Theorem tendof

Description: Functionality 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	tendof	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝑆 : 𝑇 ⟶ 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		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
5		eqid	⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6	4 1 2 5 3	istendo	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 ↔ ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ 𝑓 ) ) ( le ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ) ) )
7		simp1	⊢ ( ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆 ‘ 𝑓 ) ) ( le ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ) → 𝑆 : 𝑇 ⟶ 𝑇 )
8	6 7	syl6bi	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 → 𝑆 : 𝑇 ⟶ 𝑇 ) )
9	8	imp	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ) → 𝑆 : 𝑇 ⟶ 𝑇 )