Theorem istendod

Description: Deduce the predicate "is 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 )
	istendod.1	⊢ ( 𝜑 → ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) )
	istendod.2	⊢ ( 𝜑 → 𝑆 : 𝑇 ⟶ 𝑇 )
	istendod.3	⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇 ) → ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) )
	istendod.4	⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) )
Assertion	istendod	⊢ ( 𝜑 → 𝑆 ∈ 𝐸 )

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		istendod.1	⊢ ( 𝜑 → ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) )
7		istendod.2	⊢ ( 𝜑 → 𝑆 : 𝑇 ⟶ 𝑇 )
8		istendod.3	⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇 ) → ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) )
9		istendod.4	⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) )
10	8	3expb	⊢ ( ( 𝜑 ∧ ( 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇 ) ) → ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) )
11	10	ralrimivva	⊢ ( 𝜑 → ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) )
12	9	ralrimiva	⊢ ( 𝜑 → ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) )
13	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 ) ) ) )
14	6 13	syl	⊢ ( 𝜑 → ( 𝑆 ∈ 𝐸 ↔ ( 𝑆 : 𝑇 ⟶ 𝑇 ∧ ∀ 𝑓 ∈ 𝑇 ∀ 𝑔 ∈ 𝑇 ( 𝑆 ‘ ( 𝑓 ∘ 𝑔 ) ) = ( ( 𝑆 ‘ 𝑓 ) ∘ ( 𝑆 ‘ 𝑔 ) ) ∧ ∀ 𝑓 ∈ 𝑇 ( 𝑅 ‘ ( 𝑆 ‘ 𝑓 ) ) ≤ ( 𝑅 ‘ 𝑓 ) ) ) )
15	7 11 12 14	mpbir3and	⊢ ( 𝜑 → 𝑆 ∈ 𝐸 )