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Theorem ajval 29845
Description: Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
ajval.1 𝑋 = (BaseSetβ€˜π‘ˆ)
ajval.2 π‘Œ = (BaseSetβ€˜π‘Š)
ajval.3 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
ajval.4 𝑄 = (·𝑖OLDβ€˜π‘Š)
ajval.5 𝐴 = (π‘ˆadjπ‘Š)
Assertion
Ref Expression
ajval ((π‘ˆ ∈ CPreHilOLD ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π΄β€˜π‘‡) = (℩𝑠(𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
Distinct variable groups:   π‘₯,𝑠,𝑦,𝑇   π‘ˆ,𝑠,π‘₯,𝑦   π‘Š,𝑠,π‘₯,𝑦   𝑋,𝑠,π‘₯,𝑦   π‘Œ,𝑠,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑠)   𝑃(π‘₯,𝑦,𝑠)   𝑄(π‘₯,𝑦,𝑠)   π‘Œ(π‘₯)

Proof of Theorem ajval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 phnv 29798 . . . . 5 (π‘ˆ ∈ CPreHilOLD β†’ π‘ˆ ∈ NrmCVec)
2 ajval.1 . . . . . 6 𝑋 = (BaseSetβ€˜π‘ˆ)
3 ajval.2 . . . . . 6 π‘Œ = (BaseSetβ€˜π‘Š)
4 ajval.3 . . . . . 6 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
5 ajval.4 . . . . . 6 𝑄 = (·𝑖OLDβ€˜π‘Š)
6 ajval.5 . . . . . 6 𝐴 = (π‘ˆadjπ‘Š)
72, 3, 4, 5, 6ajfval 29793 . . . . 5 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝐴 = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))})
81, 7sylan 581 . . . 4 ((π‘ˆ ∈ CPreHilOLD ∧ π‘Š ∈ NrmCVec) β†’ 𝐴 = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))})
98fveq1d 6849 . . 3 ((π‘ˆ ∈ CPreHilOLD ∧ π‘Š ∈ NrmCVec) β†’ (π΄β€˜π‘‡) = ({βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))}β€˜π‘‡))
1093adant3 1133 . 2 ((π‘ˆ ∈ CPreHilOLD ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π΄β€˜π‘‡) = ({βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))}β€˜π‘‡))
112fvexi 6861 . . . . . 6 𝑋 ∈ V
12 fex 7181 . . . . . 6 ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑋 ∈ V) β†’ 𝑇 ∈ V)
1311, 12mpan2 690 . . . . 5 (𝑇:π‘‹βŸΆπ‘Œ β†’ 𝑇 ∈ V)
14 eqid 2737 . . . . . 6 {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))} = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))}
15 feq1 6654 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝑑:π‘‹βŸΆπ‘Œ ↔ 𝑇:π‘‹βŸΆπ‘Œ))
16 fveq1 6846 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (π‘‘β€˜π‘₯) = (π‘‡β€˜π‘₯))
1716oveq1d 7377 . . . . . . . . 9 (𝑑 = 𝑇 β†’ ((π‘‘β€˜π‘₯)𝑄𝑦) = ((π‘‡β€˜π‘₯)𝑄𝑦))
1817eqeq1d 2739 . . . . . . . 8 (𝑑 = 𝑇 β†’ (((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)) ↔ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
19182ralbidv 3213 . . . . . . 7 (𝑑 = 𝑇 β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
2015, 193anbi13d 1439 . . . . . 6 (𝑑 = 𝑇 β†’ ((𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))) ↔ (𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
2114, 20fvopab5 6985 . . . . 5 (𝑇 ∈ V β†’ ({βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))}β€˜π‘‡) = (℩𝑠(𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
2213, 21syl 17 . . . 4 (𝑇:π‘‹βŸΆπ‘Œ β†’ ({βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))}β€˜π‘‡) = (℩𝑠(𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
23 3anass 1096 . . . . . 6 ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))) ↔ (𝑇:π‘‹βŸΆπ‘Œ ∧ (𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
2423baib 537 . . . . 5 (𝑇:π‘‹βŸΆπ‘Œ β†’ ((𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))) ↔ (𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
2524iotabidv 6485 . . . 4 (𝑇:π‘‹βŸΆπ‘Œ β†’ (℩𝑠(𝑇:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))) = (℩𝑠(𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
2622, 25eqtrd 2777 . . 3 (𝑇:π‘‹βŸΆπ‘Œ β†’ ({βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))}β€˜π‘‡) = (℩𝑠(𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
27263ad2ant3 1136 . 2 ((π‘ˆ ∈ CPreHilOLD ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ ({βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))}β€˜π‘‡) = (℩𝑠(𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
2810, 27eqtrd 2777 1 ((π‘ˆ ∈ CPreHilOLD ∧ π‘Š ∈ NrmCVec ∧ 𝑇:π‘‹βŸΆπ‘Œ) β†’ (π΄β€˜π‘‡) = (℩𝑠(𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‡β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3448  {copab 5172  β„©cio 6451  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  NrmCVeccnv 29568  BaseSetcba 29570  Β·π‘–OLDcdip 29684  adjcaj 29732  CPreHilOLDccphlo 29796
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 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-aj 29734  df-ph 29797
This theorem is referenced by: (None)
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