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Theorem ajfval 30828
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 = (·𝑖OLD𝑈)
ajfval.4 𝑄 = (·𝑖OLD𝑊)
ajfval.5 𝐴 = (𝑈adj𝑊)
Assertion
Ref Expression
ajfval ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
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.
StepHypRef Expression
1 ajfval.5 . 2 𝐴 = (𝑈adj𝑊)
2 fveq2 6906 . . . . . . 7 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 ajfval.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2795 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54feq2d 6722 . . . . 5 (𝑢 = 𝑈 → (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋⟶(BaseSet‘𝑤)))
64feq3d 6723 . . . . 5 (𝑢 = 𝑈 → (𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ↔ 𝑠:(BaseSet‘𝑤)⟶𝑋))
7 fveq2 6906 . . . . . . . . . 10 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = (·𝑖OLD𝑈))
8 ajfval.3 . . . . . . . . . 10 𝑃 = (·𝑖OLD𝑈)
97, 8eqtr4di 2795 . . . . . . . . 9 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = 𝑃)
109oveqd 7448 . . . . . . . 8 (𝑢 = 𝑈 → (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) = (𝑥𝑃(𝑠𝑦)))
1110eqeq2d 2748 . . . . . . 7 (𝑢 = 𝑈 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
1211ralbidv 3178 . . . . . 6 (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
134, 12raleqbidv 3346 . . . . 5 (𝑢 = 𝑈 → (∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
145, 6, 133anbi123d 1438 . . . 4 (𝑢 = 𝑈 → ((𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)))))
1514opabbidv 5209 . . 3 (𝑢 = 𝑈 → {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))} = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)))})
16 fveq2 6906 . . . . . . 7 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 ajfval.2 . . . . . . 7 𝑌 = (BaseSet‘𝑊)
1816, 17eqtr4di 2795 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918feq3d 6723 . . . . 5 (𝑤 = 𝑊 → (𝑡:𝑋⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋𝑌))
2018feq2d 6722 . . . . 5 (𝑤 = 𝑊 → (𝑠:(BaseSet‘𝑤)⟶𝑋𝑠:𝑌𝑋))
21 fveq2 6906 . . . . . . . . . 10 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = (·𝑖OLD𝑊))
22 ajfval.4 . . . . . . . . . 10 𝑄 = (·𝑖OLD𝑊)
2321, 22eqtr4di 2795 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = 𝑄)
2423oveqd 7448 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = ((𝑡𝑥)𝑄𝑦))
2524eqeq1d 2739 . . . . . . 7 (𝑤 = 𝑊 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2618, 25raleqbidv 3346 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2726ralbidv 3178 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2819, 20, 273anbi123d 1438 . . . 4 (𝑤 = 𝑊 → ((𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))) ↔ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))))
2928opabbidv 5209 . . 3 (𝑤 = 𝑊 → {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶𝑋 ∧ ∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)))} = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
30 df-aj 30769 . . 3 adj = (𝑢 ∈ NrmCVec, 𝑤 ∈ NrmCVec ↦ {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ∧ 𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ∧ ∀𝑥 ∈ (BaseSet‘𝑢)∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)))})
31 ovex 7464 . . . . 5 (𝑌m 𝑋) ∈ V
32 ovex 7464 . . . . 5 (𝑋m 𝑌) ∈ V
3331, 32xpex 7773 . . . 4 ((𝑌m 𝑋) × (𝑋m 𝑌)) ∈ V
3417fvexi 6920 . . . . . . . . . 10 𝑌 ∈ V
353fvexi 6920 . . . . . . . . . 10 𝑋 ∈ V
3634, 35elmap 8911 . . . . . . . . 9 (𝑡 ∈ (𝑌m 𝑋) ↔ 𝑡:𝑋𝑌)
3735, 34elmap 8911 . . . . . . . . 9 (𝑠 ∈ (𝑋m 𝑌) ↔ 𝑠:𝑌𝑋)
3836, 37anbi12i 628 . . . . . . . 8 ((𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌)) ↔ (𝑡:𝑋𝑌𝑠:𝑌𝑋))
3938biimpri 228 . . . . . . 7 ((𝑡:𝑋𝑌𝑠:𝑌𝑋) → (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌)))
40393adant3 1133 . . . . . 6 ((𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) → (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌)))
4140ssopab2i 5555 . . . . 5 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ⊆ {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌))}
42 df-xp 5691 . . . . 5 ((𝑌m 𝑋) × (𝑋m 𝑌)) = {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌))}
4341, 42sseqtrri 4033 . . . 4 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ⊆ ((𝑌m 𝑋) × (𝑋m 𝑌))
4433, 43ssexi 5322 . . 3 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ∈ V
4515, 29, 30, 44ovmpo 7593 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈adj𝑊) = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
461, 45eqtrid 2789 1 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {copab 5205   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  NrmCVeccnv 30603  BaseSetcba 30605  ·𝑖OLDcdip 30719  adjcaj 30767
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-aj 30769
This theorem is referenced by:  ajfuni  30878  ajval  30880
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