Theorem tendocoval

Description: Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013)

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

Proof

Step	Hyp	Ref	Expression
1		tendof.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		tendof.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		tendof.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		simp1	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) )
5		simp2r	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑉 ∈ 𝐸 )
6	1 2 3	tendof	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑉 ∈ 𝐸 ) → 𝑉 : 𝑇 ⟶ 𝑇 )
7	4 5 6	syl2anc	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑉 : 𝑇 ⟶ 𝑇 )
8		simp3	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 )
9		fvco3	⊢ ( ( 𝑉 : 𝑇 ⟶ 𝑇 ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )
10	7 8 9	syl2anc	⊢ ( ( ( 𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑈 ∘ 𝑉 ) ‘ 𝐹 ) = ( 𝑈 ‘ ( 𝑉 ‘ 𝐹 ) ) )