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Theorem istendo 40143
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 ≀ = (leβ€˜πΎ)
tendoset.h 𝐻 = (LHypβ€˜πΎ)
tendoset.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
tendoset.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
tendoset.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
istendo ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 ↔ (𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“))))
Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,π‘Š,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 ≀ = (leβ€˜πΎ)
2 tendoset.h . . . 4 𝐻 = (LHypβ€˜πΎ)
3 tendoset.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
4 tendoset.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
5 tendoset.e . . . 4 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5tendoset 40142 . . 3 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐸 = {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
76eleq2d 2813 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 ↔ 𝑆 ∈ {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))}))
83fvexi 6898 . . . . 5 𝑇 ∈ V
9 fex 7222 . . . . 5 ((𝑆:π‘‡βŸΆπ‘‡ ∧ 𝑇 ∈ V) β†’ 𝑆 ∈ V)
108, 9mpan2 688 . . . 4 (𝑆:π‘‡βŸΆπ‘‡ β†’ 𝑆 ∈ V)
11103ad2ant1 1130 . . 3 ((𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“)) β†’ 𝑆 ∈ V)
12 feq1 6691 . . . 4 (𝑠 = 𝑆 β†’ (𝑠:π‘‡βŸΆπ‘‡ ↔ 𝑆:π‘‡βŸΆπ‘‡))
13 fveq1 6883 . . . . . 6 (𝑠 = 𝑆 β†’ (π‘ β€˜(𝑓 ∘ 𝑔)) = (π‘†β€˜(𝑓 ∘ 𝑔)))
14 fveq1 6883 . . . . . . 7 (𝑠 = 𝑆 β†’ (π‘ β€˜π‘“) = (π‘†β€˜π‘“))
15 fveq1 6883 . . . . . . 7 (𝑠 = 𝑆 β†’ (π‘ β€˜π‘”) = (π‘†β€˜π‘”))
1614, 15coeq12d 5857 . . . . . 6 (𝑠 = 𝑆 β†’ ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)))
1713, 16eqeq12d 2742 . . . . 5 (𝑠 = 𝑆 β†’ ((π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ↔ (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”))))
18172ralbidv 3212 . . . 4 (𝑠 = 𝑆 β†’ (βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ↔ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”))))
1914fveq2d 6888 . . . . . 6 (𝑠 = 𝑆 β†’ (π‘…β€˜(π‘ β€˜π‘“)) = (π‘…β€˜(π‘†β€˜π‘“)))
2019breq1d 5151 . . . . 5 (𝑠 = 𝑆 β†’ ((π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“) ↔ (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“)))
2120ralbidv 3171 . . . 4 (𝑠 = 𝑆 β†’ (βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“) ↔ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“)))
2212, 18, 213anbi123d 1432 . . 3 (𝑠 = 𝑆 β†’ ((𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“)) ↔ (𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“))))
2311, 22elab3 3671 . 2 (𝑆 ∈ {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))} ↔ (𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“)))
247, 23bitrdi 287 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑆 ∈ 𝐸 ↔ (𝑆:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘†β€˜(𝑓 ∘ 𝑔)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  Vcvv 3468   class class class wbr 5141   ∘ ccom 5673  βŸΆwf 6532  β€˜cfv 6536  lecple 17210  LHypclh 39367  LTrncltrn 39484  trLctrl 39541  TEndoctendo 40135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-tendo 40138
This theorem is referenced by:  tendotp  40144  istendod  40145  tendof  40146  tendovalco  40148
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