Theorem ajfval 31418

Description: The adjoint function. (Contributed by NM, 25-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ajfval.1	⊢ 𝑋 = (BaseSet‘𝑈)
ajfval.2	⊢ 𝑌 = (BaseSet‘𝑊)
ajfval.3	⊢ 𝑃 = (·𝑖OLD‘𝑈)
ajfval.4	⊢ 𝑄 = (·𝑖OLD‘𝑊)
ajfval.5	⊢ 𝐴 = (𝑈 adj 𝑊)

Assertion

Ref	Expression
ajfval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))})

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑈   𝑊,𝑠,𝑡,𝑥,𝑦   𝑋,𝑠,𝑡,𝑥   𝑌,𝑠,𝑡,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑡, 𝑠)   𝑃(𝑥, 𝑦, 𝑡, 𝑠)   𝑄(𝑥, 𝑦, 𝑡, 𝑠)   𝑋(𝑦)   𝑌(𝑥)

Proof of Theorem ajfval

Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ajfval.5	. 2 ⊢ 𝐴 = (𝑈 adj 𝑊)
2		fveq2 6888	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		ajfval.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2813	. . . . . 6 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5	4	feq2d 6696	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋⟶(BaseSet‘𝑤)))
6	4	feq3d 6697	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ↔ 𝑠:(BaseSet‘𝑤)⟶𝑋))
7		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → (·𝑖OLD‘𝑢) = (·𝑖OLD‘𝑈))
8		ajfval.3	. . . . . . . . . 10 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
9	7, 8	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (·𝑖OLD‘𝑢) = 𝑃)
10	9	oveqd 7439	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)) = (𝑥𝑃(𝑠‘𝑦)))
11	10	eqeq2d 2771	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)) ↔ ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦))))
12	11	ralbidv 3185	. . . . . 6 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)) ↔ ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦))))
13	4, 12	raleqbidv 3334	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦))))
14	5, 6, 13	3anbi123d 1464	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
15	14	opabbidv 5170	. . 3 ⊢ (𝑢 = 𝑈 → {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))} = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦)))})
16		fveq2 6888	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17		ajfval.2	. . . . . . 7 ⊢ 𝑌 = (BaseSet‘𝑊)
18	16, 17	eqtr4di 2813	. . . . . 6 ⊢ (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
19	18	feq3d 6697	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑡:𝑋⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋⟶𝑌))
20	18	feq2d 6696	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑠:(BaseSet‘𝑤)⟶𝑋 ↔ 𝑠:𝑌⟶𝑋))
21		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (·𝑖OLD‘𝑤) = (·𝑖OLD‘𝑊))
22		ajfval.4	. . . . . . . . . 10 ⊢ 𝑄 = (·𝑖OLD‘𝑊)
23	21, 22	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (·𝑖OLD‘𝑤) = 𝑄)
24	23	oveqd 7439	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = ((𝑡‘𝑥)𝑄𝑦))
25	24	eqeq1d 2762	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))
26	18, 25	raleqbidv 3334	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))
27	26	ralbidv 3185	. . . . 5 ⊢ (𝑤 = 𝑊 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))))
28	19, 20, 27	3anbi123d 1464	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))))
29	28	opabbidv 5170	. . 3 ⊢ (𝑤 = 𝑊 → {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥𝑃(𝑠‘𝑦)))} = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))})
30		df-aj 31359	. . 3 ⊢ adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡‘𝑥)(·𝑖OLD‘𝑤)𝑦) = (𝑥(·𝑖OLD‘𝑢)(𝑠‘𝑦)))})
31		ovex 7455	. . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V
32		ovex 7455	. . . . 5 ⊢ (𝑋 ↑m 𝑌) ∈ V
33	31, 32	xpex 7769	. . . 4 ⊢ ((𝑌 ↑m 𝑋) × (𝑋 ↑m 𝑌)) ∈ V
34	17	fvexi 6902	. . . . . . . . . 10 ⊢ 𝑌 ∈ V
35	3	fvexi 6902	. . . . . . . . . 10 ⊢ 𝑋 ∈ V
36	34, 35	elmap 8904	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑌 ↑m 𝑋) ↔ 𝑡:𝑋⟶𝑌)
37	35, 34	elmap 8904	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝑋 ↑m 𝑌) ↔ 𝑠:𝑌⟶𝑋)
38	36, 37	anbi12i 640	. . . . . . . 8 ⊢ ((𝑡 ∈ (𝑌 ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m 𝑌)) ↔ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋))
39	38	biimpri 231	. . . . . . 7 ⊢ ((𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋) → (𝑡 ∈ (𝑌 ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m 𝑌)))
40	39	3adant3 1150	. . . . . 6 ⊢ ((𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) → (𝑡 ∈ (𝑌 ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m 𝑌)))
41	40	ssopab2i 5524	. . . . 5 ⊢ {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))} ⊆ {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌 ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m 𝑌))}
42		df-xp 5656	. . . . 5 ⊢ ((𝑌 ↑m 𝑋) × (𝑋 ↑m 𝑌)) = {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌 ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m 𝑌))}
43	41, 42	sseqtrri 3958	. . . 4 ⊢ {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))} ⊆ ((𝑌 ↑m 𝑋) × (𝑋 ↑m 𝑌))
44	33, 43	ssexi 5283	. . 3 ⊢ {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))} ∈ V
45	15, 29, 30, 44	ovmpo 7582	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈 adj 𝑊) = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑡‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))})
46	1, 45	eqtrid 2807	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶𝑌 ∧ 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 1569   ∈ wcel 2144  ∀wral 3076  {copab 5166   × cxp 5648  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422   ↑_m cmap 8852  NrmCVeccnv 31193  BaseSetcba 31195  ·_𝑖OLDcdip 31309   adj caj 31357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-map 8854  df-aj 31359

This theorem is used by:  ajfuni  31468  ajval  31470