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Theorem ajfval 27504
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 6150 . . . . . . 7 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 ajfval.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2678 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54feq2d 5990 . . . . 5 (𝑢 = 𝑈 → (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋⟶(BaseSet‘𝑤)))
64feq3d 5991 . . . . 5 (𝑢 = 𝑈 → (𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ↔ 𝑠:(BaseSet‘𝑤)⟶𝑋))
7 fveq2 6150 . . . . . . . . . 10 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = (·𝑖OLD𝑈))
8 ajfval.3 . . . . . . . . . 10 𝑃 = (·𝑖OLD𝑈)
97, 8syl6eqr 2678 . . . . . . . . 9 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = 𝑃)
109oveqd 6622 . . . . . . . 8 (𝑢 = 𝑈 → (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) = (𝑥𝑃(𝑠𝑦)))
1110eqeq2d 2636 . . . . . . 7 (𝑢 = 𝑈 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
1211ralbidv 2985 . . . . . 6 (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
134, 12raleqbidv 3146 . . . . 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 1396 . . . 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 4683 . . 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 6150 . . . . . . 7 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 ajfval.2 . . . . . . 7 𝑌 = (BaseSet‘𝑊)
1816, 17syl6eqr 2678 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918feq3d 5991 . . . . 5 (𝑤 = 𝑊 → (𝑡:𝑋⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋𝑌))
2018feq2d 5990 . . . . 5 (𝑤 = 𝑊 → (𝑠:(BaseSet‘𝑤)⟶𝑋𝑠:𝑌𝑋))
21 fveq2 6150 . . . . . . . . . 10 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = (·𝑖OLD𝑊))
22 ajfval.4 . . . . . . . . . 10 𝑄 = (·𝑖OLD𝑊)
2321, 22syl6eqr 2678 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = 𝑄)
2423oveqd 6622 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = ((𝑡𝑥)𝑄𝑦))
2524eqeq1d 2628 . . . . . . 7 (𝑤 = 𝑊 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2618, 25raleqbidv 3146 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2726ralbidv 2985 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2819, 20, 273anbi123d 1396 . . . 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 4683 . . 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 27445 . . 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 6633 . . . . 5 (𝑌𝑚 𝑋) ∈ V
32 ovex 6633 . . . . 5 (𝑋𝑚 𝑌) ∈ V
3331, 32xpex 6916 . . . 4 ((𝑌𝑚 𝑋) × (𝑋𝑚 𝑌)) ∈ V
34 fvex 6160 . . . . . . . . . . 11 (BaseSet‘𝑊) ∈ V
3517, 34eqeltri 2700 . . . . . . . . . 10 𝑌 ∈ V
36 fvex 6160 . . . . . . . . . . 11 (BaseSet‘𝑈) ∈ V
373, 36eqeltri 2700 . . . . . . . . . 10 𝑋 ∈ V
3835, 37elmap 7831 . . . . . . . . 9 (𝑡 ∈ (𝑌𝑚 𝑋) ↔ 𝑡:𝑋𝑌)
3937, 35elmap 7831 . . . . . . . . 9 (𝑠 ∈ (𝑋𝑚 𝑌) ↔ 𝑠:𝑌𝑋)
4038, 39anbi12i 732 . . . . . . . 8 ((𝑡 ∈ (𝑌𝑚 𝑋) ∧ 𝑠 ∈ (𝑋𝑚 𝑌)) ↔ (𝑡:𝑋𝑌𝑠:𝑌𝑋))
4140biimpri 218 . . . . . . 7 ((𝑡:𝑋𝑌𝑠:𝑌𝑋) → (𝑡 ∈ (𝑌𝑚 𝑋) ∧ 𝑠 ∈ (𝑋𝑚 𝑌)))
42413adant3 1079 . . . . . 6 ((𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) → (𝑡 ∈ (𝑌𝑚 𝑋) ∧ 𝑠 ∈ (𝑋𝑚 𝑌)))
4342ssopab2i 4968 . . . . 5 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ⊆ {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌𝑚 𝑋) ∧ 𝑠 ∈ (𝑋𝑚 𝑌))}
44 df-xp 5085 . . . . 5 ((𝑌𝑚 𝑋) × (𝑋𝑚 𝑌)) = {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌𝑚 𝑋) ∧ 𝑠 ∈ (𝑋𝑚 𝑌))}
4543, 44sseqtr4i 3622 . . . 4 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ⊆ ((𝑌𝑚 𝑋) × (𝑋𝑚 𝑌))
4633, 45ssexi 4768 . . 3 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ∈ V
4715, 29, 30, 46ovmpt2 6750 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈adj𝑊) = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
481, 47syl5eq 2672 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 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  {copab 4677   × cxp 5077  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  NrmCVeccnv 27279  BaseSetcba 27281  ·𝑖OLDcdip 27395  adjcaj 27443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-aj 27445
This theorem is referenced by:  ajfuni  27555  ajval  27557
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