Theorem ajval 30823

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 30776	. . . . 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 30771	. . . . 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 6828	. . 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 6840	. . . . . 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 2729	. . . . . 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 6825	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
17	16	oveq1d 7368	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥)𝑄𝑦) = ((𝑇‘𝑥)𝑄𝑦))
18	17	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))
19	18	2ralbidv 3193	. . . . . . 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 6967	. . . . 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 2764	. . 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 2764	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 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438  {copab 5157  ℩cio 6440  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  NrmCVeccnv 30546  BaseSetcba 30548  ·_𝑖OLDcdip 30662  adjcaj 30710  CPreHil_OLDccphlo 30774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7356  df-oprab 7357  df-mpo 7358  df-map 8762  df-aj 30712  df-ph 30775

This theorem is referenced by: (None)