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Theorem tendovalco 37961
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 2824 . . . . 5 (le‘𝐾) = (le‘𝐾)
2 tendof.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 tendof.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 eqid 2824 . . . . 5 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendof.e . . . . 5 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5istendo 37956 . . . 4 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7 coeq1 5709 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑔) = (𝐹𝑔))
87fveq2d 6655 . . . . . . . 8 (𝑓 = 𝐹 → (𝑆‘(𝑓𝑔)) = (𝑆‘(𝐹𝑔)))
9 fveq2 6651 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
109coeq1d 5713 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑆𝑓) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)))
118, 10eqeq12d 2840 . . . . . . 7 (𝑓 = 𝐹 → ((𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔))))
12 coeq2 5710 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐹𝑔) = (𝐹𝐺))
1312fveq2d 6655 . . . . . . . 8 (𝑔 = 𝐺 → (𝑆‘(𝐹𝑔)) = (𝑆‘(𝐹𝐺)))
14 fveq2 6651 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑆𝑔) = (𝑆𝐺))
1514coeq2d 5714 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑆𝐹) ∘ (𝑆𝑔)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
1613, 15eqeq12d 2840 . . . . . . 7 (𝑔 = 𝐺 → ((𝑆‘(𝐹𝑔)) = ((𝑆𝐹) ∘ (𝑆𝑔)) ↔ (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
1711, 16rspc2v 3618 . . . . . 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 1131 . . . 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 256 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))))
21203impia 1114 . 2 ((𝐾𝑉𝑊𝐻𝑆𝐸) → ((𝐹𝑇𝐺𝑇) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺))))
2221imp 410 1 (((𝐾𝑉𝑊𝐻𝑆𝐸) ∧ (𝐹𝑇𝐺𝑇)) → (𝑆‘(𝐹𝐺)) = ((𝑆𝐹) ∘ (𝑆𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132   class class class wbr 5047  ccom 5540  wf 6332  cfv 6336  lecple 16561  LHypclh 37180  LTrncltrn 37297  trLctrl 37354  TEndoctendo 37948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-map 8391  df-tendo 37951
This theorem is referenced by:  tendoco2  37964  tendococl  37968  tendodi1  37980  tendoicl  37992  cdlemi2  38015  tendospdi1  38216
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