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 |
|---|---|---|---|---|---|---|---|---|---|---|
radical_is_radical : (radical I).is_radical | λ r ⟨n, k, hrnki⟩, ⟨n * k, (pow_mul r n k).symm ▸ hrnki⟩ | theorem | ideal.radical_is_radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"is_radical",
"pow_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_idem : radical (radical I) = radical I | (radical_is_radical I).radical | theorem | ideal.radical_idem | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical.radical_le_iff (hJ : J.is_radical) : radical I ≤ J ↔ I ≤ J | ⟨le_trans le_radical, λ h, hJ.radical ▸ radical_mono h⟩ | theorem | ideal.is_radical.radical_le_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_le_radical_iff : radical I ≤ radical J ↔ I ≤ radical J | (radical_is_radical J).radical_le_iff | theorem | ideal.radical_le_radical_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_eq_top : radical I = ⊤ ↔ I = ⊤ | ⟨λ h, (eq_top_iff_one _).2 $ let ⟨n, hn⟩ := (eq_top_iff_one _).1 h in
@one_pow R _ n ▸ hn, λ h, h.symm ▸ radical_top R⟩ | theorem | ideal.radical_eq_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"one_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.is_radical (H : is_prime I) : I.is_radical | λ r ⟨n, hrni⟩, H.mem_of_pow_mem n hrni | theorem | ideal.is_prime.is_radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.radical (H : is_prime I) : radical I = I | H.is_radical.radical | theorem | ideal.is_prime.radical | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_sup : radical (I ⊔ J) = radical (radical I ⊔ radical J) | le_antisymm (radical_mono $ sup_le_sup le_radical le_radical) $
radical_le_radical_iff.2 $ sup_le (radical_mono le_sup_left) (radical_mono le_sup_right) | theorem | ideal.radical_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"le_sup_left",
"le_sup_right",
"sup_le",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_inf : radical (I ⊓ J) = radical I ⊓ radical J | le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right))
(λ r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩, ⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm,
(pow_add r m n).symm ▸ J.mul_mem_left _ hrn⟩) | theorem | ideal.radical_inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"inf_le_left",
"inf_le_right",
"le_inf",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_mul : radical (I * J) = radical I ⊓ radical J | le_antisymm (radical_inf I J ▸ radical_mono $ @mul_le_inf _ _ I J)
(λ r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩, ⟨m + n, (pow_add r m n).symm ▸ mul_mem_mul hrm hrn⟩) | theorem | ideal.radical_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.radical_le_iff (hJ : is_prime J) :
radical I ≤ J ↔ I ≤ J | hJ.is_radical.radical_le_iff | theorem | ideal.is_prime.radical_le_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_eq_Inf (I : ideal R) :
radical I = Inf { J : ideal R | I ≤ J ∧ is_prime J } | le_antisymm (le_Inf $ λ J hJ, hJ.2.radical_le_iff.2 hJ.1) $
λ r hr, classical.by_contradiction $ λ hri,
let ⟨m, (hrm : r ∉ radical m), him, hm⟩ := zorn_nonempty_partial_order₀
{K : ideal R | r ∉ radical K}
(λ c hc hcc y hyc, ⟨Sup c, λ ⟨n, hrnc⟩, let ⟨y, hyc, hrny⟩ :=
(submodule.mem_Sup_of_directed ⟨y, hyc⟩ hc... | theorem | ideal.radical_eq_Inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"Inf_le",
"ideal",
"le_Inf",
"le_sup_left",
"le_sup_right",
"mul_assoc",
"mul_left_comm",
"pow_add",
"set.mem_singleton",
"submodule.mem_Sup_of_directed",
"zorn_nonempty_partial_order₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_radical_bot_of_no_zero_divisors {R} [comm_semiring R] [no_zero_divisors R] :
