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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|---|---|---|---|---|---|---|---|---|---|---|
coe_submodule_hom : fractional_ideal S P →+* submodule R P | ⟨coe, coe_one, coe_mul, coe_zero, coe_add⟩ | def | fractional_ideal.coe_submodule_hom | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | `fractional_ideal.submodule.has_coe` as a bundled `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_le_add_left {I J : fractional_ideal S P} (hIJ : I ≤ J) (J' : fractional_ideal S P) :
J' + I ≤ J' + J | sup_le_sup_left hIJ J' | lemma | fractional_ideal.add_le_add_left | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"sup_le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_left {I J : fractional_ideal S P} (hIJ : I ≤ J) (J' : fractional_ideal S P) :
J' * I ≤ J' * J | mul_le.mpr (λ k hk j hj, mul_mem_mul hk (hIJ hj)) | lemma | fractional_ideal.mul_le_mul_left | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"mul_le_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_self_mul_self {I : fractional_ideal S P} (hI: 1 ≤ I) : I ≤ I * I | begin
convert mul_left_mono I hI,
exact (mul_one I).symm
end | lemma | fractional_ideal.le_self_mul_self | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_self_le_self {I : fractional_ideal S P} (hI: I ≤ 1) : I * I ≤ I | begin
convert mul_left_mono I hI,
exact (mul_one I).symm
end | lemma | fractional_ideal.mul_self_le_self | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_le_one {I : ideal R} : (I : fractional_ideal S P) ≤ 1 | λ x hx, let ⟨y, _, hy⟩ := (mem_coe_ideal S).mp hx in (mem_one_iff S).mpr ⟨y, hy⟩ | lemma | fractional_ideal.coe_ideal_le_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_iff_exists_coe_ideal {J : fractional_ideal S P} :
J ≤ (1 : fractional_ideal S P) ↔ ∃ (I : ideal R), ↑I = J | begin
split,
{ intro hJ,
refine ⟨⟨{x : R | algebra_map R P x ∈ J}, _, _, _⟩, _⟩,
{ intros a b ha hb,
rw [mem_set_of_eq, ring_hom.map_add],
exact J.val.add_mem ha hb },
{ rw [mem_set_of_eq, ring_hom.map_zero],
exact J.val.zero_mem },
{ intros c x hx,
rw [smul_eq_mul, mem_set_o... | lemma | fractional_ideal.le_one_iff_exists_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"fractional_ideal",
"ideal",
"ring_hom.map_add",
"ring_hom.map_mul",
"ring_hom.map_zero",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le {I : fractional_ideal S P} :
1 ≤ I ↔ (1 : P) ∈ I | by rw [← coe_le_coe, coe_one, submodule.one_le, mem_coe] | lemma | fractional_ideal.one_le | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"one_le",
"submodule.one_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_hom : ideal R →+* fractional_ideal S P | { to_fun := coe,
map_add' := coe_ideal_sup,
map_mul' := coe_ideal_mul,
map_one' := by rw [ideal.one_eq_top, coe_ideal_top],
map_zero' := coe_ideal_bot } | def | fractional_ideal.coe_ideal_hom | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"ideal.one_eq_top"
] | `coe_ideal_hom (S : submonoid R) P` is `coe : ideal R → fractional_ideal S P` as a ring hom | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_ideal_pow (I : ideal R) (n : ℕ) : (↑(I^n) : fractional_ideal S P) = I^n | (coe_ideal_hom S P).map_pow _ n | lemma | fractional_ideal.coe_ideal_pow | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"map_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_finprod [is_localization S P] {α : Sort*} {f : α → ideal R}
(hS : S ≤ non_zero_divisors R) :
((∏ᶠ a : α, f a : ideal R) : fractional_ideal S P) = ∏ᶠ a : α, (f a : fractional_ideal S P) | monoid_hom.map_finprod_of_injective (coe_ideal_hom S P).to_monoid_hom (coe_ideal_injective' hS) f | lemma | fractional_ideal.coe_ideal_finprod | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"is_localization",
"monoid_hom.map_finprod_of_injective",
"non_zero_divisors"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_fractional.map (g : P →ₐ[R] P') {I : submodule R P} :
is_fractional S I → is_fractional S (submodule.map g.to_linear_map I) | | ⟨a, a_nonzero, hI⟩ := ⟨a, a_nonzero, λ b hb, begin
obtain ⟨b', b'_mem, hb'⟩ := submodule.mem_map.mp hb,
obtain ⟨x, hx⟩ := hI b' b'_mem,
use x,
erw [←g.commutes, hx, g.map_smul, hb']
end⟩ | lemma | is_fractional.map | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"submodule",
"submodule.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (g : P →ₐ[R] P') :
fractional_ideal S P → fractional_ideal S P' | λ I, ⟨submodule.map g.to_linear_map I, I.is_fractional.map g⟩ | def | fractional_ideal.map | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | `I.map g` is the pushforward of the fractional ideal `I` along the algebra morphism `g` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map (g : P →ₐ[R] P') (I : fractional_ideal S P) :
↑(map g I) = submodule.map g.to_linear_map I | rfl | lemma | fractional_ideal.coe_map | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_map {I : fractional_ideal S P} {g : P →ₐ[R] P'}
