Theorem: LinearIsometry.extend_apply
File: Mathlib/Analysis/InnerProductSpace/PiL2.lean

tactic: haveI : CompleteSpace S := FiniteDimensional.complete 𝕜 S
state_before: ι : Type ?u.1751792
ι' : Type ?u.1751795
𝕜 : Type u_1
inst✝¹² : IsROrC 𝕜
E : Type ?u.1751804
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace 𝕜 E
E' : Type ?u.1751824
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
F : Type ?u.1751842
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type ?u.1751862
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_2
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V
s : { x // x ∈ S }
⊢ ↑(extend L) ↑s = ↑L s

tactic: simp only [LinearIsometry.extend, ← LinearIsometry.coe_toLinearMap]
state_before: ι : Type ?u.1751792
ι' : Type ?u.1751795
𝕜 : Type u_1
inst✝¹² : IsROrC 𝕜
E : Type ?u.1751804
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace 𝕜 E
E' : Type ?u.1751824
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
F : Type ?u.1751842
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type ?u.1751862
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_2
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V
s : { x // x ∈ S }
this : CompleteSpace { x // x ∈ S }
⊢ ↑(extend L) ↑s = ↑L s

tactic: simp only [add_right_eq_self, LinearIsometry.coe_toLinearMap,
  LinearIsometryEquiv.coe_toLinearIsometry, LinearIsometry.coe_comp, Function.comp_apply,
  orthogonalProjection_mem_subspace_eq_self, LinearMap.coe_comp, ContinuousLinearMap.coe_coe,
  Submodule.coeSubtype, LinearMap.add_apply, Submodule.coe_eq_zero,
  LinearIsometryEquiv.map_eq_zero_iff, Submodule.coe_subtypeₗᵢ,
  orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.orthogonal_orthogonal,
  Submodule.coe_mem]
state_before: ι : Type ?u.1751792
ι' : Type ?u.1751795
𝕜 : Type u_1
inst✝¹² : IsROrC 𝕜
E : Type ?u.1751804
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : InnerProductSpace 𝕜 E
E' : Type ?u.1751824
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : InnerProductSpace 𝕜 E'
F : Type ?u.1751842
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : InnerProductSpace ℝ F
F' : Type ?u.1751862
inst✝⁵ : NormedAddCommGroup F'
inst✝⁴ : InnerProductSpace ℝ F'
inst✝³ : Fintype ι
V : Type u_2
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
S : Submodule 𝕜 V
L✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V
s : { x // x ∈ S }
this : CompleteSpace { x // x ∈ S }
⊢ ↑(LinearMap.comp L.toLinearMap ↑(orthogonalProjection S) +
          LinearMap.comp
            (comp (subtypeₗᵢ (LinearMap.range L.toLinearMap)ᗮ)
                (LinearIsometryEquiv.toLinearIsometry
                  (LinearIsometryEquiv.trans (stdOrthonormalBasis 𝕜 { x // x ∈ Sᗮ }).repr
                    (LinearIsometryEquiv.symm
                      (OrthonormalBasis.reindex (stdOrthonormalBasis 𝕜 { x // x ∈ (LinearMap.range L.toLinearMap)ᗮ })
                          (finCongr
                            (_ :
                              finrank 𝕜 { x // x ∈ (LinearMap.range L.toLinearMap)ᗮ } =
                                finrank 𝕜 { x // x ∈ Sᗮ }))).repr)))).toLinearMap
            ↑(orthogonalProjection Sᗮ))
      ↑s =
    ↑L.toLinearMap s