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fix: Mathematics.SO3.Basic

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jstoobysmith 2024-11-02 07:59:39 +00:00
parent 2f2f2ff3d1
commit a532c51d64
1 changed files with 8 additions and 8 deletions
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16
HepLean/Mathematics/SO3/Basic.lean
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@ -81,18 +81,18 @@ lemma toProd_continuous : Continuous toProd :=
/-- The embedding of `SO(3)` into the monoid of matrices times the opposite of
the monoid of matrices. -/
lemma toProd_embedding : Embedding toProd where
lemma toProd_embedding : IsEmbedding toProd where
inj := toProd_injective
induced := (inducing_iff ⇑toProd).mp (inducing_of_inducing_compose toProd_continuous
continuous_fst ((inducing_iff (Prod.fst ∘ ⇑toProd)).mpr rfl))
eq_induced := (isInducing_iff ⇑toProd).mp (IsInducing.of_comp toProd_continuous
continuous_fst ((isInducing_iff (Prod.fst ∘ ⇑toProd)).mpr rfl))
/-- The embedding of `SO(3)` into `GL (Fin 3) ℝ`. -/
lemma toGL_embedding : Embedding toGL.toFun where
lemma toGL_embedding : IsEmbedding toGL.toFun where
inj := toGL_injective
induced := by
eq_induced := by
refine ((fun {X} {t t'} => TopologicalSpace.ext_iff.mpr) ?_).symm
intro s
rw [TopologicalSpace.ext_iff.mp toProd_embedding.induced s]
rw [TopologicalSpace.ext_iff.mp toProd_embedding.eq_induced s]
rw [isOpen_induced_iff, isOpen_induced_iff]
apply Iff.intro ?_ ?_
· intro h
@ -111,7 +111,7 @@ lemma toGL_embedding : Embedding toGL.toFun where
exact hU2
instance : TopologicalGroup SO(3) :=
Inducing.topologicalGroup toGL toGL_embedding.toInducing
IsInducing.topologicalGroup toGL toGL_embedding.toIsInducing
lemma det_minus_id (A : SO(3)) : det (A.1 - 1) = 0 := by
have h1 : det (A.1 - 1) = - det (A.1 - 1) :=
@ -185,7 +185,7 @@ lemma exists_stationary_vec (A : SO(3)) :
lemma exists_basis_preserved (A : SO(3)) :
∃ (b : OrthonormalBasis (Fin 3) ℝ (EuclideanSpace ℝ (Fin 3))), A.toEnd (b 0) = b 0 := by
obtain ⟨v, hv⟩ := exists_stationary_vec A
have h3 : FiniteDimensional.finrank ℝ (EuclideanSpace ℝ (Fin 3)) = Fintype.card (Fin 3) := by
have h3 : Module.finrank ℝ (EuclideanSpace ℝ (Fin 3)) = Fintype.card (Fin 3) := by
simp_all only [toEnd_apply, finrank_euclideanSpace, Fintype.card_fin]
obtain ⟨b, hb⟩ := Orthonormal.exists_orthonormalBasis_extension_of_card_eq h3 hv.1
simp only [Fin.isValue, Set.mem_singleton_iff, forall_eq] at hb