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

Theorem istendo 36569
Description: The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (Contributed by NM, 8-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
istendo ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔   𝑆,𝑓,𝑔
Allowed substitution hints:   𝑅(𝑓,𝑔)   𝐸(𝑓,𝑔)   𝐻(𝑓,𝑔)   (𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem istendo
Dummy variable 𝑠 is 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, 5tendoset 36568 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))})
76eleq2d 2826 . 2 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸𝑆 ∈ {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))}))
8 fvex 6364 . . . . . 6 ((LTrn‘𝐾)‘𝑊) ∈ V
93, 8eqeltri 2836 . . . . 5 𝑇 ∈ V
10 fex 6655 . . . . 5 ((𝑆:𝑇𝑇𝑇 ∈ V) → 𝑆 ∈ V)
119, 10mpan2 709 . . . 4 (𝑆:𝑇𝑇𝑆 ∈ V)
12113ad2ant1 1128 . . 3 ((𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)) → 𝑆 ∈ V)
13 feq1 6188 . . . 4 (𝑠 = 𝑆 → (𝑠:𝑇𝑇𝑆:𝑇𝑇))
14 fveq1 6353 . . . . . 6 (𝑠 = 𝑆 → (𝑠‘(𝑓𝑔)) = (𝑆‘(𝑓𝑔)))
15 fveq1 6353 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑓) = (𝑆𝑓))
16 fveq1 6353 . . . . . . 7 (𝑠 = 𝑆 → (𝑠𝑔) = (𝑆𝑔))
1715, 16coeq12d 5443 . . . . . 6 (𝑠 = 𝑆 → ((𝑠𝑓) ∘ (𝑠𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
1814, 17eqeq12d 2776 . . . . 5 (𝑠 = 𝑆 → ((𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔))))
19182ralbidv 3128 . . . 4 (𝑠 = 𝑆 → (∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ↔ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔))))
2015fveq2d 6358 . . . . . 6 (𝑠 = 𝑆 → (𝑅‘(𝑠𝑓)) = (𝑅‘(𝑆𝑓)))
2120breq1d 4815 . . . . 5 (𝑠 = 𝑆 → ((𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
2221ralbidv 3125 . . . 4 (𝑠 = 𝑆 → (∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓) ↔ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
2313, 19, 223anbi123d 1548 . . 3 (𝑠 = 𝑆 → ((𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓)) ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
2412, 23elab3 3499 . 2 (𝑆 ∈ {𝑠 ∣ (𝑠:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑠‘(𝑓𝑔)) = ((𝑠𝑓) ∘ (𝑠𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑠𝑓)) (𝑅𝑓))} ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓)))
257, 24syl6bb 276 1 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  {cab 2747  wral 3051  Vcvv 3341   class class class wbr 4805  ccom 5271  wf 6046  cfv 6050  lecple 16171  LHypclh 35792  LTrncltrn 35909  trLctrl 35967  TEndoctendo 36561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-tendo 36564
This theorem is referenced by:  tendotp  36570  istendod  36571  tendof  36572  tendovalco  36574
  Copyright terms: Public domain W3C validator