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Theorem tendovalco 36574
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 2771 . . . . 5 (le‘𝐾) = (le‘𝐾)
2 tendof.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 tendof.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 eqid 2771 . . . . 5 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendof.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5istendo 36569 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7 coeq1 5418 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑔) = (𝐹𝑔))
87fveq2d 6336 . . . . . . . 8 (𝑓 = 𝐹 → (𝑆‘(𝑓𝑔)) = (𝑆‘(𝐹𝑔)))
9 fveq2 6332 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
109coeq1d 5422 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑆𝑓) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)))
118, 10eqeq12d 2786 . . . . . . 7 (𝑓 = 𝐹 → ((𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔))))
12 coeq2 5419 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐹𝑔) = (𝐹𝐺))
1312fveq2d 6336 . . . . . . . 8 (𝑔 = 𝐺 → (𝑆‘(𝐹𝑔)) = (𝑆‘(𝐹𝐺)))
14 fveq2 6332 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑆𝑔) = (𝑆𝐺))
1514coeq2d 5423 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑆𝐹) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
1613, 15eqeq12d 2786 . . . . . . 7 (𝑔 = 𝐺 → ((𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
1711, 16rspc2v 3472 . . . . . 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 1128 . . . 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 1109 . 2 ((𝐾𝑉𝑊𝐻𝑆𝐸) → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
2221imp 393 1 (((𝐾𝑉𝑊𝐻𝑆𝐸) ∧ (𝐹𝑇𝐺𝑇)) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4786  ccom 5253  wf 6027  cfv 6031  lecple 16156  LHypclh 35792  LTrncltrn 35909  trLctrl 35967  TEndoctendo 36561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-tendo 36564
This theorem is referenced by:  tendoco2  36577  tendococl  36581  tendodi1  36593  tendoicl  36605  cdlemi2  36628  tendospdi1  36830
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