Theorem ajval 31445

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 31398	. . . . 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 31393	. . . . 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 592	. . . 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 6879	. . 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 1150	. 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 6891	. . . . . 6 ⊢ 𝑋 ∈ V
12		fex 7224	. . . . . 6 ⊢ ((𝑇:𝑋⟶𝑌 ∧ 𝑋 ∈ V) → 𝑇 ∈ V)
13	11, 12	mpan2 704	. . . . 5 ⊢ (𝑇:𝑋⟶𝑌 → 𝑇 ∈ V)
14		eqid 2761	. . . . . 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 6679	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡:𝑋⟶𝑌 ↔ 𝑇:𝑋⟶𝑌))
16		fveq1 6876	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑥) = (𝑇‘𝑥))
17	16	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑡‘𝑥)𝑄𝑦) = ((𝑇‘𝑥)𝑄𝑦))
18	17	eqeq1d 2763	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))
19	18	2ralbidv 3227	. . . . . . 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 1466	. . . . . 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 7019	. . . . 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 18	. . . 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 1111	. . . . . 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 545	. . . . 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 6515	. . . 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 2796	. . 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 1153	. 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 2796	1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:𝑋⟶𝑌) → (𝐴‘𝑇) = (℩𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  {copab 5167  ℩cio 6485  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  NrmCVeccnv 31168  BaseSetcba 31170  ·_𝑖OLDcdip 31284  adjcaj 31332  CPreHil_OLDccphlo 31396

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8833  df-aj 31334  df-ph 31397

This theorem is used by: (None)