Theorem: NumberField.is_primitive_element_of_infinitePlace_lt
File: Mathlib/NumberTheory/NumberField/Embeddings.lean

tactic: rw [Field.primitive_element_iff_algHom_eq_of_eval ℚ ℂ ?_ _ w.embedding.toRatAlgHom]
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ℚ⟮↑x⟯ = ⊤
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (ψ : K →ₐ[ℚ] ℂ), w.embedding.toRatAlgHom ↑x = ψ ↑x → w.embedding.toRatAlgHom = ψ

k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (x : K), Polynomial.Splits (algebraMap ℚ ℂ) (minpoly ℚ x)

tactic: intro ψ hψ
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (ψ : K →ₐ[ℚ] ℂ), w.embedding.toRatAlgHom ↑x = ψ ↑x → w.embedding.toRatAlgHom = ψ
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
⊢ w.embedding.toRatAlgHom = ψ

tactic: have h : 1 ≤ w x := one_le_of_lt_one h₁ h₂
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
⊢ w.embedding.toRatAlgHom = ψ
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
⊢ w.embedding.toRatAlgHom = ψ

tactic: have main : w = InfinitePlace.mk ψ.toRingHom := by
  erw [← norm_embedding_eq, hψ] at h
  contrapose! h
  exact h₂ h.symm
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
⊢ w.embedding.toRatAlgHom = ψ
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w = mk ψ.toRingHom
⊢ w.embedding.toRatAlgHom = ψ

tactic: rw [(mk_embedding w).symm, mk_eq_iff] at main
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w = mk ψ.toRingHom
⊢ w.embedding.toRatAlgHom = ψ
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ w.embedding.toRatAlgHom = ψ

tactic: cases h₃ with
| inl hw =>
  rw [conjugate_embedding_eq_of_isReal hw, or_self] at main
  exact congr_arg RingHom.toRatAlgHom main
| inr hw =>
  refine congr_arg RingHom.toRatAlgHom (main.resolve_right fun h' ↦ hw.not_le ?_)
  have : (embedding w x).im = 0 := by
    erw [← Complex.conj_eq_iff_im, RingHom.congr_fun h' x]
    exact hψ.symm
  rwa [← norm_embedding_eq, ← Complex.re_add_im (embedding w x), this, Complex.ofReal_zero,
    zero_mul, add_zero, Complex.norm_eq_abs, Complex.abs_ofReal] at h
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ w.embedding.toRatAlgHom = ψ
state_after: no goals

tactic: erw [← norm_embedding_eq, hψ] at h
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
⊢ w = mk ψ.toRingHom
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ ‖ψ ↑x‖
⊢ w = mk ψ.toRingHom

tactic: contrapose! h
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ ‖ψ ↑x‖
⊢ w = mk ψ.toRingHom
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : w ≠ mk ψ.toRingHom
⊢ ‖ψ ↑x‖ < 1

tactic: exact h₂ h.symm
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : w ≠ mk ψ.toRingHom
⊢ ‖ψ ↑x‖ < 1
state_after: no goals

tactic: rw [conjugate_embedding_eq_of_isReal hw, or_self] at main
state_before: case inl
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : w.IsReal
⊢ w.embedding.toRatAlgHom = ψ
state_after: case inl
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom
hw : w.IsReal
⊢ w.embedding.toRatAlgHom = ψ

tactic: exact congr_arg RingHom.toRatAlgHom main
state_before: case inl
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom
hw : w.IsReal
⊢ w.embedding.toRatAlgHom = ψ
state_after: no goals

tactic: refine congr_arg RingHom.toRatAlgHom (main.resolve_right fun h' ↦ hw.not_le ?_)
state_before: case inr
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
⊢ w.embedding.toRatAlgHom = ψ
state_after: case inr
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ 1 ≤ |(w.embedding ↑x).re|

tactic: have : (embedding w x).im = 0 := by
  erw [← Complex.conj_eq_iff_im, RingHom.congr_fun h' x]
  exact hψ.symm
state_before: case inr
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ 1 ≤ |(w.embedding ↑x).re|
state_after: case inr
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
this : (w.embedding ↑x).im = 0
⊢ 1 ≤ |(w.embedding ↑x).re|

tactic: rwa [← norm_embedding_eq, ← Complex.re_add_im (embedding w x), this, Complex.ofReal_zero,
  zero_mul, add_zero, Complex.norm_eq_abs, Complex.abs_ofReal] at h
state_before: case inr
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
this : (w.embedding ↑x).im = 0
⊢ 1 ≤ |(w.embedding ↑x).re|
state_after: no goals

tactic: erw [← Complex.conj_eq_iff_im, RingHom.congr_fun h' x]
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ (w.embedding ↑x).im = 0
state_after: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ ψ.toRingHom ↑x = w.embedding ↑x

tactic: exact hψ.symm
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
ψ : K →ₐ[ℚ] ℂ
hψ : w.embedding.toRatAlgHom ↑x = ψ ↑x
h : 1 ≤ w ↑x
main : w.embedding = ψ.toRingHom ∨ ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
hw : |(w.embedding ↑x).re| < 1
h' : ComplexEmbedding.conjugate w.embedding = ψ.toRingHom
⊢ ψ.toRingHom ↑x = w.embedding ↑x
state_after: no goals

tactic: exact fun x ↦ IsAlgClosed.splits_codomain (minpoly ℚ x)
state_before: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
F : Type u_3
inst✝¹ : Field F
inst✝ : NumberField K
x : 𝓞 K
w : InfinitePlace K
h₁ : x ≠ 0
h₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1
h₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1
⊢ ∀ (x : K), Polynomial.Splits (algebraMap ℚ ℂ) (minpoly ℚ x)
state_after: no goals