Theorems · Definition · algebraic topology

Bundle.ContinuousRiemannianMetric.mk.noConfusion

{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} →
                        {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} →
                                {isVonNBounded : ∀ (b : B), Bornology.IsVonNBounded ℝ {v | ((inner b) v) v < 1}} →
                                  {continuous : Continuous fun b => ⟨b, inner b⟩} →
                                    {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 := inner, symm := symm, pos := pos,
                                                    isVonNBounded := isVonNBounded, continuous := continuous } =
                                                  { inner := inner', symm := symm', pos := pos',
                                                    isVonNBounded := isVonNBounded', continuous := continuous' } →
                                                (inner ≍ inner' → P) → P

Defined in

Mathlib.Topology.VectorBundle.Riemannian

Cited by

1 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound