Theorem: NumberField.canonicalEmbedding.integralBasis_repr_apply
File: Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean

tactic: rw [← Basis.restrictScalars_repr_apply ℚ _ ⟨_, mem_rat_span_latticeBasis K x⟩, eq_ratCast,
  Rat.cast_inj]
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ ((latticeBasis K).repr ((canonicalEmbedding K) x)) i = ↑(((integralBasis K).repr x) i)
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(canonicalEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i

tactic: let f := (canonicalEmbedding K).toRatAlgHom.toLinearMap.codRestrict _
  (fun x ↦ mem_rat_span_latticeBasis K x)
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(canonicalEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(canonicalEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i

tactic: suffices ((latticeBasis K).restrictScalars ℚ).repr.toLinearMap ∘ₗ f =
  (integralBasis K).repr.toLinearMap from DFunLike.congr_fun (LinearMap.congr_fun this x) i
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(canonicalEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
⊢ ↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f = ↑(integralBasis K).repr

tactic: refine Basis.ext (integralBasis K) (fun i ↦ ?_)
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
⊢ ↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f = ↑(integralBasis K).repr
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ (↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)

tactic: have : f (integralBasis K i) = ((latticeBasis K).restrictScalars ℚ) i := by
  apply Subtype.val_injective
  rw [LinearMap.codRestrict_apply, AlgHom.toLinearMap_apply, Basis.restrictScalars_apply,
    latticeBasis_apply]
  rfl
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ (↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)
state_after: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
this : f ((integralBasis K) i) = (Basis.restrictScalars ℚ (latticeBasis K)) i
⊢ (↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)

tactic: simp_rw [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, this, Basis.repr_self]
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
this : f ((integralBasis K) i) = (Basis.restrictScalars ℚ (latticeBasis K)) i
⊢ (↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)
state_after: no goals

tactic: apply Subtype.val_injective
state_before: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ f ((integralBasis K) i) = (Basis.restrictScalars ℚ (latticeBasis K)) i
state_after: case a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ ↑(f ((integralBasis K) i)) = ↑((Basis.restrictScalars ℚ (latticeBasis K)) i)

tactic: rw [LinearMap.codRestrict_apply, AlgHom.toLinearMap_apply, Basis.restrictScalars_apply,
  latticeBasis_apply]
state_before: case a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ ↑(f ((integralBasis K) i)) = ↑((Basis.restrictScalars ℚ (latticeBasis K)) i)
state_after: case a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ (canonicalEmbedding K).toRatAlgHom ((integralBasis K) i) = (canonicalEmbedding K) ((integralBasis K) i)

tactic: rfl
state_before: case a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ (canonicalEmbedding K).toRatAlgHom ((integralBasis K) i) = (canonicalEmbedding K) ((integralBasis K) i)
state_after: no goals