Theorems · Theorem · functional analysis

LinearIsometryEquiv.toIsometryEquiv_trans

∀ {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} {E : Type u_5} {E₂ : Type u_6} {E₃ : Type u_7} [inst : Semiring R]
  [inst_1 : Semiring R₂] [inst_2 : Semiring R₃] {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R} {σ₁₃ : R →+* R₃} {σ₃₁ : R₃ →+* R}
  {σ₂₃ : R₂ →+* R₃} {σ₃₂ : R₃ →+* R₂} [inst_3 : RingHomInvPair σ₁₂ σ₂₁] [inst_4 : RingHomInvPair σ₂₁ σ₁₂]
  [inst_5 : RingHomInvPair σ₁₃ σ₃₁] [inst_6 : RingHomInvPair σ₃₁ σ₁₃] [inst_7 : RingHomInvPair σ₂₃ σ₃₂]
  [inst_8 : RingHomInvPair σ₃₂ σ₂₃] [inst_9 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [inst_10 : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁]
  [inst_11 : SeminormedAddCommGroup E] [inst_12 : SeminormedAddCommGroup E₂] [inst_13 : SeminormedAddCommGroup E₃]
  [inst_14 : Module R E] [inst_15 : Module R₂ E₂] [inst_16 : Module R₃ E₃] (e : E ≃ₛₗᵢ[σ₁₂] E₂) (e' : E₂ ≃ₛₗᵢ[σ₂₃] E₃),
  (e.trans e').toIsometryEquiv = e.toIsometryEquiv.trans e'.toIsometryEquiv

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

0 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringSemiringRingHomInvPairRingHomInvPairRingHomInvPairRingHomInvPairRingHomInvPairRingHomInvPairRingHomCompTripleRingHomCompTripleSeminormedAddCommGroupSeminormedAddCommGroupSeminormedAddCommGroupModuleModuleModule