@@ -169,8 +169,8 @@ lemma comap_map_eq_map_of_isLocalization_algebraMapSubmonoid :
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rw [IsScalarTower.algebraMap_eq R S Sₚ, ← Ideal.map_map, eq_comm]
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apply Ideal.le_comap_map.antisymm
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intro x hx
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- obtain ⟨α, hα, hαx⟩ : ∃ α ∉ p, α • x ∈ pS
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- · have ⟨⟨y, s⟩, hy⟩ := (IsLocalization.mem_map_algebraMap_iff
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+ obtain ⟨α, hα, hαx⟩ : ∃ α ∉ p, α • x ∈ pS := by
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+ have ⟨⟨y, s⟩, hy⟩ := (IsLocalization.mem_map_algebraMap_iff
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(Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ).mp hx
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rw [← map_mul,
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IsLocalization.eq_iff_exists (Algebra.algebraMapSubmonoid S p.primeCompl)] at hy
@@ -179,8 +179,8 @@ lemma comap_map_eq_map_of_isLocalization_algebraMapSubmonoid :
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refine ⟨α, hα, ?_⟩
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rw [Algebra.smul_def, e, Submonoid.coe_mul, mul_assoc, mul_comm _ x, hc]
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exact Ideal.mul_mem_left _ _ y.prop
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- obtain ⟨β, γ, hγ, hβ⟩ : ∃ β γ, γ ∈ p ∧ β * α = 1 + γ
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- · obtain ⟨β, hβ⟩ := Ideal.Quotient.mk_surjective (I := p) (Ideal.Quotient.mk p α)⁻¹
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+ obtain ⟨β, γ, hγ, hβ⟩ : ∃ β γ, γ ∈ p ∧ β * α = 1 + γ := by
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+ obtain ⟨β, hβ⟩ := Ideal.Quotient.mk_surjective (I := p) (Ideal.Quotient.mk p α)⁻¹
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refine ⟨β, β * α - 1 , ?_, ?_⟩
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· rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one,
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map_mul, hβ, inv_mul_cancel, sub_self]
@@ -210,8 +210,8 @@ def quotMapEquivQuotMapMaximalIdealOfIsLocalization : S ⧸ pS ≃+* Sₚ ⧸ pS
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obtain ⟨x, s, rfl⟩ := IsLocalization.mk'_surjective
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(Algebra.algebraMapSubmonoid S p.primeCompl) x
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obtain ⟨α, hα : α ∉ p, e⟩ := s.prop
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- obtain ⟨β, γ, hγ, hβ⟩ : ∃ β γ, γ ∈ p ∧ α * β = 1 + γ
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- · obtain ⟨β, hβ⟩ := Ideal.Quotient.mk_surjective (I := p) (Ideal.Quotient.mk p α)⁻¹
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+ obtain ⟨β, γ, hγ, hβ⟩ : ∃ β γ, γ ∈ p ∧ α * β = 1 + γ := by
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+ obtain ⟨β, hβ⟩ := Ideal.Quotient.mk_surjective (I := p) (Ideal.Quotient.mk p α)⁻¹
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refine ⟨β, α * β - 1 , ?_, ?_⟩
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· rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one,
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map_mul, hβ, mul_inv_cancel, sub_self]
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