Theorems · Theorem · algebraic topology

Bundle.ContinuousRiemannianMetric.mk.inj

∀ {B : Type u_4} {inst : TopologicalSpace B} {F : Type u_5} {inst_1 : NormedAddCommGroup F} {inst_2 : NormedSpace ℝ F}
  {E : B → Type u_6} {inst_3 : TopologicalSpace (Bundle.TotalSpace F E)} {inst_4 : (b : B) → TopologicalSpace (E b)}
  {inst_5 : (b : B) → AddCommGroup (E b)} {inst_6 : (b : B) → Module ℝ (E b)} {inst_7 : FiberBundle F E}
  {inst_8 : VectorBundle ℝ F E} {inner : (b : B) → E b →L[ℝ] E b →L[ℝ] ℝ}
  {symm : ∀ (b : B) (v w : E b), ((inner b) v) w = ((inner b) w) v}
  {pos : ∀ (b : B) (v : E b), v ≠ 0 → 0 < ((inner b) v) v}
  {isVonNBounded : ∀ (b : B), Bornology.IsVonNBounded ℝ {v | ((inner b) v) v < 1}}
  {continuous : Continuous fun b => ⟨b, inner b⟩} {inner_1 : (b : B) → E b →L[ℝ] E b →L[ℝ] ℝ}
  {symm_1 : ∀ (b : B) (v w : E b), ((inner_1 b) v) w = ((inner_1 b) w) v}
  {pos_1 : ∀ (b : B) (v : E b), v ≠ 0 → 0 < ((inner_1 b) v) v}
  {isVonNBounded_1 : ∀ (b : B), Bornology.IsVonNBounded ℝ {v | ((inner_1 b) v) v < 1}}
  {continuous_1 : Continuous fun b => ⟨b, inner_1 b⟩},
  { inner := inner, symm := symm, pos := pos, isVonNBounded := isVonNBounded, continuous := continuous } =
      { inner := inner_1, symm := symm_1, pos := pos_1, isVonNBounded := isVonNBounded_1, continuous := continuous_1 } →
    inner = inner_1

Defined in

Mathlib.Topology.VectorBundle.Riemannian

Cited by

1 results in Mathlib

Foundations

Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound