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Theorem tendovalco 35560
Description: Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
Hypotheses
Ref Expression
tendof.h 𝐻 = (LHyp‘𝐾)
tendof.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendof.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
tendovalco (((𝐾𝑉𝑊𝐻𝑆𝐸) ∧ (𝐹𝑇𝐺𝑇)) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))

Proof of Theorem tendovalco
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
2 tendof.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 tendof.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 eqid 2621 . . . . 5 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendof.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5istendo 35555 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7 coeq1 5244 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑔) = (𝐹𝑔))
87fveq2d 6157 . . . . . . . 8 (𝑓 = 𝐹 → (𝑆‘(𝑓𝑔)) = (𝑆‘(𝐹𝑔)))
9 fveq2 6153 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
109coeq1d 5248 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑆𝑓) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)))
118, 10eqeq12d 2636 . . . . . . 7 (𝑓 = 𝐹 → ((𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔))))
12 coeq2 5245 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐹𝑔) = (𝐹𝐺))
1312fveq2d 6157 . . . . . . . 8 (𝑔 = 𝐺 → (𝑆‘(𝐹𝑔)) = (𝑆‘(𝐹𝐺)))
14 fveq2 6153 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑆𝑔) = (𝑆𝐺))
1514coeq2d 5249 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑆𝐹) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
1613, 15eqeq12d 2636 . . . . . . 7 (𝑔 = 𝐺 → ((𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
1711, 16rspc2v 3310 . . . . . 6 ((𝐹𝑇𝐺𝑇) → (∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
1817com12 32 . . . . 5 (∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
19183ad2ant2 1081 . . . 4 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓)) → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
206, 19syl6bi 243 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))))
21203impia 1258 . 2 ((𝐾𝑉𝑊𝐻𝑆𝐸) → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
2221imp 445 1 (((𝐾𝑉𝑊𝐻𝑆𝐸) ∧ (𝐹𝑇𝐺𝑇)) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4618  ccom 5083  wf 5848  cfv 5852  lecple 15876  LHypclh 34777  LTrncltrn 34894  trLctrl 34952  TEndoctendo 35547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-tendo 35550
This theorem is referenced by:  tendoco2  35563  tendococl  35567  tendodi1  35579  tendoicl  35591  cdlemi2  35614  tendospdi1  35816
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