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Theorem tendovalco 39231
Description: Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
Hypotheses
Ref Expression
tendof.h 𝐻 = (LHypβ€˜πΎ)
tendof.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
tendof.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
tendovalco (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ)))

Proof of Theorem tendovalco
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
2 tendof.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
3 tendof.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
4 eqid 2737 . . . . 5 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
5 tendof.e . . . . 5 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5istendo 39226 . . . 4 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 ↔ (𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜(π‘†β€˜π‘“))(leβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“))))
7 coeq1 5814 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
87fveq2d 6847 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘†β€˜(𝑓 ∘ 𝑔)) = (π‘†β€˜(𝐹 ∘ 𝑔)))
9 fveq2 6843 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘†β€˜π‘“) = (π‘†β€˜πΉ))
109coeq1d 5818 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”)))
118, 10eqeq12d 2753 . . . . . . 7 (𝑓 = 𝐹 β†’ ((π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ↔ (π‘†β€˜(𝐹 ∘ 𝑔)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”))))
12 coeq2 5815 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺))
1312fveq2d 6847 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘†β€˜(𝐹 ∘ 𝑔)) = (π‘†β€˜(𝐹 ∘ 𝐺)))
14 fveq2 6843 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘†β€˜π‘”) = (π‘†β€˜πΊ))
1514coeq2d 5819 . . . . . . . 8 (𝑔 = 𝐺 β†’ ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ)))
1613, 15eqeq12d 2753 . . . . . . 7 (𝑔 = 𝐺 β†’ ((π‘†β€˜(𝐹 ∘ 𝑔)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”)) ↔ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ))))
1711, 16rspc2v 3591 . . . . . 6 ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ))))
1817com12 32 . . . . 5 (βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) β†’ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ))))
19183ad2ant2 1135 . . . 4 ((𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜(π‘†β€˜π‘“))(leβ€˜πΎ)(((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)) β†’ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ))))
206, 19syl6bi 253 . . 3 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 β†’ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ)))))
21203impia 1118 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) β†’ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ))))
2221imp 408 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ (π‘†β€˜(𝐹 ∘ 𝐺)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   class class class wbr 5106   ∘ ccom 5638  βŸΆwf 6493  β€˜cfv 6497  lecple 17141  LHypclh 38450  LTrncltrn 38567  trLctrl 38624  TEndoctendo 39218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-tendo 39221
This theorem is referenced by:  tendoco2  39234  tendococl  39238  tendodi1  39250  tendoicl  39262  cdlemi2  39285  tendospdi1  39486
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