Theorem ajval 30843

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 = (·𝑖OLD‘𝑈)
ajval.4	⊢ 𝑄 = (·𝑖OLD‘𝑊)
ajval.5	⊢ 𝐴 = (𝑈adj𝑊)

Assertion

Ref	Expression
ajval	⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))

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.

Step	Hyp	Ref	Expression
1		phnv 30796	. . . . 5 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
2		ajval.1	. . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈)
3		ajval.2	. . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊)
4		ajval.3	. . . . . 6 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
5		ajval.4	. . . . . 6 ⊢ 𝑄 = (·𝑖OLD‘𝑊)
6		ajval.5	. . . . . 6 ⊢ 𝐴 = (𝑈adj𝑊)
7	2, 3, 4, 5, 6	ajfval 30791	. . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))})
8	1, 7	sylan 580	. . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))})
9	8	fveq1d 6830	. . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec) → (𝐴‘𝑇) = ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇))
10	9	3adant3 1132	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇))
11	2	fvexi 6842	. . . . . 6 ⊢ 𝑋 ∈ V
12		fex 7166	. . . . . 6 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑋 ∈ V) → 𝑇 ∈ V)
13	11, 12	mpan2 691	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → 𝑇 ∈ V)
14		eqid 2733	. . . . . 6 ⊢ {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))} = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}
15		feq1 6634	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡:𝑋⟶𝑌 ↔ 𝑇:𝑋⟶𝑌))
16		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
17	16	oveq1d 7367	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥)𝑄𝑦) = ((𝑇‘𝑥)𝑄𝑦))
18	17	eqeq1d 2735	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))
19	18	2ralbidv 3197	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))
20	15, 19	3anbi13d 1440	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
21	14, 20	fvopab5 6968	. . . . 5 ⊢ (𝑇 ∈ V → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
22	13, 21	syl 17	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
23		3anass 1094	. . . . . 6 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑇:𝑋⟶𝑌 ∧ (𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
24	23	baib 535	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → ((𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
25	24	iotabidv 6470	. . . 4 ⊢ (𝑇:𝑋⟶𝑌 → (℩𝑠(𝑇:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
26	22, 25	eqtrd 2768	. . 3 ⊢ (𝑇:𝑋⟶𝑌 → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
27	26	3ad2ant3 1135	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → ({⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))}‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
28	10, 27	eqtrd 2768	1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  Vcvv 3437  {copab 5155  ℩cio 6440  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  NrmCVeccnv 30566  BaseSetcba 30568  ·_𝑖OLDcdip 30682  adjcaj 30730  CPreHil_OLDccphlo 30794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758  df-aj 30732  df-ph 30795

This theorem is referenced by: (None)