Theorem tendocl

Description: Closure of 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	tendocl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 )

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	1 2 3	tendof	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝑆 : 𝑇 ⟶ 𝑇 )
5	4	3adant3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → 𝑆 : 𝑇 ⟶ 𝑇 )
6		simp3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 )
7	5 6	ffvelcdmd	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 )