Theorem tendospass

Description: Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013)

	Ref	Expression
Hypotheses	tendosp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendosp.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendosp.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion	tendospass	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tendosp.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tendosp.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tendosp.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4	1 2 3	tendof	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ) → 𝑉 : 𝑇 ⟶ 𝑇 )
5	4	3ad2antr2	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → 𝑉 : 𝑇 ⟶ 𝑇 )
6		simpr3	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → 𝐹 ∈ 𝑇 )
7		fvco3	⊢ ( ( 𝑉 : 𝑇 ⟶ 𝑇 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )
8	5 6 7	syl2anc	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )