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Update Mathlib/Geometry/Manifold/ExistsRiemannianMetric.lean
Co-authored-by: Rida Hamadani <mridahamadani@gmail.com>
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Mathlib/Geometry/Manifold/ExistsRiemannianMetric.lean

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@@ -125,10 +125,10 @@ lemma seminormOfBilinearForm_sub_self {x : B} (φ : E x →L[ℝ] E x →L[ℝ]
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simp
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lemma seminormOfBilinearForm_sub_comm {x : B} (φ : E x →L[ℝ] E x →L[ℝ] ℝ) (hpos : ∀ v, 0 ≤ φ v v)
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(hsymm : ∀ u v, φ u v = φ v u) (hdef : ∀ v, φ v v = 0 → v = 0)
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(u v : VectorSpaceAux x φ hpos hsymm hdef) :
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seminormOfBilinearForm φ hpos hsymm (u.val - v.val) =
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seminormOfBilinearForm φ hpos hsymm (v.val - u.val) := by
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(hsymm : ∀ u v, φ u v = φ v u) (hdef : ∀ v, φ v v = 0 → v = 0)
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(u v : VectorSpaceAux x φ hpos hsymm hdef) :
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seminormOfBilinearForm φ hpos hsymm (u.val - v.val) =
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seminormOfBilinearForm φ hpos hsymm (v.val - u.val) := by
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change √((φ (u.val - v.val)) (u.val - v.val)) = √((φ (v.val - u.val)) (v.val - u.val))
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simp only [map_sub, ContinuousLinearMap.coe_sub', Pi.sub_apply]
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ring_nf

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