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Theorem tendovalco 36840
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 2825 . . . . 5 (le‘𝐾) = (le‘𝐾)
2 tendof.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 tendof.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 eqid 2825 . . . . 5 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendof.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5istendo 36835 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7 coeq1 5512 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑔) = (𝐹𝑔))
87fveq2d 6437 . . . . . . . 8 (𝑓 = 𝐹 → (𝑆‘(𝑓𝑔)) = (𝑆‘(𝐹𝑔)))
9 fveq2 6433 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
109coeq1d 5516 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑆𝑓) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)))
118, 10eqeq12d 2840 . . . . . . 7 (𝑓 = 𝐹 → ((𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔))))
12 coeq2 5513 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐹𝑔) = (𝐹𝐺))
1312fveq2d 6437 . . . . . . . 8 (𝑔 = 𝐺 → (𝑆‘(𝐹𝑔)) = (𝑆‘(𝐹𝐺)))
14 fveq2 6433 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑆𝑔) = (𝑆𝐺))
1514coeq2d 5517 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑆𝐹) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
1613, 15eqeq12d 2840 . . . . . . 7 (𝑔 = 𝐺 → ((𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
1711, 16rspc2v 3539 . . . . . 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 1170 . . . 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 245 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))))
21203impia 1151 . 2 ((𝐾𝑉𝑊𝐻𝑆𝐸) → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
2221imp 397 1 (((𝐾𝑉𝑊𝐻𝑆𝐸) ∧ (𝐹𝑇𝐺𝑇)) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117   class class class wbr 4873  ccom 5346  wf 6119  cfv 6123  lecple 16312  LHypclh 36059  LTrncltrn 36176  trLctrl 36233  TEndoctendo 36827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-tendo 36830
This theorem is referenced by:  tendoco2  36843  tendococl  36847  tendodi1  36859  tendoicl  36871  cdlemi2  36894  tendospdi1  37095
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