{y : P'} : y ∈ I.map g ↔ ∃ x, x ∈ I ∧ g x = y | submodule.mem_map | lemma | fractional_ideal.mem_map | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"mem_map",
"submodule.mem_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : I.map (alg_hom.id _ _) = I | coe_to_submodule_injective (submodule.map_id I) | lemma | fractional_ideal.map_id | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"alg_hom.id",
"map_id",
"submodule.map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (g' : P' →ₐ[R] P'') :
I.map (g'.comp g) = (I.map g).map g' | coe_to_submodule_injective (submodule.map_comp g.to_linear_map g'.to_linear_map I) | lemma | fractional_ideal.map_comp | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"map_comp",
"submodule.map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe_ideal (I : ideal R) :
(I : fractional_ideal S P).map g = I | begin
ext x,
simp only [mem_coe_ideal],
split,
{ rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩,
exact ⟨y, hy, (g.commutes y).symm⟩ },
{ rintro ⟨y, hy, rfl⟩,
exact ⟨_, ⟨y, hy, rfl⟩, g.commutes y⟩ },
end | lemma | fractional_ideal.map_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one :
(1 : fractional_ideal S P).map g = 1 | map_coe_ideal g ⊤ | lemma | fractional_ideal.map_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zero :
(0 : fractional_ideal S P).map g = 0 | map_coe_ideal g 0 | lemma | fractional_ideal.map_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add : (I + J).map g = I.map g + J.map g | coe_to_submodule_injective (submodule.map_sup _ _ _) | lemma | fractional_ideal.map_add | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"submodule.map_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul : (I * J).map g = I.map g * J.map g | begin
simp only [mul_def],
exact coe_to_submodule_injective (submodule.map_mul _ _ _)
end | lemma | fractional_ideal.map_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"map_mul",
"submodule.map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_map_symm (g : P ≃ₐ[R] P') :
(I.map (g : P →ₐ[R] P')).map (g.symm : P' →ₐ[R] P) = I | by rw [←map_comp, g.symm_comp, map_id] | lemma | fractional_ideal.map_map_symm | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_symm_map (I : fractional_ideal S P') (g : P ≃ₐ[R] P') :
(I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I | by rw [←map_comp, g.comp_symm, map_id] | lemma | fractional_ideal.map_symm_map | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mem_map {f : P →ₐ[R] P'} (h : function.injective f) {x : P} {I : fractional_ideal S P} :
f x ∈ map f I ↔ x ∈ I | mem_map.trans ⟨λ ⟨x', hx', x'_eq⟩, h x'_eq ▸ hx', λ h, ⟨x, h, rfl⟩⟩ | lemma | fractional_ideal.map_mem_map | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_injective (f : P →ₐ[R] P') (h : function.injective f) :
function.injective (map f : fractional_ideal S P → fractional_ideal S P') | λ I J hIJ, ext (λ x, (map_mem_map h).symm.trans (hIJ.symm ▸ map_mem_map h)) | lemma | fractional_ideal.map_injective | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv (g : P ≃ₐ[R] P') :
fractional_ideal S P ≃+* fractional_ideal S P' | { to_fun := map g,
inv_fun := map g.symm,
map_add' := λ I J, map_add I J _,
map_mul' := λ I J, map_mul I J _,
left_inv := λ I, by { rw [←map_comp, alg_equiv.symm_comp, map_id] },
right_inv := λ I, by { rw [←map_comp, alg_equiv.comp_symm, map_id] } } | def | fractional_ideal.map_equiv | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"alg_equiv.comp_symm",
"alg_equiv.symm_comp",
"fractional_ideal",
"inv_fun",
"map_id",
"map_mul"
] | If `g` is an equivalence, `map g` is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fun_map_equiv (g : P ≃ₐ[R] P') :
(map_equiv g : fractional_ideal S P → fractional_ideal S P') = map g | rfl | lemma | fractional_ideal.coe_fun_map_equiv | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv_apply (g : P ≃ₐ[R] P') (I : fractional_ideal S P) :
map_equiv g I = map ↑g I | rfl | lemma | fractional_ideal.map_equiv_apply | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv_symm (g : P ≃ₐ[R] P') :
((map_equiv g).symm : fractional_ideal S P' ≃+* _) = map_equiv g.symm | rfl | lemma | fractional_ideal.map_equiv_symm | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_equiv_refl :
map_equiv alg_equiv.refl = ring_equiv.refl (fractional_ideal S P) | ring_equiv.ext (λ x, by simp) | lemma | fractional_ideal.map_equiv_refl | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"alg_equiv.refl",
"fractional_ideal",
"ring_equiv.ext",
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fractional_span_iff {s : set P} :
is_fractional S (span R s) ↔ ∃ a ∈ S, ∀ (b : P), b ∈ s → is_integer R (a • b) | ⟨λ ⟨a, a_mem, h⟩, ⟨a, a_mem, λ b hb, h b (subset_span hb)⟩,
