MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ajfval Structured version   Visualization version   GIF version

Theorem ajfval 28570
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 6643 . . . . . . 7 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 ajfval.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2874 . . . . . 6 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
54feq2d 6473 . . . . 5 (𝑢 = 𝑈 → (𝑡:(BaseSet‘𝑢)⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋⟶(BaseSet‘𝑤)))
64feq3d 6474 . . . . 5 (𝑢 = 𝑈 → (𝑠:(BaseSet‘𝑤)⟶(BaseSet‘𝑢) ↔ 𝑠:(BaseSet‘𝑤)⟶𝑋))
7 fveq2 6643 . . . . . . . . . 10 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = (·𝑖OLD𝑈))
8 ajfval.3 . . . . . . . . . 10 𝑃 = (·𝑖OLD𝑈)
97, 8syl6eqr 2874 . . . . . . . . 9 (𝑢 = 𝑈 → (·𝑖OLD𝑢) = 𝑃)
109oveqd 7147 . . . . . . . 8 (𝑢 = 𝑈 → (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) = (𝑥𝑃(𝑠𝑦)))
1110eqeq2d 2832 . . . . . . 7 (𝑢 = 𝑈 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
1211ralbidv 3185 . . . . . 6 (𝑢 = 𝑈 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥(·𝑖OLD𝑢)(𝑠𝑦)) ↔ ∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦))))
134, 12raleqbidv 3386 . . . . 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 1433 . . . 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 5105 . . 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 6643 . . . . . . 7 (𝑤 = 𝑊 → (BaseSet‘𝑤) = (BaseSet‘𝑊))
17 ajfval.2 . . . . . . 7 𝑌 = (BaseSet‘𝑊)
1816, 17syl6eqr 2874 . . . . . 6 (𝑤 = 𝑊 → (BaseSet‘𝑤) = 𝑌)
1918feq3d 6474 . . . . 5 (𝑤 = 𝑊 → (𝑡:𝑋⟶(BaseSet‘𝑤) ↔ 𝑡:𝑋𝑌))
2018feq2d 6473 . . . . 5 (𝑤 = 𝑊 → (𝑠:(BaseSet‘𝑤)⟶𝑋𝑠:𝑌𝑋))
21 fveq2 6643 . . . . . . . . . 10 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = (·𝑖OLD𝑊))
22 ajfval.4 . . . . . . . . . 10 𝑄 = (·𝑖OLD𝑊)
2321, 22syl6eqr 2874 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖OLD𝑤) = 𝑄)
2423oveqd 7147 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = ((𝑡𝑥)𝑄𝑦))
2524eqeq1d 2823 . . . . . . 7 (𝑤 = 𝑊 → (((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2618, 25raleqbidv 3386 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2726ralbidv 3185 . . . . 5 (𝑤 = 𝑊 → (∀𝑥𝑋𝑦 ∈ (BaseSet‘𝑤)((𝑡𝑥)(·𝑖OLD𝑤)𝑦) = (𝑥𝑃(𝑠𝑦)) ↔ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))))
2819, 20, 273anbi123d 1433 . . . 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 5105 . . 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 28511 . . 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 7163 . . . . 5 (𝑌m 𝑋) ∈ V
32 ovex 7163 . . . . 5 (𝑋m 𝑌) ∈ V
3331, 32xpex 7451 . . . 4 ((𝑌m 𝑋) × (𝑋m 𝑌)) ∈ V
3417fvexi 6657 . . . . . . . . . 10 𝑌 ∈ V
353fvexi 6657 . . . . . . . . . 10 𝑋 ∈ V
3634, 35elmap 8410 . . . . . . . . 9 (𝑡 ∈ (𝑌m 𝑋) ↔ 𝑡:𝑋𝑌)
3735, 34elmap 8410 . . . . . . . . 9 (𝑠 ∈ (𝑋m 𝑌) ↔ 𝑠:𝑌𝑋)
3836, 37anbi12i 629 . . . . . . . 8 ((𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌)) ↔ (𝑡:𝑋𝑌𝑠:𝑌𝑋))
3938biimpri 231 . . . . . . 7 ((𝑡:𝑋𝑌𝑠:𝑌𝑋) → (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌)))
40393adant3 1129 . . . . . 6 ((𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦))) → (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌)))
4140ssopab2i 5410 . . . . 5 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ⊆ {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌))}
42 df-xp 5534 . . . . 5 ((𝑌m 𝑋) × (𝑋m 𝑌)) = {⟨𝑡, 𝑠⟩ ∣ (𝑡 ∈ (𝑌m 𝑋) ∧ 𝑠 ∈ (𝑋m 𝑌))}
4341, 42sseqtrri 3980 . . . 4 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ⊆ ((𝑌m 𝑋) × (𝑋m 𝑌))
4433, 43ssexi 5199 . . 3 {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))} ∈ V
4515, 29, 30, 44ovmpo 7284 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑈adj𝑊) = {⟨𝑡, 𝑠⟩ ∣ (𝑡:𝑋𝑌𝑠:𝑌𝑋 ∧ ∀𝑥𝑋𝑦𝑌 ((𝑡𝑥)𝑄𝑦) = (𝑥𝑃(𝑠𝑦)))})
461, 45syl5eq 2868 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 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  {copab 5101   × cxp 5526  wf 6324  cfv 6328  (class class class)co 7130  m cmap 8381  NrmCVeccnv 28345  BaseSetcba 28347  ·𝑖OLDcdip 28461  adjcaj 28509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-aj 28511
This theorem is referenced by:  ajfuni  28620  ajval  28622
  Copyright terms: Public domain W3C validator