(⊥ : ideal R).is_radical | λ x hx, hx.rec_on (λ n hn, pow_eq_zero hn) | lemma | ideal.is_radical_bot_of_no_zero_divisors | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"comm_semiring",
"ideal",
"is_radical",
"no_zero_divisors",
"pow_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_bot_of_no_zero_divisors {R : Type u} [comm_semiring R] [no_zero_divisors R] :
radical (⊥ : ideal R) = ⊥ | eq_bot_iff.2 is_radical_bot_of_no_zero_divisors | lemma | ideal.radical_bot_of_no_zero_divisors | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"comm_semiring",
"ideal",
"no_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_pow (n : ℕ) : (⊤ ^ n : ideal R) = ⊤ | nat.rec_on n one_eq_top $ λ n ih, by rw [pow_succ, ih, top_mul] | theorem | ideal.top_pow | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ih",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radical_pow (n : ℕ) (H : n > 0) : radical (I^n) = radical I | nat.rec_on n (not.elim dec_trivial) (λ n ih H,
or.cases_on (lt_or_eq_of_le $ nat.le_of_lt_succ H)
(λ H, calc radical (I^(n+1))
= radical I ⊓ radical (I^n) : by { rw pow_succ, exact radical_mul _ _ }
... = radical I ⊓ radical I : by rw ih H
... = radical I : inf_idem)
(λ H, H ▸ (pow_one I).s... | theorem | ideal.radical_pow | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ih",
"inf_idem",
"not.elim",
"pow_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.mul_le {I J P : ideal R} (hp : is_prime P) :
I * J ≤ P ↔ I ≤ P ∨ J ≤ P | ⟨λ h, or_iff_not_imp_left.2 $ λ hip j hj, let ⟨i, hi, hip⟩ := set.not_subset.1 hip in
(hp.mem_or_mem $ h $ mul_mem_mul hi hj).resolve_left hip,
λ h, or.cases_on h (le_trans $ le_trans mul_le_inf inf_le_left)
(le_trans $ le_trans mul_le_inf inf_le_right)⟩ | theorem | ideal.is_prime.mul_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"inf_le_left",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.inf_le {I J P : ideal R} (hp : is_prime P) :
I ⊓ J ≤ P ↔ I ≤ P ∨ J ≤ P | ⟨λ h, hp.mul_le.1 $ le_trans mul_le_inf h,
λ h, or.cases_on h (le_trans inf_le_left) (le_trans inf_le_right)⟩ | theorem | ideal.is_prime.inf_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"inf_le_left",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.multiset_prod_le {s : multiset (ideal R)} {P : ideal R}
(hp : is_prime P) (hne : s ≠ 0) :
s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P | suffices s.prod ≤ P → ∃ I ∈ s, I ≤ P,
from ⟨this, λ ⟨i, his, hip⟩, le_trans multiset_prod_le_inf $
le_trans (multiset.inf_le his) hip⟩,
begin
classical,
obtain ⟨b, hb⟩ : ∃ b, b ∈ s := multiset.exists_mem_of_ne_zero hne,
obtain ⟨t, rfl⟩ : ∃ t, s = b ::ₘ t,
from ⟨s.erase b, (multiset.cons_erase hb).symm⟩,
... | theorem | ideal.is_prime.multiset_prod_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"exists_eq_left",
"exists_prop",
"ideal",
"ih",
"imp_self",
"multiset",
"multiset.cons_erase",
"multiset.cons_swap",
"multiset.cons_zero",
"multiset.exists_mem_of_ne_zero",
"multiset.inf_le",
"multiset.mem_cons_of_mem",
"multiset.mem_cons_self",
"multiset.mem_singleton",
"multiset.prod_c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.multiset_prod_map_le {s : multiset ι} (f : ι → ideal R) {P : ideal R}
(hp : is_prime P) (hne : s ≠ 0) :
(s.map f).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P | begin
rw hp.multiset_prod_le (mt multiset.map_eq_zero.mp hne),
simp_rw [exists_prop, multiset.mem_map, exists_exists_and_eq_and],
end | theorem | ideal.is_prime.multiset_prod_map_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"exists_exists_and_eq_and",
