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Theorem istendo 39226
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 39225 . . 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 2824 . 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 6857 . . . . 5 𝑇 ∈ V
9 fex 7177 . . . . 5 ((𝑆:π‘‡βŸΆπ‘‡ ∧ 𝑇 ∈ V) β†’ 𝑆 ∈ V)
108, 9mpan2 690 . . . 4 (𝑆:π‘‡βŸΆπ‘‡ β†’ 𝑆 ∈ V)
11103ad2ant1 1134 . . 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 6650 . . . 4 (𝑠 = 𝑆 β†’ (𝑠:π‘‡βŸΆπ‘‡ ↔ 𝑆:π‘‡βŸΆπ‘‡))
13 fveq1 6842 . . . . . 6 (𝑠 = 𝑆 β†’ (π‘ β€˜(𝑓 ∘ 𝑔)) = (π‘†β€˜(𝑓 ∘ 𝑔)))
14 fveq1 6842 . . . . . . 7 (𝑠 = 𝑆 β†’ (π‘ β€˜π‘“) = (π‘†β€˜π‘“))
15 fveq1 6842 . . . . . . 7 (𝑠 = 𝑆 β†’ (π‘ β€˜π‘”) = (π‘†β€˜π‘”))
1614, 15coeq12d 5821 . . . . . 6 (𝑠 = 𝑆 β†’ ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) = ((π‘†β€˜π‘“) ∘ (π‘†β€˜π‘”)))
1713, 16eqeq12d 2753 . . . . 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 3213 . . . 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 6847 . . . . . 6 (𝑠 = 𝑆 β†’ (π‘…β€˜(π‘ β€˜π‘“)) = (π‘…β€˜(π‘†β€˜π‘“)))
2019breq1d 5116 . . . . 5 (𝑠 = 𝑆 β†’ ((π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“) ↔ (π‘…β€˜(π‘†β€˜π‘“)) ≀ (π‘…β€˜π‘“)))
2120ralbidv 3175 . . . 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 1437 . . 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 3639 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  Vcvv 3446   class class class wbr 5106   ∘ ccom 5638  βŸΆwf 6493  β€˜cfv 6497  lecple 17141  LHypclh 38450  LTrncltrn 38567  trLctrl 38624  TEndoctendo 39218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-tendo 39221
This theorem is referenced by:  tendotp  39227  istendod  39228  tendof  39229  tendovalco  39231
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