Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendotp Structured version   Visualization version   GIF version

Theorem tendotp 36571
Description: Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
Hypotheses
Ref Expression
tendoset.l = (le‘𝐾)
tendoset.h 𝐻 = (LHyp‘𝐾)
tendoset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoset.r 𝑅 = ((trL‘𝐾)‘𝑊)
tendoset.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
tendotp (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸𝐹𝑇) → (𝑅‘(𝑆𝐹)) (𝑅𝐹))

Proof of Theorem tendotp
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tendoset.l . . . 4 = (le‘𝐾)
2 tendoset.h . . . 4 𝐻 = (LHyp‘𝐾)
3 tendoset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 tendoset.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
5 tendoset.e . . . 4 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5istendo 36570 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
7 fveq2 6333 . . . . . . 7 (𝑓 = 𝐹 → (𝑆𝑓) = (𝑆𝐹))
87fveq2d 6337 . . . . . 6 (𝑓 = 𝐹 → (𝑅‘(𝑆𝑓)) = (𝑅‘(𝑆𝐹)))
9 fveq2 6333 . . . . . 6 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
108, 9breq12d 4800 . . . . 5 (𝑓 = 𝐹 → ((𝑅‘(𝑆𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
1110rspccv 3457 . . . 4 (∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓) → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
12113ad2ant3 1129 . . 3 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)) → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹)))
136, 12syl6bi 243 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 → (𝐹𝑇 → (𝑅‘(𝑆𝐹)) (𝑅𝐹))))
14133imp 1101 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸𝐹𝑇) → (𝑅‘(𝑆𝐹)) (𝑅𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4787  ccom 5254  wf 6026  cfv 6030  lecple 16156  LHypclh 35793  LTrncltrn 35910  trLctrl 35968  TEndoctendo 36562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-tendo 36565
This theorem is referenced by:  tendococl  36582  tendoid  36583  tendopltp  36590  tendoicl  36606  cdlemi1  36628  tendotr  36640  cdleml1N  36786  dva1dim  36795  dialss  36856  diblss  36980
  Copyright terms: Public domain W3C validator