Theorem tendotp

Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013)

	Ref	Expression
Hypotheses	tendoset.l	⊢ ≤ = ( le ‘ 𝐾 )
	tendoset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	tendoset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	tendoset.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	tendoset.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
Assertion	tendotp	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		tendoset.l	⊢ ≤ = ( le ‘ 𝐾 )
2		tendoset.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		tendoset.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		tendoset.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5		tendoset.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6	1 2 3 4 5	istendo	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 ↔ ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) )
7		2fveq3	⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) = ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) )
8		fveq2	⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐹 ) )
9	7 8	breq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ↔ ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) )
10	9	rspccv	⊢ ( ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) )
11	10	3ad2ant3	⊢ ( ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) )
12	6 11	syl6bi	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 → ( 𝐹 ∈ 𝑇 → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) ) ) )
13	12	3imp	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑆 ‘ 𝐹 ) ) ≤ ( 𝑅 ‘ 𝐹 ) )