λ ⟨a, a_mem, h⟩, ⟨a, a_mem, λ b hb, span_induction hb
h
(by { rw smul_zero, exact is_integer_zero })
(λ x y hx hy, by { rw smul_add, exact is_integer_add hx hy })
(λ s x hx, by { rw smul_comm, exact is_integer_smul hx })⟩⟩ | lemma | fractional_ideal.is_fractional_span_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"smul_add",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fractional_of_fg {I : submodule R P} (hI : I.fg) :
is_fractional S I | begin
rcases hI with ⟨I, rfl⟩,
rcases exist_integer_multiples_of_finset S I with ⟨⟨s, hs1⟩, hs⟩,
rw is_fractional_span_iff,
exact ⟨s, hs1, hs⟩,
end | lemma | fractional_ideal.is_fractional_of_fg | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_mul_finite_of_mem_mul {I J : fractional_ideal S P} {x : P} (hx : x ∈ I * J) :
∃ (T T' : finset P), (T : set P) ⊆ I ∧ (T' : set P) ⊆ J ∧ x ∈ span R (T * T' : set P) | submodule.mem_span_mul_finite_of_mem_mul (by simpa using mem_coe.mpr hx) | lemma | fractional_ideal.mem_span_mul_finite_of_mem_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"finset",
"fractional_ideal",
"submodule.mem_span_mul_finite_of_mem_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_fg (inj : function.injective (algebra_map R P)) (I : ideal R) :
fg ((I : fractional_ideal S P) : submodule R P) ↔ I.fg | coe_submodule_fg _ inj _ | lemma | fractional_ideal.coe_ideal_fg | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"fractional_ideal",
"ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_unit (I : (fractional_ideal S P)ˣ) :
fg (I : submodule R P) | submodule.fg_unit $ units.map (coe_submodule_hom S P).to_monoid_hom I | lemma | fractional_ideal.fg_unit | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule",
"submodule.fg_unit",
"units.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fg_of_is_unit (I : fractional_ideal S P) (h : is_unit I) :
fg (I : submodule R P) | fg_unit h.unit | lemma | fractional_ideal.fg_of_is_unit | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_unit",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.ideal.fg_of_is_unit (inj : function.injective (algebra_map R P))
(I : ideal R) (h : is_unit (I : fractional_ideal S P)) :
I.fg | by { rw ← coe_ideal_fg S inj I, exact fg_of_is_unit I h } | lemma | ideal.fg_of_is_unit | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"fractional_ideal",
"ideal",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv :
fractional_ideal S P ≃+* fractional_ideal S P' | map_equiv
{ commutes' := λ r, ring_equiv_of_ring_equiv_eq _ _,
..ring_equiv_of_ring_equiv P P' (ring_equiv.refl R)
(show S.map _ = S, by rw [ring_equiv.to_monoid_hom_refl, submonoid.map_id]) } | def | fractional_ideal.canonical_equiv | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ring_equiv.refl",
"ring_equiv.to_monoid_hom_refl",
"submonoid.map_id"
] | `canonical_equiv f f'` is the canonical equivalence between the fractional
ideals in `P` and in `P'` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_canonical_equiv_apply {I : fractional_ideal S P} {x : P'} :
x ∈ canonical_equiv S P P' I ↔
∃ y ∈ I, is_localization.map P' (ring_hom.id R)
(λ y (hy : y ∈ S), show ring_hom.id R y ∈ S, from hy) (y : P) = x | begin
rw [canonical_equiv, map_equiv_apply, mem_map],
exact ⟨λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩, λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩⟩
end | lemma | fractional_ideal.mem_canonical_equiv_apply | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_localization.map",
"mem_map",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv_symm :
(canonical_equiv S P P').symm = canonical_equiv S P' P | ring_equiv.ext $ λ I, set_like.ext_iff.mpr $ λ x,
by { rw [mem_canonical_equiv_apply, canonical_equiv, map_equiv_symm, map_equiv,
ring_equiv.coe_mk, mem_map],
exact ⟨λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩, λ ⟨y, mem, eq⟩, ⟨y, mem, eq⟩⟩ } | lemma | fractional_ideal.canonical_equiv_symm | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"mem_map",
"ring_equiv.coe_mk",
"ring_equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv_flip (I) :
canonical_equiv S P P' (canonical_equiv S P' P I) = I | by rw [←canonical_equiv_symm, ring_equiv.symm_apply_apply] | lemma | fractional_ideal.canonical_equiv_flip | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"ring_equiv.symm_apply_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv_canonical_equiv (P'' : Type*) [comm_ring P''] [algebra R P'']
[is_localization S P''] (I : fractional_ideal S P) :
canonical_equiv S P' P'' (canonical_equiv S P P' I) = canonical_equiv S P P'' I | begin
ext,
simp only [is_localization.map_map, ring_hom_inv_pair.comp_eq₂, mem_canonical_equiv_apply,
exists_prop, exists_exists_and_eq_and],
refl
end | lemma | fractional_ideal.canonical_equiv_canonical_equiv | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra",
"comm_ring",
