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Theorem ajval 30691
Description: Value of the adjoint function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
ajval.1 𝑋 = (BaseSet‘𝑈)
ajval.2 𝑌 = (BaseSet‘𝑊)
ajval.3 𝑃 = (·𝑖OLD𝑈)
ajval.4 𝑄 = (·𝑖OLD𝑊)
ajval.5 𝐴 = (𝑈adj𝑊)
Assertion
Ref Expression
ajval ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝐴𝑇) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
Distinct variable groups:   𝑥,𝑠,𝑦,𝑇   𝑈,𝑠,𝑥,𝑦   𝑊,𝑠,𝑥,𝑦   𝑋,𝑠,𝑥,𝑦   𝑌,𝑠,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑠)   𝑃(𝑥,𝑦,𝑠)   𝑄(𝑥,𝑦,𝑠)   𝑌(𝑥)

Proof of Theorem ajval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 phnv 30644 . . . . 5 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
2 ajval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
3 ajval.2 . . . . . 6 𝑌 = (BaseSet‘𝑊)
4 ajval.3 . . . . . 6 𝑃 = (·𝑖OLD𝑈)
5 ajval.4 . . . . . 6 𝑄 = (·𝑖OLD𝑊)
6 ajval.5 . . . . . 6 𝐴 = (𝑈adj𝑊)
72, 3, 4, 5, 6ajfval 30639 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
81, 7sylan 578 . . . 4 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
98fveq1d 6904 . . 3 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → (𝐴𝑇) = ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇))
1093adant3 1129 . 2 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝐴𝑇) = ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇))
112fvexi 6916 . . . . . 6 𝑋 ∈ V
12 fex 7244 . . . . . 6 ((𝑇:𝑋𝑌𝑋 ∈ V) → 𝑇 ∈ V)
1311, 12mpan2 689 . . . . 5 (𝑇:𝑋𝑌𝑇 ∈ V)
14 eqid 2728 . . . . . 6 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}
15 feq1 6708 . . . . . . 7 (𝑡 = 𝑇 → (𝑡:𝑋𝑌𝑇:𝑋𝑌))
16 fveq1 6901 . . . . . . . . . 10 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
1716oveq1d 7441 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑡𝑥)𝑄𝑦) = ((𝑇𝑥)𝑄𝑦))
1817eqeq1d 2730 . . . . . . . 8 (𝑡 = 𝑇 → (((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
19182ralbidv 3216 . . . . . . 7 (𝑡 = 𝑇 → (∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2015, 193anbi13d 1434 . . . . . 6 (𝑡 = 𝑇 → ((𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) ↔ (𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2114, 20fvopab5 7043 . . . . 5 (𝑇 ∈ V → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇) = (℩𝑠(𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2213, 21syl 17 . . . 4 (𝑇:𝑋𝑌 → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇) = (℩𝑠(𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
23 3anass 1092 . . . . . 6 ((𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) ↔ (𝑇:𝑋𝑌 ∧ (𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2423baib 534 . . . . 5 (𝑇:𝑋𝑌 → ((𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) ↔ (𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2524iotabidv 6537 . . . 4 (𝑇:𝑋𝑌 → (℩𝑠(𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2622, 25eqtrd 2768 . . 3 (𝑇:𝑋𝑌 → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
27263ad2ant3 1132 . 2 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2810, 27eqtrd 2768 1 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝐴𝑇) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  Vcvv 3473  {copab 5214  cio 6503  wf 6549  cfv 6553  (class class class)co 7426  NrmCVeccnv 30414  BaseSetcba 30416  ·𝑖OLDcdip 30530  adjcaj 30578  CPreHilOLDccphlo 30642
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-aj 30580  df-ph 30643
This theorem is referenced by: (None)
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