Metamath Proof Explorer


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 ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ 𝑆𝐸𝐹𝑇 ) → ( 𝑅 ‘ ( 𝑆𝐹 ) ) ( 𝑅𝐹 ) )

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 ( ( 𝐾𝑉𝑊𝐻 ) → ( 𝑆𝐸 ↔ ( 𝑆 : 𝑇𝑇 ∧ ∀ 𝑓𝑇𝑔𝑇 ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) ∧ ∀ 𝑓𝑇 ( 𝑅 ‘ ( 𝑆𝑓 ) ) ( 𝑅𝑓 ) ) ) )
7 2fveq3 ( 𝑓 = 𝐹 → ( 𝑅 ‘ ( 𝑆𝑓 ) ) = ( 𝑅 ‘ ( 𝑆𝐹 ) ) )
8 fveq2 ( 𝑓 = 𝐹 → ( 𝑅𝑓 ) = ( 𝑅𝐹 ) )
9 7 8 breq12d ( 𝑓 = 𝐹 → ( ( 𝑅 ‘ ( 𝑆𝑓 ) ) ( 𝑅𝑓 ) ↔ ( 𝑅 ‘ ( 𝑆𝐹 ) ) ( 𝑅𝐹 ) ) )
10 9 rspccv ( ∀ 𝑓𝑇 ( 𝑅 ‘ ( 𝑆𝑓 ) ) ( 𝑅𝑓 ) → ( 𝐹𝑇 → ( 𝑅 ‘ ( 𝑆𝐹 ) ) ( 𝑅𝐹 ) ) )
11 10 3ad2ant3 ( ( 𝑆 : 𝑇𝑇 ∧ ∀ 𝑓𝑇𝑔𝑇 ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) ∧ ∀ 𝑓𝑇 ( 𝑅 ‘ ( 𝑆𝑓 ) ) ( 𝑅𝑓 ) ) → ( 𝐹𝑇 → ( 𝑅 ‘ ( 𝑆𝐹 ) ) ( 𝑅𝐹 ) ) )
12 6 11 syl6bi ( ( 𝐾𝑉𝑊𝐻 ) → ( 𝑆𝐸 → ( 𝐹𝑇 → ( 𝑅 ‘ ( 𝑆𝐹 ) ) ( 𝑅𝐹 ) ) ) )
13 12 3imp ( ( ( 𝐾𝑉𝑊𝐻 ) ∧ 𝑆𝐸𝐹𝑇 ) → ( 𝑅 ‘ ( 𝑆𝐹 ) ) ( 𝑅𝐹 ) )