"exists_exists_and_eq_and",
"exists_prop",
"fractional_ideal",
"is_localization",
"is_localization.map_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv_trans_canonical_equiv (P'' : Type*) [comm_ring P'']
[algebra R P''] [is_localization S P''] :
(canonical_equiv S P P').trans (canonical_equiv S P' P'') = canonical_equiv S P P'' | ring_equiv.ext (canonical_equiv_canonical_equiv S P P' P'') | lemma | fractional_ideal.canonical_equiv_trans_canonical_equiv | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra",
"comm_ring",
"is_localization",
"ring_equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv_coe_ideal (I : ideal R) :
canonical_equiv S P P' I = I | by { ext, simp [is_localization.map_eq] } | lemma | fractional_ideal.canonical_equiv_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"ideal",
"is_localization.map_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv_self : canonical_equiv S P P = ring_equiv.refl _ | begin
rw ← canonical_equiv_trans_canonical_equiv S P P,
convert (canonical_equiv S P P).symm_trans_self,
exact (canonical_equiv_symm S P P).symm
end | lemma | fractional_ideal.canonical_equiv_self | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_ne_zero_mem_is_integer [nontrivial R] (hI : I ≠ 0) :
∃ x ≠ (0 : R), algebra_map R K x ∈ I | begin
obtain ⟨y, y_mem, y_not_mem⟩ := set_like.exists_of_lt
(by simpa only using bot_lt_iff_ne_bot.mpr hI),
have y_ne_zero : y ≠ 0 := by simpa using y_not_mem,
obtain ⟨z, ⟨x, hx⟩⟩ := exists_integer_multiple R⁰ y,
refine ⟨x, _, _⟩,
{ rw [ne.def, ← @is_fraction_ring.to_map_eq_zero_iff R _ K, hx, algebra.smu... | lemma | fractional_ideal.exists_ne_zero_mem_is_integer | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"is_fraction_ring.to_map_eq_zero_iff",
"is_fraction_ring.to_map_ne_zero_of_mem_non_zero_divisors",
"mul_ne_zero",
"nontrivial",
"set_like.exists_of_lt"
] | Nonzero fractional ideals contain a nonzero integer. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_ne_zero [nontrivial R] (hI : I ≠ 0) : I.map h ≠ 0 | begin
obtain ⟨x, x_ne_zero, hx⟩ := exists_ne_zero_mem_is_integer hI,
contrapose! x_ne_zero with map_eq_zero,
refine is_fraction_ring.to_map_eq_zero_iff.mp (eq_zero_iff.mp map_eq_zero _ (mem_map.mpr _)),
exact ⟨algebra_map R K x, hx, h.commutes x⟩,
end | lemma | fractional_ideal.map_ne_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"map_eq_zero",
"map_ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_zero_iff [nontrivial R] : I.map h = 0 ↔ I = 0 | ⟨imp_of_not_imp_not _ _ (map_ne_zero _), λ hI, hI.symm ▸ map_zero h⟩ | lemma | fractional_ideal.map_eq_zero_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"map_ne_zero",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_injective : function.injective (coe : ideal R → fractional_ideal R⁰ K) | coe_ideal_injective' le_rfl | lemma | fractional_ideal.coe_ideal_injective | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_inj {I J : ideal R} :
(I : fractional_ideal R⁰ K) = (J : fractional_ideal R⁰ K) ↔ I = J | coe_ideal_inj' le_rfl | lemma | fractional_ideal.coe_ideal_inj | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_eq_zero {I : ideal R} : (I : fractional_ideal R⁰ K) = 0 ↔ I = ⊥ | coe_ideal_eq_zero' le_rfl | lemma | fractional_ideal.coe_ideal_eq_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_ne_zero {I : ideal R} : (I : fractional_ideal R⁰ K) ≠ 0 ↔ I ≠ ⊥ | coe_ideal_ne_zero' le_rfl | lemma | fractional_ideal.coe_ideal_ne_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_eq_one {I : ideal R} : (I : fractional_ideal R⁰ K) = 1 ↔ I = 1 | by simpa only [ideal.one_eq_top] using coe_ideal_inj | lemma | fractional_ideal.coe_ideal_eq_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"ideal.one_eq_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_ne_one {I : ideal R} : (I : fractional_ideal R⁰ K) ≠ 1 ↔ I ≠ 1 | not_iff_not.mpr coe_ideal_eq_one | lemma | fractional_ideal.coe_ideal_ne_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero_of_mul_eq_one (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) : I ≠ 0 | λ hI, zero_ne_one' (fractional_ideal R₁⁰ K) (by { convert h, simp [hI], }) | lemma | fractional_ideal.ne_zero_of_mul_eq_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"zero_ne_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_fractional.div_of_nonzero {I J : submodule R₁ K} :
is_fractional R₁⁰ I → is_fractional R₁⁰ J → J ≠ 0 → is_fractional R₁⁰ (I / J) | | ⟨aI, haI, hI⟩ ⟨aJ, haJ, hJ⟩ h := begin
obtain ⟨y, mem_J, not_mem_zero⟩ := set_like.exists_of_lt
(by simpa only using bot_lt_iff_ne_bot.mpr h),
obtain ⟨y', hy'⟩ := hJ y mem_J,
use (aI * y'),
split,
{ apply (non_zero_divisors R₁).mul_mem haI (mem_non_zero_divisors_iff_ne_zero.mpr _),
intro y'_eq_zero,... | lemma | is_fractional.div_of_nonzero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"is_fraction_ring.injective",
