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Theorem tendovalco 40293
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 2725 . . . . 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 2725 . . . . 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 40288 . . . 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 5854 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
87fveq2d 6895 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘†β€˜(𝑓 ∘ 𝑔)) = (π‘†β€˜(𝐹 ∘ 𝑔)))
9 fveq2 6891 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘†β€˜π‘“) = (π‘†β€˜πΉ))
109coeq1d 5858 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”)))
118, 10eqeq12d 2741 . . . . . . 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 5855 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺))
1312fveq2d 6895 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘†β€˜(𝐹 ∘ 𝑔)) = (π‘†β€˜(𝐹 ∘ 𝐺)))
14 fveq2 6891 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘†β€˜π‘”) = (π‘†β€˜πΊ))
1514coeq2d 5859 . . . . . . . 8 (𝑔 = 𝐺 β†’ ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ)))
1613, 15eqeq12d 2741 . . . . . . 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 3613 . . . . . 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 1131 . . . 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, 19biimtrdi 252 . . 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 1114 . 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 405 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051   class class class wbr 5143   ∘ ccom 5676  βŸΆwf 6538  β€˜cfv 6542  lecple 17237  LHypclh 39512  LTrncltrn 39629  trLctrl 39686  TEndoctendo 40280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-map 8843  df-tendo 40283
This theorem is referenced by:  tendoco2  40296  tendococl  40300  tendodi1  40312  tendoicl  40324  cdlemi2  40347  tendospdi1  40548
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