"exists_prop",
"ideal",
"multiset",
"multiset.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.prod_le {s : finset ι} {f : ι → ideal R} {P : ideal R}
(hp : is_prime P) (hne : s.nonempty) :
s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P | hp.multiset_prod_map_le f (mt finset.val_eq_zero.mp hne.ne_empty) | theorem | ideal.is_prime.prod_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime.inf_le' {s : finset ι} {f : ι → ideal R} {P : ideal R} (hp : is_prime P)
(hsne: s.nonempty) :
s.inf f ≤ P ↔ ∃ i ∈ s, f i ≤ P | ⟨λ h, (hp.prod_le hsne).1 $ le_trans prod_le_inf h,
λ ⟨i, his, hip⟩, le_trans (finset.inf_le his) hip⟩ | theorem | ideal.is_prime.inf_le' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"finset",
"finset.inf_le",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_union {R : Type u} [ring R] {I J K : ideal R} :
(I : set R) ⊆ J ∪ K ↔ I ≤ J ∨ I ≤ K | ⟨λ h, or_iff_not_imp_left.2 $ λ hij s hsi,
let ⟨r, hri, hrj⟩ := set.not_subset.1 hij in classical.by_contradiction $ λ hsk,
or.cases_on (h $ I.add_mem hri hsi)
(λ hj, hrj $ add_sub_cancel r s ▸ J.sub_mem hj ((h hsi).resolve_right hsk))
(λ hk, hsk $ add_sub_cancel' r s ▸ K.sub_mem hk ((h hri).resolve_left hr... | theorem | ideal.subset_union | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring",
"set.subset.trans",
"set.subset_union_left",
"set.subset_union_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_union_prime' {R : Type u} [comm_ring R] {s : finset ι} {f : ι → ideal R} {a b : ι}
(hp : ∀ i ∈ s, is_prime (f i)) {I : ideal R} :
(I : set R) ⊆ f a ∪ f b ∪ (⋃ i ∈ (↑s : set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | suffices (I : set R) ⊆ f a ∪ f b ∪ (⋃ i ∈ (↑s : set ι), f i) →
I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i,
from ⟨this, λ h, or.cases_on h (λ h, set.subset.trans h $ set.subset.trans
(set.subset_union_left _ _) (set.subset_union_left _ _)) $
λ h, or.cases_on h (λ h, set.subset.trans h $ set.subset.trans
(s... | theorem | ideal.subset_union_prime' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"comm_ring",
"exists_false",
"exists_prop",
"finset",
"finset.card_eq_zero",
"finset.card_insert_of_not_mem",
"finset.coe_empty",
"finset.coe_insert",
"finset.forall_mem_insert",
"finset.inf_empty",
"finset.inf_eq_infi",
"finset.insert_erase",
"finset.insert_subset_insert",
"finset.mem_ins... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_union_prime {R : Type u} [comm_ring R] {s : finset ι} {f : ι → ideal R} (a b : ι)
(hp : ∀ i ∈ s, i ≠ a → i ≠ b → is_prime (f i)) {I : ideal R} :
(I : set R) ⊆ (⋃ i ∈ (↑s : set ι), f i) ↔ ∃ i ∈ s, I ≤ f i | suffices (I : set R) ⊆ (⋃ i ∈ (↑s : set ι), f i) → ∃ i, i ∈ s ∧ I ≤ f i,
from ⟨λ h, bex_def.2 $ this h, λ ⟨i, his, hi⟩, set.subset.trans hi $ set.subset_bUnion_of_mem $
show i ∈ (↑s : set ι), from his⟩,
assume h : (I : set R) ⊆ (⋃ i ∈ (↑s : set ι), f i),
begin
classical,
by_cases has : a ∈ s,
{ unfreezingI ... | theorem | ideal.subset_union_prime | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"bex_def",
"comm_ring",
"finset",
"finset.coe_empty",
"finset.coe_insert",
"finset.exists_mem_insert",
"finset.insert_erase",
"finset.mem_insert_of_mem",
"finset.not_mem_erase",
"ideal",
"set.bUnion_empty",
"set.bUnion_insert",
"set.nonempty_of_mem",
"set.subset.trans",
"set.subset_bUnio... | Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_of_dvd {I J : ideal R} : I ∣ J → J ≤ I | | ⟨K, h⟩ := h.symm ▸ le_trans mul_le_inf inf_le_left | lemma | ideal.le_of_dvd | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"inf_le_left"
] | If `I` divides `J`, then `I` contains `J`.
