Theorems · Theorem · functional analysis

ContinuousLinearMap.comp_assoc

∀ {R₁ : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [inst : Semiring R₁] [inst_1 : Semiring R₂] [inst_2 : Semiring R₃]
  {σ₁₂ : R₁ →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R₁ →+* R₃} {M₁ : Type u_4} [inst_3 : TopologicalSpace M₁]
  [inst_4 : AddCommMonoid M₁] {M₂ : Type u_6} [inst_5 : TopologicalSpace M₂] [inst_6 : AddCommMonoid M₂] {M₃ : Type u_7}
  [inst_7 : TopologicalSpace M₃] [inst_8 : AddCommMonoid M₃] {M₄ : Type u_8} [inst_9 : TopologicalSpace M₄]
  [inst_10 : AddCommMonoid M₄] [inst_11 : Module R₁ M₁] [inst_12 : Module R₂ M₂] [inst_13 : Module R₃ M₃]
  [inst_14 : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] {R₄ : Type u_9} [inst_15 : Semiring R₄] [inst_16 : Module R₄ M₄]
  {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄} {σ₃₄ : R₃ →+* R₄} [inst_17 : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄]
  [inst_18 : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄] [inst_19 : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄)
  (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂), (h ∘SL g) ∘SL f = h ∘SL g ∘SL f

Defined in

Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic

Cited by

14 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringSemiringTopologicalSpaceAddCommMonoidTopologicalSpaceAddCommMonoidTopologicalSpaceAddCommMonoidTopologicalSpaceAddCommMonoidModuleModuleModuleRingHomCompTripleSemiringModuleRingHomCompTripleRingHomCompTripleRingHomCompTriple