"is_fractional",
"mul_comm",
"non_zero_divisors",
"ring_hom.map_zero",
"set_like.exists_of_lt",
"submodule",
"submodule.smul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fractional_div_of_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
is_fractional R₁⁰ (I / J : submodule R₁ K) | I.is_fractional.div_of_nonzero J.is_fractional $ λ H, h $
coe_to_submodule_injective $ H.trans coe_zero.symm | lemma | fractional_ideal.fractional_div_of_nonzero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"is_fractional",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_zero {I : fractional_ideal R₁⁰ K} :
I / 0 = 0 | dif_pos rfl | lemma | fractional_ideal.div_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"div_zero",
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) :
(I / J) = ⟨I / J, fractional_div_of_nonzero h⟩ | dif_neg h | lemma | fractional_ideal.div_nonzero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div {I J : fractional_ideal R₁⁰ K} (hJ : J ≠ 0) :
(↑(I / J) : submodule R₁ K) = ↑I / (↑J : submodule R₁ K) | congr_arg _ (dif_neg hJ) | lemma | fractional_ideal.coe_div | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_div_iff_of_nonzero {I J : fractional_ideal R₁⁰ K} (h : J ≠ 0) {x} :
x ∈ I / J ↔ ∀ y ∈ J, x * y ∈ I | by { rw div_nonzero h, exact submodule.mem_div_iff_forall_mul_mem } | lemma | fractional_ideal.mem_div_iff_of_nonzero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule.mem_div_iff_forall_mul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_div_le_one {I : fractional_ideal R₁⁰ K} : I * (1 / I) ≤ 1 | begin
by_cases hI : I = 0,
{ rw [hI, div_zero, mul_zero],
exact zero_le 1 },
{ rw [← coe_le_coe, coe_mul, coe_div hI, coe_one],
apply submodule.mul_one_div_le_one },
end | lemma | fractional_ideal.mul_one_div_le_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"div_zero",
"fractional_ideal",
"mul_zero",
"submodule.mul_one_div_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_self_mul_one_div {I : fractional_ideal R₁⁰ K} (hI : I ≤ (1 : fractional_ideal R₁⁰ K)) :
I ≤ I * (1 / I) | begin
by_cases hI_nz : I = 0,
{ rw [hI_nz, div_zero, mul_zero], exact zero_le 0 },
{ rw [← coe_le_coe, coe_mul, coe_div hI_nz, coe_one],
rw [← coe_le_coe, coe_one] at hI,
exact submodule.le_self_mul_one_div hI },
end | lemma | fractional_ideal.le_self_mul_one_div | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"div_zero",
"fractional_ideal",
"mul_zero",
"submodule.le_self_mul_one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff_of_nonzero {I J J' : fractional_ideal R₁⁰ K} (hJ' : J' ≠ 0) :
I ≤ J / J' ↔ ∀ (x ∈ I) (y ∈ J'), x * y ∈ J | ⟨ λ h x hx, (mem_div_iff_of_nonzero hJ').mp (h hx),
λ h x hx, (mem_div_iff_of_nonzero hJ').mpr (h x hx) ⟩ | lemma | fractional_ideal.le_div_iff_of_nonzero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div_iff_mul_le {I J J' : fractional_ideal R₁⁰ K} (hJ' : J' ≠ 0) :
I ≤ J / J' ↔ I * J' ≤ J | begin
rw div_nonzero hJ',
convert submodule.le_div_iff_mul_le using 1,
rw [← coe_mul, coe_le_coe]
end | lemma | fractional_ideal.le_div_iff_mul_le | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"le_div_iff_mul_le",
"submodule.le_div_iff_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_one {I : fractional_ideal R₁⁰ K} : I / 1 = I | begin
rw [div_nonzero (one_ne_zero' (fractional_ideal R₁⁰ K))],
ext,
split; intro h,
{ simpa using mem_div_iff_forall_mul_mem.mp h 1
((algebra_map R₁ K).map_one ▸ coe_mem_one R₁⁰ 1) },
{ apply mem_div_iff_forall_mul_mem.mpr,
rintros y ⟨y', _, rfl⟩,
rw mul_comm,
convert submodule.smul_mem _ y... | lemma | fractional_ideal.div_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"div_one",
"fractional_ideal",
"map_one",
"mul_comm",
"one_ne_zero'",
"submodule.smul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_div_of_mul_eq_one_right (I J : fractional_ideal R₁⁰ K) (h : I * J = 1) :
J = 1 / I | begin
have hI : I ≠ 0 := ne_zero_of_mul_eq_one I J h,
suffices h' : I * (1 / I) = 1,
{ exact (congr_arg units.inv $
@units.ext _ _ (units.mk_of_mul_eq_one _ _ h) (units.mk_of_mul_eq_one _ _ h') rfl) },
apply le_antisymm,
{ apply mul_le.mpr _,
intros x hx y hy,
rw mul_comm,
exact (mem_div_iff... | theorem | fractional_ideal.eq_one_div_of_mul_eq_one_right | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"eq_one_div_of_mul_eq_one_right",
"fractional_ideal",
"mul_comm",
"units.ext",
"units.mk_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_self_cancel_iff {I : fractional_ideal R₁⁰ K} :
I * (1 / I) = 1 ↔ ∃ J, I * J = 1 | ⟨λ h, ⟨(1 / I), h⟩, λ ⟨J, hJ⟩, by rwa [← eq_one_div_of_mul_eq_one_right I J hJ]⟩ | theorem | fractional_ideal.mul_div_self_cancel_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"eq_one_div_of_mul_eq_one_right",
"fractional_ideal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_div (I J : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(I / J).map (h : K →ₐ[R₁] K') = I.map h / J.map h | begin