In a Dedekind domain, to divide and contain are equivalent, see `ideal.dvd_iff_le`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_unit_iff {I : ideal R} :
is_unit I ↔ I = ⊤ | is_unit_iff_dvd_one.trans ((@one_eq_top R _).symm ▸
⟨λ h, eq_top_iff.mpr (ideal.le_of_dvd h), λ h, ⟨⊤, by rw [mul_top, h]⟩⟩) | lemma | ideal.is_unit_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.le_of_dvd",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_units : unique ((ideal R)ˣ) | { default := 1,
uniq := λ u, units.ext
(show (u : ideal R) = 1, by rw [is_unit_iff.mp u.is_unit, one_eq_top]) } | instance | ideal.unique_units | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"unique",
"units.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (I : ideal R) : ideal S | span (f '' I) | def | ideal.map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than
the image itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap (I : ideal S) : ideal R | { carrier := f ⁻¹' I,
add_mem' := λ x y hx hy, by simp only [set.mem_preimage, set_like.mem_coe,
map_add, add_mem hx hy] at *,
zero_mem' := by simp only [set.mem_preimage, map_zero, set_like.mem_coe, submodule.zero_mem],
smul_mem' := λ c x hx, by { simp only [smul_eq_mul, ... | def | ideal.comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"map_mul",
"set.mem_preimage",
"set_like.mem_coe",
"smul_eq_mul",
"submodule.zero_mem"
] | `I.comap f` is the preimage of `I` under `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mono (h : I ≤ J) : map f I ≤ map f J | span_mono $ set.image_subset _ h | theorem | ideal.map_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set.image_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_of_mem (f : F) {I : ideal R} {x : R} (h : x ∈ I) : f x ∈ map f I | subset_span ⟨x, h, rfl⟩ | theorem | ideal.mem_map_of_mem | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_coe_mem_map (f : F) (I : ideal R) (x : I) : f x ∈ I.map f | mem_map_of_mem f x.prop | lemma | ideal.apply_coe_mem_map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_iff_le_comap :
map f I ≤ K ↔ I ≤ comap f K | span_le.trans set.image_subset_iff | theorem | ideal.map_le_iff_le_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set.image_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {x} : x ∈ comap f K ↔ f x ∈ K | iff.rfl | theorem | ideal.mem_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mono (h : K ≤ L) : comap f K ≤ comap f L | set.preimage_mono (λ x hx, h hx) | theorem | ideal.comap_mono | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"set.preimage_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_ne_top (hK : K ≠ ⊤) : comap f K ≠ ⊤ | (ne_top_iff_one _).2 $ by rw [mem_comap, map_one];
exact (ne_top_iff_one _).1 hK | theorem | ideal.comap_ne_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_comap_of_inv_on (g : G) (I : ideal R) (hf : set.left_inv_on g f I) :
I.map f ≤ I.comap g | begin
refine ideal.span_le.2 _,
rintros x ⟨x, hx, rfl⟩,
rw [set_like.mem_coe, mem_comap, hf hx],
exact hx,
end | lemma | ideal.map_le_comap_of_inv_on | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"set.left_inv_on",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_le_map_of_inv_on (g : G) (I : ideal S) (hf : set.left_inv_on g f (f ⁻¹' I)) :
I.comap f ≤ I.map g | λ x (hx : f x ∈ I), hf hx ▸ ideal.mem_map_of_mem g hx | lemma | ideal.comap_le_map_of_inv_on | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.mem_map_of_mem",
"set.left_inv_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_comap_of_inverse (g : G) (I : ideal R) (h : function.left_inverse g f) :
I.map f ≤ I.comap g | map_le_comap_of_inv_on _ _ _ $ h.left_inv_on _ | lemma | ideal.map_le_comap_of_inverse | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | The `ideal` version of `set.image_subset_preimage_of_inverse`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_le_map_of_inverse (g : G) (I : ideal S) (h : function.left_inverse g f) :
I.comap f ≤ I.map g | comap_le_map_of_inv_on _ _ _ $ h.left_inv_on _ | lemma | ideal.comap_le_map_of_inverse | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | The `ideal` version of `set.preimage_subset_image_of_inverse`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_prime.comap [hK : K.is_prime] : (comap f K).is_prime | ⟨comap_ne_top _ hK.1, λ x y,
