Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendovalco Structured version   Visualization version   GIF version
Theorem tendovalco 40149
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 2726 . . . . 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 2726 . . . . 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 40144 . . . 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 5851 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (𝑓 ∘ 𝑔) = (𝐹 ∘ 𝑔))
87fveq2d 6889 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘†β€˜(𝑓 ∘ 𝑔)) = (π‘†β€˜(𝐹 ∘ 𝑔)))
9 fveq2 6885 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘†β€˜π‘“) = (π‘†β€˜πΉ))
109coeq1d 5855 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”)))
118, 10eqeq12d 2742 . . . . . . 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 5852 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺))
1312fveq2d 6889 . . . . . . . 8 (𝑔 = 𝐺 β†’ (π‘†β€˜(𝐹 ∘ 𝑔)) = (π‘†β€˜(𝐹 ∘ 𝐺)))
14 fveq2 6885 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (π‘†β€˜π‘”) = (π‘†β€˜πΊ))
1514coeq2d 5856 . . . . . . . 8 (𝑔 = 𝐺 β†’ ((π‘†β€˜πΉ) ∘ (π‘†β€˜π‘”)) = ((π‘†β€˜πΉ) ∘ (π‘†β€˜πΊ)))
1613, 15eqeq12d 2742 . . . . . . 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 3617 . . . . . 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, 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 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 406 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141   ∘ ccom 5673  βŸΆwf 6533  β€˜cfv 6537  lecple 17213  LHypclh 39368  LTrncltrn 39485  trLctrl 39542  TEndoctendo 40136
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-tendo 40139
This theorem is referenced by:  tendoco2  40152  tendococl  40156  tendodi1  40168  tendoicl  40180  cdlemi2  40203  tendospdi1  40404
  Copyright terms: Public domain W3C validator