Theorems · Theorem · algebraic topology

Bundle.ContinuousRiemannianMetric.mk.injEq

∀ {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

0 results in Mathlib

Foundations

Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceAddCommGroupModuleFiberBundleVectorBundle