by simp only [mem_comap, map_mul]; apply hK.2⟩ | instance | ideal.is_prime.comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_top : map f ⊤ = ⊤ | (eq_top_iff_one _).2 $ subset_span ⟨1, trivial, map_one f⟩ | theorem | ideal.map_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_map_comap : galois_connection (ideal.map f) (ideal.comap f) | λ I J, ideal.map_le_iff_le_comap | lemma | ideal.gc_map_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection",
"ideal.comap",
"ideal.map",
"ideal.map_le_iff_le_comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_id : I.comap (ring_hom.id R) = I | ideal.ext $ λ _, iff.rfl | lemma | ideal.comap_id | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.ext",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : I.map (ring_hom.id R) = I | (gc_map_comap (ring_hom.id R)).l_unique galois_connection.id comap_id | lemma | ideal.map_id | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection.id",
"map_id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap {T : Type*} [semiring T] {I : ideal T} (f : R →+* S)
(g : S →+* T) : (I.comap g).comap f = I.comap (g.comp f) | rfl | lemma | ideal.comap_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map {T : Type*} [semiring T] {I : ideal R} (f : R →+* S)
(g : S →+* T) : (I.map f).map g = I.map (g.comp f) | ((gc_map_comap f).compose (gc_map_comap g)).l_unique
(gc_map_comap (g.comp f)) (λ _, comap_comap _ _) | lemma | ideal.map_map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_span (f : F) (s : set R) :
map f (span s) = span (f '' s) | symm $ submodule.span_eq_of_le _
(λ y ⟨x, hy, x_eq⟩, x_eq ▸ mem_map_of_mem f (subset_span hy))
(map_le_iff_le_comap.2 $ span_le.2 $ set.image_subset_iff.1 subset_span) | lemma | ideal.map_span | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"submodule.span_eq_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_le_of_le_comap : I ≤ K.comap f → I.map f ≤ K | (gc_map_comap f).l_le | lemma | ideal.map_le_of_le_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_of_map_le : I.map f ≤ K → I ≤ K.comap f | (gc_map_comap f).le_u | lemma | ideal.le_comap_of_map_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_map : I ≤ (I.map f).comap f | (gc_map_comap f).le_u_l _ | lemma | ideal.le_comap_map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_le : (K.comap f).map f ≤ K | (gc_map_comap f).l_u_le _ | lemma | ideal.map_comap_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_top : (⊤ : ideal S).comap f = ⊤ | (gc_map_comap f).u_top | lemma | ideal.comap_top | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_eq_top_iff {I : ideal S} : I.comap f = ⊤ ↔ I = ⊤ | ⟨ λ h, I.eq_top_iff_one.mpr (map_one f ▸ mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)),
λ h, by rw [h, comap_top] ⟩ | lemma | ideal.comap_eq_top_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_bot : (⊥ : ideal R).map f = ⊥ | (gc_map_comap f).l_bot | lemma | ideal.map_bot | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comap_map : ((I.map f).comap f).map f = I.map f | (gc_map_comap f).l_u_l_eq_l I | lemma | ideal.map_comap_map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_map_comap : ((K.comap f).map f).comap f = K.comap f | (gc_map_comap f).u_l_u_eq_u K | lemma | ideal.comap_map_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup : (I ⊔ J).map f = I.map f ⊔ J.map f | (gc_map_comap f : galois_connection (map f) (comap f)).l_sup | lemma | ideal.map_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_inf : comap f (K ⊓ L) = comap f K ⊓ comap f L | rfl | theorem | ideal.comap_inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr (K : ι → ideal R) : (supr K).map f = ⨆ i, (K i).map f | (gc_map_comap f : galois_connection (map f) (comap f)).l_supr | lemma | ideal.map_supr | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection",
"ideal",
"map_supr",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_infi (K : ι → ideal S) : (infi K).comap f = ⨅ i, (K i).comap f | (gc_map_comap f : galois_connection (map f) (comap f)).u_infi | lemma | ideal.comap_infi | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection",
"ideal",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_Sup (s : set (ideal R)): (Sup s).map f = ⨆ I ∈ s, (I : ideal R).map f | (gc_map_comap f : galois_connection (map f) (comap f)).l_Sup | lemma | ideal.map_Sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_Inf (s : set (ideal S)): (Inf s).comap f = ⨅ I ∈ s, (I : ideal S).comap f | (gc_map_comap f : galois_connection (map f) (comap f)).u_Inf | lemma | ideal.comap_Inf | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_Inf' (s : set (ideal S)) : (Inf s).comap f = ⨅ I ∈ (comap f '' s), I | trans (comap_Inf f s) (by rw infi_image) | lemma | ideal.comap_Inf' | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"infi_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_is_prime [H : is_prime K] : is_prime (comap f K) | ⟨comap_ne_top f H.ne_top,
