Theorems · Definition · algebraic topology

Bundle.RiemannianMetric.mk.noConfusion

{B : Type u_4} →
  {E : B → Type u_6} →
    {inst : (b : B) → TopologicalSpace (E b)} →
      {inst_1 : (b : B) → AddCommGroup (E b)} →
        {inst_2 : (b : B) → Module ℝ (E b)} →
          {P : Sort u} →
            {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} →
                  {continuousAt : ∀ (b : B), ContinuousAt (fun v => ((inner b) v) v) 0} →
                    {isVonNBounded : ∀ (b : B), Bornology.IsVonNBounded ℝ {v | ((inner b) v) v < 1}} →
                      {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} →
                            {continuousAt' : ∀ (b : B), ContinuousAt (fun v => ((inner' b) v) v) 0} →
                              {isVonNBounded' : ∀ (b : B), Bornology.IsVonNBounded ℝ {v | ((inner' b) v) v < 1}} →
                                { inner := inner, symm := symm, pos := pos, continuousAt := continuousAt,
                                      isVonNBounded := isVonNBounded } =
                                    { inner := inner', symm := symm', pos := pos', continuousAt := continuousAt',
                                      isVonNBounded := isVonNBounded' } →
                                  (inner ≍ inner' → P) → P

Defined in

Mathlib.Topology.VectorBundle.Riemannian

Cited by

1 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound