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Theorem istendod 40765
Description: Deduce the predicate "is 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)
istendod.1 (𝜑 → (𝐾𝑉𝑊𝐻))
istendod.2 (𝜑𝑆:𝑇𝑇)
istendod.3 ((𝜑𝑓𝑇𝑔𝑇) → (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
istendod.4 ((𝜑𝑓𝑇) → (𝑅‘(𝑆𝑓)) (𝑅𝑓))
Assertion
Ref Expression
istendod (𝜑𝑆𝐸)
Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔   𝑆,𝑓,𝑔   ,𝑓   𝑅,𝑓   𝜑,𝑓,𝑔
Allowed substitution hints:   𝑅(𝑔)   𝐸(𝑓,𝑔)   𝐻(𝑓,𝑔)   (𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem istendod
StepHypRef Expression
1 istendod.2 . 2 (𝜑𝑆:𝑇𝑇)
2 istendod.3 . . . 4 ((𝜑𝑓𝑇𝑔𝑇) → (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
323expb 1120 . . 3 ((𝜑 ∧ (𝑓𝑇𝑔𝑇)) → (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
43ralrimivva 3201 . 2 (𝜑 → ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
5 istendod.4 . . 3 ((𝜑𝑓𝑇) → (𝑅‘(𝑆𝑓)) (𝑅𝑓))
65ralrimiva 3145 . 2 (𝜑 → ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))
7 istendod.1 . . 3 (𝜑 → (𝐾𝑉𝑊𝐻))
8 tendoset.l . . . 4 = (le‘𝐾)
9 tendoset.h . . . 4 𝐻 = (LHyp‘𝐾)
10 tendoset.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
11 tendoset.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
12 tendoset.e . . . 4 𝐸 = ((TEndo‘𝐾)‘𝑊)
138, 9, 10, 11, 12istendo 40763 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
147, 13syl 17 . 2 (𝜑 → (𝑆𝐸 ↔ (𝑆:𝑇𝑇 ∧ ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)) ∧ ∀𝑓𝑇 (𝑅‘(𝑆𝑓)) (𝑅𝑓))))
151, 4, 6, 14mpbir3and 1342 1 (𝜑𝑆𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060   class class class wbr 5142  ccom 5688  wf 6556  cfv 6560  lecple 17305  LHypclh 39987  LTrncltrn 40104  trLctrl 40161  TEndoctendo 40755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-tendo 40758
This theorem is referenced by:  tendoidcl  40772  tendococl  40775  tendoplcl  40784  tendo0cl  40793  tendoicl  40799  cdlemk56  40974
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