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Theorem ajfval 30838
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 6907 . . . . . . 7 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 ajfval.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2793 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54feq2d 6723 . . . . 5 (𝑢 = 𝑈 → (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋⟶(BaseSet‘𝑤)))
64feq3d 6724 . . . . 5 (𝑢 = 𝑈 → (𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ↔ 𝑠:(BaseSet‘𝑤)⟶𝑋))
7 fveq2 6907 . . . . . . . . . 10 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = (·𝑖OLD𝑈))
8 ajfval.3 . . . . . . . . . 10 𝑃 = (·𝑖OLD𝑈)
97, 8eqtr4di 2793 . . . . . . . . 9 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = 𝑃)
109oveqd 7448 . . . . . . . 8 (𝑢 = 𝑈 → (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) = (𝑥𝑃(𝑠𝑦)))
1110eqeq2d 2746 . . . . . . 7 (𝑢 = 𝑈 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
1211ralbidv 3176 . . . . . 6 (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
134, 12raleqbidv 3344 . . . . 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 1435 . . . 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 5214 . . 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 6907 . . . . . . 7 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 ajfval.2 . . . . . . 7 𝑌 = (BaseSet‘𝑊)
1816, 17eqtr4di 2793 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918feq3d 6724 . . . . 5 (𝑤 = 𝑊 → (𝑡:𝑋⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋𝑌))
2018feq2d 6723 . . . . 5 (𝑤 = 𝑊 → (𝑠:(BaseSet‘𝑤)⟶𝑋𝑠:𝑌𝑋))
21 fveq2 6907 . . . . . . . . . 10 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = (·𝑖OLD𝑊))
22 ajfval.4 . . . . . . . . . 10 𝑄 = (·𝑖OLD𝑊)
2321, 22eqtr4di 2793 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = 𝑄)
2423oveqd 7448 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = ((𝑡𝑥)𝑄𝑦))
2524eqeq1d 2737 . . . . . . 7 (𝑤 = 𝑊 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2618, 25raleqbidv 3344 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2726ralbidv 3176 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2819, 20, 273anbi123d 1435 . . . 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 5214 . . 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 30779 . . 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 7772 . . . 4 ((𝑌m 𝑋) × (𝑋m 𝑌)) ∈ V
3417fvexi 6921 . . . . . . . . . 10 𝑌 ∈ V
353fvexi 6921 . . . . . . . . . 10 𝑋 ∈ V
3634, 35elmap 8910 . . . . . . . . 9 (𝑡 ∈ (𝑌m 𝑋) ↔ 𝑡:𝑋𝑌)
3735, 34elmap 8910 . . . . . . . . 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 1131 . . . . . 6 ((𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) → (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌)))
4140ssopab2i 5560 . . . . 5 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ⊆ {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌))}
42 df-xp 5695 . . . . 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 5328 . . 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 2787 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 1086   = wceq 1537  wcel 2106  wral 3059  {copab 5210   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  NrmCVeccnv 30613  BaseSetcba 30615  ·𝑖OLDcdip 30729  adjcaj 30777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-aj 30779
This theorem is referenced by:  ajfuni  30888  ajval  30890
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