by_cases H : J = 0,
{ rw [H, div_zero, map_zero, div_zero] },
{ apply coe_to_submodule_injective,
simp [div_nonzero H, div_nonzero (map_ne_zero _ H), submodule.map_div] }
end | lemma | fractional_ideal.map_div | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"div_zero",
"fractional_ideal",
"map_div",
"map_ne_zero",
"submodule.map_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one_div (I : fractional_ideal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
(1 / I).map (h : K →ₐ[R₁] K') = 1 / I.map h | by rw [map_div, map_one] | lemma | fractional_ideal.map_one_div | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"map_div",
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_or_one (I : fractional_ideal K⁰ L) : I = 0 ∨ I = 1 | begin
rw or_iff_not_imp_left,
intro hI,
simp_rw [@set_like.ext_iff _ _ _ I 1, mem_one_iff],
intro x,
split,
{ intro x_mem,
obtain ⟨n, d, rfl⟩ := is_localization.mk'_surjective K⁰ x,
refine ⟨n / d, _⟩,
rw [map_div₀, is_fraction_ring.mk'_eq_div] },
{ rintro ⟨x, rfl⟩,
obtain ⟨y, y_ne, y_mem⟩ ... | lemma | fractional_ideal.eq_zero_or_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"div_mul_cancel",
"fractional_ideal",
"is_fraction_ring.mk'_eq_div",
"is_localization.mk'_surjective",
"map_div₀",
"or_iff_not_imp_left",
"ring_hom.map_mul",
"set_like.ext_iff",
"submodule.smul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_or_one_of_is_field (hF : is_field R₁) (I : fractional_ideal R₁⁰ K) : I = 0 ∨ I = 1 | by letI : field R₁ := hF.to_field; exact eq_zero_or_one I | lemma | fractional_ideal.eq_zero_or_one_of_is_field | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"field",
"fractional_ideal",
"is_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_finset {ι : Type*} (s : finset ι) (f : ι → K) : fractional_ideal R₁⁰ K | ⟨submodule.span R₁ (f '' s), begin
obtain ⟨a', ha'⟩ := is_localization.exist_integer_multiples R₁⁰ s f,
refine ⟨a', a'.2, λ x hx, submodule.span_induction hx _ _ _ _⟩,
{ rintro _ ⟨i, hi, rfl⟩, exact ha' i hi },
{ rw smul_zero, exact is_localization.is_integer_zero },
{ intros x y hx hy, rw smul_add, exact is_... | def | fractional_ideal.span_finset | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"finset",
"fractional_ideal",
"is_localization.exist_integer_multiples",
"is_localization.is_integer_add",
"is_localization.is_integer_smul",
"is_localization.is_integer_zero",
"smul_add",
"smul_zero",
"submodule.span_induction"
] | `fractional_ideal.span_finset R₁ s f` is the fractional ideal of `R₁` generated by `f '' s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
span_finset_eq_zero {ι : Type*} {s : finset ι} {f : ι → K} :
span_finset R₁ s f = 0 ↔ ∀ j ∈ s, f j = 0 | by simp only [← coe_to_submodule_inj, span_finset_coe, coe_zero, submodule.span_eq_bot,
set.mem_image, finset.mem_coe, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] | lemma | fractional_ideal.span_finset_eq_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"and_imp",
"finset",
"finset.mem_coe",
"forall_apply_eq_imp_iff₂",
"forall_exists_index",
"set.mem_image",
"submodule.span_eq_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_finset_ne_zero {ι : Type*} {s : finset ι} {f : ι → K} :
span_finset R₁ s f ≠ 0 ↔ ∃ j ∈ s, f j ≠ 0 | by simp | lemma | fractional_ideal.span_finset_ne_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fractional_span_singleton (x : P) : is_fractional S (span R {x} : submodule R P) | let ⟨a, ha⟩ := exists_integer_multiple S x in
is_fractional_span_iff.mpr ⟨a, a.2, λ x' hx', (set.mem_singleton_iff.mp hx').symm ▸ ha⟩ | lemma | fractional_ideal.is_fractional_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"is_fractional",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton (x : P) : fractional_ideal S P | ⟨span R {x}, is_fractional_span_singleton x⟩ | def | fractional_ideal.span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal"
] | `span_singleton x` is the fractional ideal generated by `x` if `0 ∉ S` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_span_singleton (x : P) :
(span_singleton S x : submodule R P) = span R {x} | by { rw span_singleton, refl } | lemma | fractional_ideal.coe_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton {x y : P} :
x ∈ span_singleton S y ↔ ∃ (z : R), z • y = x | by { rw span_singleton, exact submodule.mem_span_singleton } | lemma | fractional_ideal.mem_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"submodule.mem_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_span_singleton_self (x : P) :
x ∈ span_singleton S x | (mem_span_singleton S).mpr ⟨1, one_smul _ _⟩ | lemma | fractional_ideal.mem_span_singleton_self | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_le_iff_mem {x : P} {I : fractional_ideal S P} :
