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

Theorem tendof 38007
Description: Functionality of 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
tendof (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸) → 𝑆:𝑇𝑇)

Proof of Theorem tendof
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2824 . . . 4 (le‘𝐾) = (le‘𝐾)
2 tendof.h . . . 4 𝐻 = (LHyp‘𝐾)
3 tendof.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 eqid 2824 . . . 4 ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
5 tendof.e . . . 4 𝐸 = ((TEndo‘𝐾)‘𝑊)
61, 2, 3, 4, 5istendo 38004 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓))))
7 simp1 1133 . . 3 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (((trL‘𝐾)‘𝑊)‘(𝑆𝑓))(le‘𝐾)(((trL‘𝐾)‘𝑊)‘𝑓)) → 𝑆:𝑇𝑇)
86, 7syl6bi 256 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸𝑆:𝑇𝑇))
98imp 410 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑆𝐸) → 𝑆:𝑇𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133   class class class wbr 5052  ccom 5546  wf 6339  cfv 6343  lecple 16572  LHypclh 37228  LTrncltrn 37345  trLctrl 37402  TEndoctendo 37996
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 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-tendo 37999
This theorem is referenced by:  tendoeq1  38008  tendocoval  38010  tendocl  38011  tendo1mul  38014  tendo1mulr  38015  tendococl  38016  tendoconid  38073  tendospass  38263  dvhlveclem  38352  dicvscacl  38435
  Copyright terms: Public domain W3C validator