λ x y h, H.mem_or_mem $ by rwa [mem_comap, map_mul] at h⟩ | theorem | ideal.comap_is_prime | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inf_le : map f (I ⊓ J) ≤ map f I ⊓ map f J | (gc_map_comap f : galois_connection (map f) (comap f)).monotone_l.map_inf_le _ _ | theorem | ideal.map_inf_le | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_sup : comap f K ⊔ comap f L ≤ comap f (K ⊔ L) | (gc_map_comap f : galois_connection (map f) (comap f)).monotone_u.le_map_sup _ _ | theorem | ideal.le_comap_sup | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_top_eq_map {R S : Type*} [comm_semiring R] [comm_semiring S] [algebra R S]
(I : ideal R) : I • (⊤ : submodule R S) = (I.map (algebra_map R S)).restrict_scalars R | begin
refine le_antisymm (submodule.smul_le.mpr (λ r hr y _, _) )
(λ x hx, submodule.span_induction hx _ _ _ _),
{ rw algebra.smul_def,
exact mul_mem_right _ _ (mem_map_of_mem _ hr) },
{ rintros _ ⟨x, hx, rfl⟩,
rw [← mul_one (algebra_map R S x), ← algebra.smul_def],
exact submodule.smul_mem_sm... | lemma | ideal.smul_top_eq_map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra",
"algebra.smul_def",
"algebra_map",
"comm_semiring",
"ideal",
"mul_one",
"restrict_scalars",
"smul_add",
"submodule",
"submodule.add_mem",
"submodule.mem_top",
"submodule.smul_induction_on",
"submodule.smul_mem_smul",
"submodule.span_induction",
"submodule.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_restrict_scalars {R S : Type*} [comm_semiring R] [semiring S] [algebra R S]
(I : ideal S) : ((I.restrict_scalars R) : set S) = ↑I | rfl | lemma | ideal.coe_restrict_scalars | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra",
"comm_semiring",
"ideal",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_mul {R S : Type*} [comm_semiring R] [comm_semiring S] [algebra R S]
(I J : ideal S) : (I * J).restrict_scalars R = I.restrict_scalars R * J.restrict_scalars R | le_antisymm (λ x hx, submodule.mul_induction_on hx
(λ x hx y hy, submodule.mul_mem_mul hx hy)
(λ x y, submodule.add_mem _))
(submodule.mul_le.mpr (λ x hx y hy, ideal.mul_mem_mul hx hy)) | lemma | ideal.restrict_scalars_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"algebra",
"comm_semiring",
"ideal",
"ideal.mul_mem_mul",
"restrict_scalars",
"submodule.add_mem",
"submodule.mul_induction_on",
"submodule.mul_mem_mul"
] | The smallest `S`-submodule that contains all `x ∈ I * y ∈ J`
is also the smallest `R`-submodule that does so. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comap_of_surjective (I : ideal S) :
map f (comap f I) = I | le_antisymm (map_le_iff_le_comap.2 le_rfl)
(λ s hsi, let ⟨r, hfrs⟩ := hf s in
hfrs ▸ (mem_map_of_mem f $ show f r ∈ I, from hfrs.symm ▸ hsi)) | theorem | ideal.map_comap_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi_map_comap : galois_insertion (map f) (comap f) | galois_insertion.monotone_intro
((gc_map_comap f).monotone_u)
((gc_map_comap f).monotone_l)
(λ _, le_comap_map)
(map_comap_of_surjective _ hf) | def | ideal.gi_map_comap | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"galois_insertion",
"galois_insertion.monotone_intro"
] | `map` and `comap` are adjoint, and the composition `map f ∘ comap f` is the
identity | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_surjective_of_surjective : surjective (map f) | (gi_map_comap f hf).l_surjective | lemma | ideal.map_surjective_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_injective_of_surjective : injective (comap f) | (gi_map_comap f hf).u_injective | lemma | ideal.comap_injective_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sup_comap_of_surjective (I J : ideal S) : (I.comap f ⊔ J.comap f).map f = I ⊔ J | (gi_map_comap f hf).l_sup_u _ _ | lemma | ideal.map_sup_comap_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr_comap_of_surjective (K : ι → ideal S) : (⨆i, (K i).comap f).map f = supr K | (gi_map_comap f hf).l_supr_u _ | lemma | ideal.map_supr_comap_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inf_comap_of_surjective (I J : ideal S) : (I.comap f ⊓ J.comap f).map f = I ⊓ J | (gi_map_comap f hf).l_inf_u _ _ | lemma | ideal.map_inf_comap_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_infi_comap_of_surjective (K : ι → ideal S) : (⨅i, (K i).comap f).map f = infi K | (gi_map_comap f hf).l_infi_u _ | lemma | ideal.map_infi_comap_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_image_of_mem_map_of_surjective {I : ideal R} {y}
