Theorems · Definition · functional analysis

LinearIsometryEquiv.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] →
                                          [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] →
                                            [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 ≃ₛₗᵢ[σ₁₂] E₂) → (E₂ ≃ₛₗᵢ[σ₂₃] E₃) → E ≃ₛₗᵢ[σ₁₃] E₃

Composition of LinearIsometryEquivs as a LinearIsometryEquiv.

Defined in

Mathlib.Analysis.Normed.Operator.LinearIsometry

Cited by

45 results in Mathlib

Foundations

Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringSemiringRingHomInvPairRingHomInvPairRingHomInvPairRingHomInvPairRingHomInvPairRingHomInvPairRingHomCompTripleRingHomCompTripleSeminormedAddCommGroupSeminormedAddCommGroupSeminormedAddCommGroupModuleModuleModule