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Theorem ajval 29803
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 29756 . . . . 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 29751 . . . . 5 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
81, 7sylan 580 . . . 4 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
98fveq1d 6844 . . 3 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec) → (𝐴𝑇) = ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇))
1093adant3 1132 . 2 ((𝑈 ∈ CPreHilOLD𝑊 ∈ NrmCVec ∧ 𝑇:𝑋𝑌) → (𝐴𝑇) = ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇))
112fvexi 6856 . . . . . 6 𝑋 ∈ V
12 fex 7176 . . . . . 6 ((𝑇:𝑋𝑌𝑋 ∈ V) → 𝑇 ∈ V)
1311, 12mpan2 689 . . . . 5 (𝑇:𝑋𝑌𝑇 ∈ V)
14 eqid 2736 . . . . . 6 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}
15 feq1 6649 . . . . . . 7 (𝑡 = 𝑇 → (𝑡:𝑋𝑌𝑇:𝑋𝑌))
16 fveq1 6841 . . . . . . . . . 10 (𝑡 = 𝑇 → (𝑡𝑥) = (𝑇𝑥))
1716oveq1d 7372 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑡𝑥)𝑄𝑦) = ((𝑇𝑥)𝑄𝑦))
1817eqeq1d 2738 . . . . . . . 8 (𝑡 = 𝑇 → (((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
19182ralbidv 3212 . . . . . . 7 (𝑡 = 𝑇 → (∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2015, 193anbi13d 1438 . . . . . 6 (𝑡 = 𝑇 → ((𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) ↔ (𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2114, 20fvopab5 6980 . . . . 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 1095 . . . . . 6 ((𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) ↔ (𝑇:𝑋𝑌 ∧ (𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2423baib 536 . . . . 5 (𝑇:𝑋𝑌 → ((𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) ↔ (𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2524iotabidv 6480 . . . 4 (𝑇:𝑋𝑌 → (℩𝑠(𝑇:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2622, 25eqtrd 2776 . . 3 (𝑇:𝑋𝑌 → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))}‘𝑇) = (℩𝑠(𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑇𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
27263ad2ant3 1135 . 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 2776 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  {copab 5167  cio 6446  wf 6492  cfv 6496  (class class class)co 7357  NrmCVeccnv 29526  BaseSetcba 29528  ·𝑖OLDcdip 29642  adjcaj 29690  CPreHilOLDccphlo 29754
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-aj 29692  df-ph 29755
This theorem is referenced by: (None)
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