(H : y ∈ map f I) : y ∈ f '' I | submodule.span_induction H (λ _, id) ⟨0, I.zero_mem, map_zero f⟩
(λ y1 y2 ⟨x1, hx1i, hxy1⟩ ⟨x2, hx2i, hxy2⟩,
⟨x1 + x2, I.add_mem hx1i hx2i, hxy1 ▸ hxy2 ▸ map_add f _ _⟩)
(λ c y ⟨x, hxi, hxy⟩,
let ⟨d, hdc⟩ := hf c in ⟨d * x, I.mul_mem_left _ hxi, hdc ▸ hxy ▸ map_mul f _ _⟩) | theorem | ideal.mem_image_of_mem_map_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"map_mul",
"submodule.span_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map_iff_of_surjective {I : ideal R} {y} :
y ∈ map f I ↔ ∃ x, x ∈ I ∧ f x = y | ⟨λ h, (set.mem_image _ _ _).2 (mem_image_of_mem_map_of_surjective f hf h),
λ ⟨x, hx⟩, hx.right ▸ (mem_map_of_mem f hx.left)⟩ | lemma | ideal.mem_map_iff_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"set.mem_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_map_of_comap_le_of_surjective : comap f K ≤ I → K ≤ map f I | λ h, (map_comap_of_surjective f hf K) ▸ map_mono h | lemma | ideal.le_map_of_comap_le_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_submodule_map (f : R →+* S) [h : ring_hom_surjective f] (I : ideal R) :
I.map f = submodule.map f.to_semilinear_map I | submodule.ext (λ x, mem_map_iff_of_surjective f h.1) | lemma | ideal.map_eq_submodule_map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring_hom_surjective",
"submodule.ext",
"submodule.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bot_le_of_injective : comap f ⊥ ≤ I | begin
refine le_trans (λ x hx, _) bot_le,
rw [mem_comap, submodule.mem_bot, ← map_zero f] at hx,
exact eq.symm (hf hx) ▸ (submodule.zero_mem ⊥)
end | lemma | ideal.comap_bot_le_of_injective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"bot_le",
"submodule.mem_bot",
"submodule.zero_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_bot_of_injective : ideal.comap f ⊥ = ⊥ | le_bot_iff.mp (ideal.comap_bot_le_of_injective f hf) | lemma | ideal.comap_bot_of_injective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal.comap",
"ideal.comap_bot_le_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_map_of_surjective (I : ideal R) : comap f (map f I) = I ⊔ comap f ⊥ | le_antisymm (assume r h, let ⟨s, hsi, hfsr⟩ := mem_image_of_mem_map_of_surjective f hf h in
submodule.mem_sup.2 ⟨s, hsi, r - s, (submodule.mem_bot S).2 $ by rw [map_sub, hfsr, sub_self],
add_sub_cancel'_right s r⟩)
(sup_le (map_le_iff_le_comap.1 le_rfl) (comap_mono bot_le)) | theorem | ideal.comap_map_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"bot_le",
"ideal",
"le_rfl",
"submodule.mem_bot",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_iso_of_surjective : ideal S ≃o { p : ideal R // comap f ⊥ ≤ p } | { to_fun := λ J, ⟨comap f J, comap_mono bot_le⟩,
inv_fun := λ I, map f I.1,
left_inv := λ J, map_comap_of_surjective f hf J,
right_inv := λ I, subtype.eq $ show comap f (map f I.1) = I.1,
from (comap_map_of_surjective f hf I).symm ▸ le_antisymm
(sup_le le_rfl I.2) le_sup_left,
map_rel_iff' := λ I1 I2,... | def | ideal.rel_iso_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"inv_fun",
"le_rfl",
"le_sup_left",
"sup_le"
] | Correspondence theorem | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_embedding_of_surjective : ideal S ↪o ideal R | (rel_iso_of_surjective f hf).to_rel_embedding.trans (subtype.rel_embedding _ _) | def | ideal.order_embedding_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"subtype.rel_embedding"
] | The map on ideals induced by a surjective map preserves inclusion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq_top_or_is_maximal_of_surjective {I : ideal R} (H : is_maximal I) :
(map f I) = ⊤ ∨ is_maximal (map f I) | begin
refine or_iff_not_imp_left.2 (λ ne_top, ⟨⟨λ h, ne_top h, λ J hJ, _⟩⟩),
{ refine (rel_iso_of_surjective f hf).injective
(subtype.ext_iff.2 (eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne _ _)) comap_top.symm)),
{ exact (map_le_iff_le_comap).1 (le_of_lt hJ) },
{ exact λ h, hJ.right (le_map_of_comap_l... | theorem | ideal.map_eq_top_or_is_maximal_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_is_maximal_of_surjective {K : ideal S} [H : is_maximal K] : is_maximal (comap f K) | begin
refine ⟨⟨comap_ne_top _ H.1.1, λ J hJ, _⟩⟩,
suffices : map f J = ⊤,
{ replace this := congr_arg (comap f) this,
rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this,
rw eq_top_iff,
exact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono (bot_le)) (le_of_lt hJ))) },
refine ... | theorem | ideal.comap_is_maximal_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"bot_le",
