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Theorem tendovalco 40747
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 2734 . . . . 5 (le‘𝐾) = (le‘𝐾)
2 tendof.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 tendof.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 eqid 2734 . . . . 5 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendof.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5istendo 40742 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7 coeq1 5870 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑔) = (𝐹𝑔))
87fveq2d 6910 . . . . . . . 8 (𝑓 = 𝐹 → (𝑆‘(𝑓𝑔)) = (𝑆‘(𝐹𝑔)))
9 fveq2 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
109coeq1d 5874 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑆𝑓) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)))
118, 10eqeq12d 2750 . . . . . . 7 (𝑓 = 𝐹 → ((𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔))))
12 coeq2 5871 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐹𝑔) = (𝐹𝐺))
1312fveq2d 6910 . . . . . . . 8 (𝑔 = 𝐺 → (𝑆‘(𝐹𝑔)) = (𝑆‘(𝐹𝐺)))
14 fveq2 6906 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑆𝑔) = (𝑆𝐺))
1514coeq2d 5875 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑆𝐹) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
1613, 15eqeq12d 2750 . . . . . . 7 (𝑔 = 𝐺 → ((𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
1711, 16rspc2v 3632 . . . . . 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 1133 . . . 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, 19biimtrdi 253 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))))
21203impia 1116 . 2 ((𝐾𝑉𝑊𝐻𝑆𝐸) → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
2221imp 406 1 (((𝐾𝑉𝑊𝐻𝑆𝐸) ∧ (𝐹𝑇𝐺𝑇)) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058   class class class wbr 5147  ccom 5692  wf 6558  cfv 6562  lecple 17304  LHypclh 39966  LTrncltrn 40083  trLctrl 40140  TEndoctendo 40734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-tendo 40737
This theorem is referenced by:  tendoco2  40750  tendococl  40754  tendodi1  40766  tendoicl  40778  cdlemi2  40801  tendospdi1  41002
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