Metamath Proof Explorer


Theorem tendovalco

Description: Value of composition of translations in 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 tendovalco ( ( ( 𝐾𝑉𝑊𝐻𝑆𝐸 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) )

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 ( ( 𝐾𝑉𝑊𝐻 ) → ( 𝑆𝐸 ↔ ( 𝑆 : 𝑇𝑇 ∧ ∀ 𝑓𝑇𝑔𝑇 ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) ∧ ∀ 𝑓𝑇 ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆𝑓 ) ) ( le ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ) ) )
7 coeq1 ( 𝑓 = 𝐹 → ( 𝑓𝑔 ) = ( 𝐹𝑔 ) )
8 7 fveq2d ( 𝑓 = 𝐹 → ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( 𝑆 ‘ ( 𝐹𝑔 ) ) )
9 fveq2 ( 𝑓 = 𝐹 → ( 𝑆𝑓 ) = ( 𝑆𝐹 ) )
10 9 coeq1d ( 𝑓 = 𝐹 → ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝑔 ) ) )
11 8 10 eqeq12d ( 𝑓 = 𝐹 → ( ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) ↔ ( 𝑆 ‘ ( 𝐹𝑔 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝑔 ) ) ) )
12 coeq2 ( 𝑔 = 𝐺 → ( 𝐹𝑔 ) = ( 𝐹𝐺 ) )
13 12 fveq2d ( 𝑔 = 𝐺 → ( 𝑆 ‘ ( 𝐹𝑔 ) ) = ( 𝑆 ‘ ( 𝐹𝐺 ) ) )
14 fveq2 ( 𝑔 = 𝐺 → ( 𝑆𝑔 ) = ( 𝑆𝐺 ) )
15 14 coeq2d ( 𝑔 = 𝐺 → ( ( 𝑆𝐹 ) ∘ ( 𝑆𝑔 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) )
16 13 15 eqeq12d ( 𝑔 = 𝐺 → ( ( 𝑆 ‘ ( 𝐹𝑔 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝑔 ) ) ↔ ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) ) )
17 11 16 rspc2v ( ( 𝐹𝑇𝐺𝑇 ) → ( ∀ 𝑓𝑇𝑔𝑇 ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) → ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) ) )
18 17 com12 ( ∀ 𝑓𝑇𝑔𝑇 ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) → ( ( 𝐹𝑇𝐺𝑇 ) → ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) ) )
19 18 3ad2ant2 ( ( 𝑆 : 𝑇𝑇 ∧ ∀ 𝑓𝑇𝑔𝑇 ( 𝑆 ‘ ( 𝑓𝑔 ) ) = ( ( 𝑆𝑓 ) ∘ ( 𝑆𝑔 ) ) ∧ ∀ 𝑓𝑇 ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑆𝑓 ) ) ( le ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑓 ) ) → ( ( 𝐹𝑇𝐺𝑇 ) → ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) ) )
20 6 19 syl6bi ( ( 𝐾𝑉𝑊𝐻 ) → ( 𝑆𝐸 → ( ( 𝐹𝑇𝐺𝑇 ) → ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) ) ) )
21 20 3impia ( ( 𝐾𝑉𝑊𝐻𝑆𝐸 ) → ( ( 𝐹𝑇𝐺𝑇 ) → ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) ) )
22 21 imp ( ( ( 𝐾𝑉𝑊𝐻𝑆𝐸 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝑆 ‘ ( 𝐹𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) )