"eq_top_iff",
"ideal",
"sup_eq_left",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_le_comap_iff_of_surjective (I J : ideal S) : comap f I ≤ comap f J ↔ I ≤ J | ⟨λ h, (map_comap_of_surjective f hf I).symm.le.trans (map_le_of_le_comap h),
λ h, le_comap_of_map_le ((map_comap_of_surjective f hf I).le.trans h)⟩ | theorem | ideal.comap_le_comap_iff_of_surjective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_of_equiv (I : ideal R) (f : R ≃+* S) : (I.map (f : R →+* S)).map (f.symm : S →+* R) = I | by simp [← ring_equiv.to_ring_hom_eq_coe, map_map] | lemma | ideal.map_of_equiv | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring_equiv.to_ring_hom_eq_coe"
] | If `f : R ≃+* S` is a ring isomorphism and `I : ideal R`, then `map f (map f.symm) = I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_of_equiv (I : ideal R) (f : R ≃+* S) :
(I.comap (f.symm : S →+* R)).comap (f : R →+* S) = I | by simp [← ring_equiv.to_ring_hom_eq_coe, comap_comap] | lemma | ideal.comap_of_equiv | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ring_equiv.to_ring_hom_eq_coe"
] | If `f : R ≃+* S` is a ring isomorphism and `I : ideal R`, then `comap f.symm (comap f) = I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_comap_of_equiv (I : ideal R) (f : R ≃+* S) : I.map (f : R →+* S) = I.comap f.symm | le_antisymm (le_comap_of_map_le (map_of_equiv I f).le)
(le_map_of_comap_le_of_surjective _ f.surjective (comap_of_equiv I f).le) | lemma | ideal.map_comap_of_equiv | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | If `f : R ≃+* S` is a ring isomorphism and `I : ideal R`, then `map f I = comap f.symm I`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_iso_of_bijective : ideal S ≃o ideal R | { to_fun := comap f,
inv_fun := map f,
left_inv := (rel_iso_of_surjective f hf.right).left_inv,
right_inv := λ J, subtype.ext_iff.1
((rel_iso_of_surjective f hf.right).right_inv ⟨J, comap_bot_le_of_injective f hf.left⟩),
map_rel_iff' := λ _ _, (rel_iso_of_surjective f hf.right).map_rel_iff' } | def | ideal.rel_iso_of_bijective | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"inv_fun"
] | Special case of the correspondence theorem for isomorphic rings | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_le_iff_le_map {I : ideal R} {K : ideal S} : comap f K ≤ I ↔ K ≤ map f I | ⟨λ h, le_map_of_comap_le_of_surjective f hf.right h,
λ h, ((rel_iso_of_bijective f hf).right_inv I) ▸ comap_mono h⟩ | lemma | ideal.comap_le_iff_le_map | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map.is_maximal {I : ideal R} (H : is_maximal I) : is_maximal (map f I) | by refine or_iff_not_imp_left.1
(map_eq_top_or_is_maximal_of_surjective f hf.right H) (λ h, H.1.1 _);
calc I = comap f (map f I) : ((rel_iso_of_bijective f hf).right_inv I).symm
... = comap f ⊤ : by rw h
... = ⊤ : by rw comap_top | theorem | ideal.map.is_maximal | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ring_equiv.bot_maximal_iff (e : R ≃+* S) :
(⊥ : ideal R).is_maximal ↔ (⊥ : ideal S).is_maximal | ⟨λ h, (@map_bot _ _ _ _ _ _ e.to_ring_hom) ▸ map.is_maximal e.to_ring_hom e.bijective h,
λ h, (@map_bot _ _ _ _ _ _ e.symm.to_ring_hom) ▸ map.is_maximal e.symm.to_ring_hom
e.symm.bijective h⟩ | lemma | ideal.ring_equiv.bot_maximal_iff | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul : map f (I * J) = map f I * map f J | le_antisymm (map_le_iff_le_comap.2 $ mul_le.2 $ λ r hri s hsj,
show f (r * s) ∈ _, by rw map_mul;
exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj))
(trans_rel_right _ (span_mul_span _ _) $ span_le.2 $
set.Union₂_subset $ λ i ⟨r, hri, hfri⟩,
set.Union₂_subset $ λ j ⟨s, hsj, hfsj⟩,
set.singleton_... | theorem | ideal.map_mul | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"map_mul",
"set.Union₂_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hom : ideal R →*₀ ideal S | { to_fun := map f,
map_mul' := λ I J, ideal.map_mul f I J,
map_one' := by convert ideal.map_top f; exact one_eq_top,
map_zero' := ideal.map_bot } | def | ideal.map_hom | ring_theory.ideal | src/ring_theory/ideal/operations.lean | [
"algebra.algebra.operations",
"algebra.ring.equiv",
"data.nat.choose.sum",
"linear_algebra.basis.bilinear",
"ring_theory.coprime.lemmas",
"ring_theory.ideal.basic",
"ring_theory.non_zero_divisors"
] | [
"ideal",
"ideal.map_bot",
"ideal.map_mul",
"ideal.map_top"
] | The pushforward `ideal.map` as a monoid-with-zero homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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