span_singleton S x ≤ I ↔ x ∈ I | by rw [← coe_le_coe, coe_span_singleton, submodule.span_singleton_le_iff_mem x ↑I, mem_coe] | lemma | fractional_ideal.span_singleton_le_iff_mem | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule.span_singleton_le_iff_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_eq_span_singleton [no_zero_smul_divisors R P] {x y : P} :
span_singleton S x = span_singleton S y ↔ ∃ z : Rˣ, z • x = y | by { rw [← submodule.span_singleton_eq_span_singleton, span_singleton, span_singleton],
exact subtype.mk_eq_mk } | lemma | fractional_ideal.span_singleton_eq_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"no_zero_smul_divisors",
"submodule.span_singleton_eq_span_singleton",
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_span_singleton_of_principal (I : fractional_ideal S P)
[is_principal (I : submodule R P)] :
I = span_singleton S (generator (I : submodule R P)) | by { rw span_singleton, exact coe_to_submodule_injective (span_singleton_generator ↑I).symm } | lemma | fractional_ideal.eq_span_singleton_of_principal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_principal_iff (I : fractional_ideal S P) :
is_principal (I : submodule R P) ↔ ∃ x, I = span_singleton S x | ⟨λ h, ⟨@generator _ _ _ _ _ ↑I h, @eq_span_singleton_of_principal _ _ _ _ _ _ _ I h⟩,
λ ⟨x, hx⟩, { principal := ⟨x, trans (congr_arg _ hx) (coe_span_singleton _ x)⟩ } ⟩ | lemma | fractional_ideal.is_principal_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_zero : span_singleton S (0 : P) = 0 | by { ext, simp [submodule.mem_span_singleton, eq_comm] } | lemma | fractional_ideal.span_singleton_zero | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"submodule.mem_span_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_eq_zero_iff {y : P} : span_singleton S y = 0 ↔ y = 0 | ⟨λ h, span_eq_bot.mp (by simpa using congr_arg subtype.val h : span R {y} = ⊥) y (mem_singleton y),
λ h, by simp [h] ⟩ | lemma | fractional_ideal.span_singleton_eq_zero_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_ne_zero_iff {y : P} : span_singleton S y ≠ 0 ↔ y ≠ 0 | not_congr span_singleton_eq_zero_iff | lemma | fractional_ideal.span_singleton_ne_zero_iff | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_one : span_singleton S (1 : P) = 1 | begin
ext,
refine (mem_span_singleton S).trans ((exists_congr _).trans (mem_one_iff S).symm),
intro x',
rw [algebra.smul_def, mul_one]
end | lemma | fractional_ideal.span_singleton_one | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_span_singleton (x y : P) :
span_singleton S x * span_singleton S y = span_singleton S (x * y) | begin
apply coe_to_submodule_injective,
simp only [coe_mul, coe_span_singleton, span_mul_span, singleton_mul_singleton],
end | lemma | fractional_ideal.span_singleton_mul_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_pow (x : P) (n : ℕ) : span_singleton S x ^ n = span_singleton S (x ^ n) | begin
induction n with n hn,
{ rw [pow_zero, pow_zero, span_singleton_one] },
{ rw [pow_succ, hn, span_singleton_mul_span_singleton, pow_succ] }
end | lemma | fractional_ideal.span_singleton_pow | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ideal_span_singleton (x : R) :
(↑(ideal.span {x} : ideal R) : fractional_ideal S P) = span_singleton S (algebra_map R P x) | begin
ext y,
refine (mem_coe_ideal S).trans (iff.trans _ (mem_span_singleton S).symm),
split,
{ rintros ⟨y', hy', rfl⟩,
obtain ⟨x', rfl⟩ := submodule.mem_span_singleton.mp hy',
use x',
rw [smul_eq_mul, ring_hom.map_mul, algebra.smul_def] },
{ rintros ⟨y', rfl⟩,
refine ⟨y' * x, submodule.mem_sp... | lemma | fractional_ideal.coe_ideal_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.smul_def",
"algebra_map",
"fractional_ideal",
"ideal",
"ideal.span",
"ring_hom.map_mul",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
canonical_equiv_span_singleton {P'} [comm_ring P'] [algebra R P'] [is_localization S P']
(x : P) :
canonical_equiv S P P' (span_singleton S x) =
span_singleton S (is_localization.map P' (ring_hom.id R)
(λ y (hy : y ∈ S), show ring_hom.id R y ∈ S, from hy) x) | begin
apply set_like.ext_iff.mpr,
intro y,
split; intro h,
{ rw mem_span_singleton,
obtain ⟨x', hx', rfl⟩ := (mem_canonical_equiv_apply _ _ _).mp h,
obtain ⟨z, rfl⟩ := (mem_span_singleton _).mp hx',
use z,
rw is_localization.map_smul,
refl },
{ rw mem_canonical_equiv_apply,
obtain ⟨z, ... | lemma | fractional_ideal.canonical_equiv_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra",
"comm_ring",
"is_localization",
"is_localization.map",
"is_localization.map_smul",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_singleton_mul {x y : P} {I : fractional_ideal S P} :
y ∈ span_singleton S x * I ↔ ∃ y' ∈ I, y = x * y' | begin
split,
{ intro h,
apply fractional_ideal.mul_induction_on h,
{ intros x' hx' y' hy',
obtain ⟨a, ha⟩ := (mem_span_singleton S).mp hx',
use [a • y', submodule.smul_mem I a hy'],
