_eq_top | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"eq_top_iff",
"is_unit",
"is_unit_iff_dvd_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_prime {p : α} (hp : p ≠ 0) :
is_prime (span ({p} : set α)) ↔ prime p | by simp [is_prime_iff, prime, span_singleton_eq_top, hp, mem_span_singleton] | theorem | ideal.span_singleton_prime | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal.is_prime {I : ideal α} (H : I.is_maximal) : I.is_prime | ⟨H.1.1, λ x y hxy, or_iff_not_imp_left.2 $ λ hx, begin
let J : ideal α := submodule.span α (insert x ↑I),
have IJ : I ≤ J := (set.subset.trans (subset_insert _ _) subset_span),
have xJ : x ∈ J := ideal.subset_span (set.mem_insert x I),
cases is_maximal_iff.1 H with _ oJ,
specialize oJ J x IJ hx xJ,
rcases ... | theorem | ideal.is_maximal.is_prime | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal",
"ideal.subset_span",
"mul_assoc",
"mul_comm",
"mul_one",
"set.mem_insert",
"set.subset.trans",
"smul_eq_mul",
"submodule.add_mem",
"submodule.smul_mem",
"submodule.span",
"submodule.span_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_maximal.is_prime' (I : ideal α) : ∀ [H : I.is_maximal], I.is_prime | is_maximal.is_prime | instance | ideal.is_maximal.is_prime' | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_lt_span_singleton [comm_ring β] [is_domain β] {x y : β} :
span ({x} : set β) < span ({y} : set β) ↔ dvd_not_unit y x | by rw [lt_iff_le_not_le, span_singleton_le_span_singleton, span_singleton_le_span_singleton,
dvd_and_not_dvd_iff] | lemma | ideal.span_singleton_lt_span_singleton | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_ring",
"dvd_and_not_dvd_iff",
"dvd_not_unit",
"is_domain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
factors_decreasing [comm_ring β] [is_domain β]
(b₁ b₂ : β) (h₁ : b₁ ≠ 0) (h₂ : ¬ is_unit b₂) :
span ({b₁ * b₂} : set β) < span {b₁} | lt_of_le_not_le (ideal.span_le.2 $ singleton_subset_iff.2 $
ideal.mem_span_singleton.2 ⟨b₂, rfl⟩) $ λ h,
h₂ $ is_unit_of_dvd_one _ $ (mul_dvd_mul_iff_left h₁).1 $
by rwa [mul_one, ← ideal.span_singleton_le_span_singleton] | lemma | ideal.factors_decreasing | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"comm_ring",
"ideal.span_singleton_le_span_singleton",
"is_domain",
"is_unit",
"is_unit_of_dvd_one",
"mul_dvd_mul_iff_left",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mem_right (h : a ∈ I) : a * b ∈ I | mul_comm b a ▸ I.mul_mem_left b h | lemma | ideal.mul_mem_right | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_mem_of_mem (ha : a ∈ I) (n : ℕ) (hn : 0 < n) : a ^ n ∈ I | nat.cases_on n (not.elim dec_trivial) (λ m hm, (pow_succ a m).symm ▸ I.mul_mem_right (a^m) ha) hn | lemma | ideal.pow_mem_of_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"not.elim",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.mul_mem_iff_mem_or_mem {I : ideal α} (hI : I.is_prime) :
∀ {x y : α}, x * y ∈ I ↔ x ∈ I ∨ y ∈ I | λ x y, ⟨hI.mem_or_mem, by { rintro (h | h), exacts [I.mul_mem_right y h, I.mul_mem_left x h] }⟩ | theorem | ideal.is_prime.mul_mem_iff_mem_or_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.pow_mem_iff_mem {I : ideal α} (hI : I.is_prime)
{r : α} (n : ℕ) (hn : 0 < n) : r ^ n ∈ I ↔ r ∈ I | ⟨hI.mem_of_pow_mem n, (λ hr, I.pow_mem_of_mem hr n hn)⟩ | theorem | ideal.is_prime.pow_mem_iff_mem | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_multiset_sum_mem_span_pow (s : multiset α) (n : ℕ) :
s.sum ^ (s.card * n + 1) ∈ span ((s.map (λ x, x ^ (n + 1))).to_finset : set α) | begin
induction s using multiset.induction_on with a s hs,
{ simp },
simp only [finset.coe_insert, multiset.map_cons, multiset.to_finset_cons, multiset.sum_cons,
multiset.card_cons, add_pow],
refine submodule.sum_mem _ _,
intros c hc,
rw mem_span_insert,
by_cases h : n+1 ≤ c,
{ refine ⟨a ^ (c - (n +... | theorem | ideal.pow_multiset_sum_mem_span_pow | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"add_pow",
"add_tsub_assoc_of_le",
"add_tsub_cancel_of_le",
"finset.coe_insert",
"mul_assoc",
"mul_comm",
"multiset",
"multiset.card_cons",
"multiset.induction_on",
"multiset.map_cons",
"multiset.to_finset_cons",
"one_mul",
"pow_add",
"submodule.sum_mem",
"submodule.zero_mem",
"zero_mu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_pow_mem_span_pow {ι} (s : finset ι) (f : ι → α) (n : ℕ) :
(∑ i in s, f i) ^ (s.card * n + 1) ∈ span ((λ i, f i ^ (n + 1)) '' s) | begin
convert pow_multiset_sum_mem_span_pow (s.1.map f) n,
{ rw multiset.card_map, refl },
rw [multiset.map_map, multiset.to_finset_map, finset.val_to_finset, finset.coe_image]
end | theorem | ideal.sum_pow_mem_span_pow | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"finset",
"finset.coe_image",
"finset.val_to_finset",
"multiset.card_map",
"multiset.map_map",
"multiset.to_finset_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_pow_eq_top (s : set α)
(hs : span s = ⊤) (n : ℕ) : span ((λ x, x ^ n) '' s) = ⊤ | begin
rw eq_top_iff_one,
cases n,
{ obtain rfl | ⟨x, hx⟩ := eq_empty_or_nonempty s,
{ rw [set.image_empty, hs],
trivial },
{ exact subset_span ⟨_, hx, pow_zero _⟩ } },
rw [eq_top_iff_one, span, finsupp.mem_span_iff_total] at hs,
rcases hs with ⟨f, hf⟩,
change f.support.sum (λ a, f a * a) = 1 a... | theorem | ideal.span_pow_eq_top | ring_theory.ideal | src/ring_theory/ideal/basic.lean | [
"algebra.associated",
"linear_algebra.basic",
"order.atoms",
"order.compactly_generated",
"tactic.abel",
"data.nat.choose.sum",
"linear_algebra.finsupp"
] | [
"finset.mem_coe",
"finsupp.mem_span_iff_total",
"mul_comm",
"mul_pow",
"one_pow",
"pow_zero",
"set.image_empty",
"set.mem_image",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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