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Theorem istendod 38057
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 1117 . . 3 ((𝜑 ∧ (𝑓𝑇𝑔𝑇)) → (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
43ralrimivva 3159 . 2 (𝜑 → ∀𝑓𝑇𝑔𝑇 (𝑆‘(𝑓𝑔)) = ((𝑆𝑓) ∘ (𝑆𝑔)))
5 istendod.4 . . 3 ((𝜑𝑓𝑇) → (𝑅‘(𝑆𝑓)) (𝑅𝑓))
65ralrimiva 3152 . 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 38055 . . 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 1339 1 (𝜑𝑆𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109   class class class wbr 5033  ccom 5527  wf 6324  cfv 6328  lecple 16568  LHypclh 37279  LTrncltrn 37396  trLctrl 37453  TEndoctendo 38047
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-tendo 38050
This theorem is referenced by:  tendoidcl  38064  tendococl  38067  tendoplcl  38076  tendo0cl  38085  tendoicl  38091  cdlemk56  38266
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