rw [←ha, algebra.mul_smul_comm, algebra.smul_mul_assoc] },
{ rintros _ _ ⟨y, hy, rfl⟩ ⟨y', hy', rfl⟩,
... | lemma | fractional_ideal.mem_singleton_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.mul_smul_comm",
"algebra.smul_mul_assoc",
"fractional_ideal",
"fractional_ideal.mul_induction_on",
"one_smul",
"submodule.add_mem",
"submodule.smul_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'_mul_coe_ideal_eq_coe_ideal {I J : ideal R₁} {x y : R₁} (hy : y ∈ R₁⁰) :
span_singleton R₁⁰ (is_localization.mk' K x ⟨y, hy⟩) * I = (J : fractional_ideal R₁⁰ K) ↔
ideal.span {x} * I = ideal.span {y} * J | begin
have : span_singleton R₁⁰ (is_localization.mk' _ (1 : R₁) ⟨y, hy⟩) *
span_singleton R₁⁰ (algebra_map R₁ K y) = 1,
{ rw [span_singleton_mul_span_singleton, mul_comm, ← is_localization.mk'_eq_mul_mk'_one,
is_localization.mk'_self, span_singleton_one] },
let y' : (fractional_ideal R₁⁰ K)ˣ :=... | lemma | fractional_ideal.mk'_mul_coe_ideal_eq_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra_map",
"fractional_ideal",
"ideal",
"ideal.span",
"is_localization.mk'",
"is_localization.mk'_eq_mul_mk'_one",
"is_localization.mk'_self",
"mk'",
"mul_comm",
"one_mul",
"units.mk_of_mul_eq_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
span_singleton_mul_coe_ideal_eq_coe_ideal {I J : ideal R₁} {z : K} :
span_singleton R₁⁰ z * (I : fractional_ideal R₁⁰ K) = J ↔
ideal.span {((is_localization.sec R₁⁰ z).1 : R₁)} * I =
ideal.span {(is_localization.sec R₁⁰ z).2} * J | -- `erw` to deal with the distinction between `y` and `⟨y.1, y.2⟩`
by erw [← mk'_mul_coe_ideal_eq_coe_ideal K (is_localization.sec R₁⁰ z).2.prop,
is_localization.mk'_sec K z] | lemma | fractional_ideal.span_singleton_mul_coe_ideal_eq_coe_ideal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"fractional_ideal",
"ideal",
"ideal.span",
"is_localization.mk'_sec",
"is_localization.sec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_div_span_singleton (x : K) :
1 / span_singleton R₁⁰ x = span_singleton R₁⁰ (x⁻¹) | if h : x = 0 then by simp [h] else (eq_one_div_of_mul_eq_one_right _ _ (by simp [h])).symm | lemma | fractional_ideal.one_div_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"eq_one_div_of_mul_eq_one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_span_singleton (J : fractional_ideal R₁⁰ K) (d : K) :
J / span_singleton R₁⁰ d = span_singleton R₁⁰ (d⁻¹) * J | begin
rw ← one_div_span_singleton,
by_cases hd : d = 0,
{ simp only [hd, span_singleton_zero, div_zero, zero_mul] },
have h_spand : span_singleton R₁⁰ d ≠ 0 := mt span_singleton_eq_zero_iff.mp hd,
apply le_antisymm,
{ intros x hx,
rw [← mem_coe, coe_div h_spand, submodule.mem_div_iff_forall_mul_mem] at ... | lemma | fractional_ideal.div_span_singleton | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"div_zero",
"fractional_ideal",
"inv_mul_cancel",
"le_div_iff_mul_le",
"mul_assoc",
"mul_left_comm",
"mul_one",
"submodule.mem_div_iff_forall_mul_mem",
"submodule.mul_mem_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_eq_span_singleton_mul (I : fractional_ideal R₁⁰ K) :
∃ (a : R₁) (aI : ideal R₁), a ≠ 0 ∧ I = span_singleton R₁⁰ (algebra_map R₁ K a)⁻¹ * aI | begin
obtain ⟨a_inv, nonzero, ha⟩ := I.is_fractional,
have nonzero := mem_non_zero_divisors_iff_ne_zero.mp nonzero,
have map_a_nonzero : algebra_map R₁ K a_inv ≠ 0 :=
mt is_fraction_ring.to_map_eq_zero_iff.mp nonzero,
refine ⟨a_inv,
submodule.comap (algebra.linear_map R₁ K)
↑(span_sing... | lemma | fractional_ideal.exists_eq_span_singleton_mul | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra.linear_map",
"algebra.linear_map_apply",
"algebra.smul_def",
"algebra_map",
"fractional_ideal",
"ideal",
"inv_mul_cancel",
"mul_assoc",
"one_mul",
"submodule.comap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_principal {R} [comm_ring R] [is_domain R] [is_principal_ideal_ring R]
[algebra R K] [is_fraction_ring R K]
(I : fractional_ideal R⁰ K) : (I : submodule R K).is_principal | begin
obtain ⟨a, aI, -, ha⟩ := exists_eq_span_singleton_mul I,
use (algebra_map R K a)⁻¹ * algebra_map R K (generator aI),
suffices : I = span_singleton R⁰ ((algebra_map R K a)⁻¹ * algebra_map R K (generator aI)),
{ rw span_singleton at this, exact congr_arg subtype.val this },
conv_lhs { rw [ha, ←span_single... | instance | fractional_ideal.is_principal | ring_theory | src/ring_theory/fractional_ideal.lean | [
"algebra.big_operators.finprod",
"ring_theory.integral_closure",
"ring_theory.localization.integer",
"ring_theory.localization.submodule",
"ring_theory.noetherian",
"ring_theory.principal_ideal_domain",
"tactic.field_simp"
] | [
"algebra",
"algebra_map",
"comm_ring",
"fractional_ideal",
"ideal.submodule_span_eq",
"is_domain",
"is_fraction_ring",
"is_principal_ideal_ring",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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