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Theorem ajfval 30057
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 = (BaseSetβ€˜π‘ˆ)
ajfval.2 π‘Œ = (BaseSetβ€˜π‘Š)
ajfval.3 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
ajfval.4 𝑄 = (·𝑖OLDβ€˜π‘Š)
ajfval.5 𝐴 = (π‘ˆadjπ‘Š)
Assertion
Ref Expression
ajfval ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝐴 = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))})
Distinct variable groups:   𝑑,𝑠,π‘₯,𝑦,π‘ˆ   π‘Š,𝑠,𝑑,π‘₯,𝑦   𝑋,𝑠,𝑑,π‘₯   π‘Œ,𝑠,𝑑,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑑,𝑠)   𝑃(π‘₯,𝑦,𝑑,𝑠)   𝑄(π‘₯,𝑦,𝑑,𝑠)   𝑋(𝑦)   π‘Œ(π‘₯)

Proof of Theorem ajfval
Dummy variables 𝑀 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ajfval.5 . 2 𝐴 = (π‘ˆadjπ‘Š)
2 fveq2 6891 . . . . . . 7 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = (BaseSetβ€˜π‘ˆ))
3 ajfval.1 . . . . . . 7 𝑋 = (BaseSetβ€˜π‘ˆ)
42, 3eqtr4di 2790 . . . . . 6 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = 𝑋)
54feq2d 6703 . . . . 5 (𝑒 = π‘ˆ β†’ (𝑑:(BaseSetβ€˜π‘’)⟢(BaseSetβ€˜π‘€) ↔ 𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€)))
64feq3d 6704 . . . . 5 (𝑒 = π‘ˆ β†’ (𝑠:(BaseSetβ€˜π‘€)⟢(BaseSetβ€˜π‘’) ↔ 𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹))
7 fveq2 6891 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ (·𝑖OLDβ€˜π‘’) = (·𝑖OLDβ€˜π‘ˆ))
8 ajfval.3 . . . . . . . . . 10 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
97, 8eqtr4di 2790 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (·𝑖OLDβ€˜π‘’) = 𝑃)
109oveqd 7425 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦)) = (π‘₯𝑃(π‘ β€˜π‘¦)))
1110eqeq2d 2743 . . . . . . 7 (𝑒 = π‘ˆ β†’ (((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦)) ↔ ((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
1211ralbidv 3177 . . . . . 6 (𝑒 = π‘ˆ β†’ (βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
134, 12raleqbidv 3342 . . . . 5 (𝑒 = π‘ˆ β†’ (βˆ€π‘₯ ∈ (BaseSetβ€˜π‘’)βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
145, 6, 133anbi123d 1436 . . . 4 (𝑒 = π‘ˆ β†’ ((𝑑:(BaseSetβ€˜π‘’)⟢(BaseSetβ€˜π‘€) ∧ 𝑠:(BaseSetβ€˜π‘€)⟢(BaseSetβ€˜π‘’) ∧ βˆ€π‘₯ ∈ (BaseSetβ€˜π‘’)βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦))) ↔ (𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€) ∧ 𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
1514opabbidv 5214 . . 3 (𝑒 = π‘ˆ β†’ {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:(BaseSetβ€˜π‘’)⟢(BaseSetβ€˜π‘€) ∧ 𝑠:(BaseSetβ€˜π‘€)⟢(BaseSetβ€˜π‘’) ∧ βˆ€π‘₯ ∈ (BaseSetβ€˜π‘’)βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦)))} = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€) ∧ 𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))})
16 fveq2 6891 . . . . . . 7 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = (BaseSetβ€˜π‘Š))
17 ajfval.2 . . . . . . 7 π‘Œ = (BaseSetβ€˜π‘Š)
1816, 17eqtr4di 2790 . . . . . 6 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = π‘Œ)
1918feq3d 6704 . . . . 5 (𝑀 = π‘Š β†’ (𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€) ↔ 𝑑:π‘‹βŸΆπ‘Œ))
2018feq2d 6703 . . . . 5 (𝑀 = π‘Š β†’ (𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹ ↔ 𝑠:π‘ŒβŸΆπ‘‹))
21 fveq2 6891 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (·𝑖OLDβ€˜π‘€) = (·𝑖OLDβ€˜π‘Š))
22 ajfval.4 . . . . . . . . . 10 𝑄 = (·𝑖OLDβ€˜π‘Š)
2321, 22eqtr4di 2790 . . . . . . . . 9 (𝑀 = π‘Š β†’ (·𝑖OLDβ€˜π‘€) = 𝑄)
2423oveqd 7425 . . . . . . . 8 (𝑀 = π‘Š β†’ ((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = ((π‘‘β€˜π‘₯)𝑄𝑦))
2524eqeq1d 2734 . . . . . . 7 (𝑀 = π‘Š β†’ (((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)) ↔ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
2618, 25raleqbidv 3342 . . . . . 6 (𝑀 = π‘Š β†’ (βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
2726ralbidv 3177 . . . . 5 (𝑀 = π‘Š β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))))
2819, 20, 273anbi123d 1436 . . . 4 (𝑀 = π‘Š β†’ ((𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€) ∧ 𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))) ↔ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))))
2928opabbidv 5214 . . 3 (𝑀 = π‘Š β†’ {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€) ∧ 𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))} = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))})
30 df-aj 29998 . . 3 adj = (𝑒 ∈ NrmCVec, 𝑀 ∈ NrmCVec ↦ {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:(BaseSetβ€˜π‘’)⟢(BaseSetβ€˜π‘€) ∧ 𝑠:(BaseSetβ€˜π‘€)⟢(BaseSetβ€˜π‘’) ∧ βˆ€π‘₯ ∈ (BaseSetβ€˜π‘’)βˆ€π‘¦ ∈ (BaseSetβ€˜π‘€)((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦)))})
31 ovex 7441 . . . . 5 (π‘Œ ↑m 𝑋) ∈ V
32 ovex 7441 . . . . 5 (𝑋 ↑m π‘Œ) ∈ V
3331, 32xpex 7739 . . . 4 ((π‘Œ ↑m 𝑋) Γ— (𝑋 ↑m π‘Œ)) ∈ V
3417fvexi 6905 . . . . . . . . . 10 π‘Œ ∈ V
353fvexi 6905 . . . . . . . . . 10 𝑋 ∈ V
3634, 35elmap 8864 . . . . . . . . 9 (𝑑 ∈ (π‘Œ ↑m 𝑋) ↔ 𝑑:π‘‹βŸΆπ‘Œ)
3735, 34elmap 8864 . . . . . . . . 9 (𝑠 ∈ (𝑋 ↑m π‘Œ) ↔ 𝑠:π‘ŒβŸΆπ‘‹)
3836, 37anbi12i 627 . . . . . . . 8 ((𝑑 ∈ (π‘Œ ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m π‘Œ)) ↔ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹))
3938biimpri 227 . . . . . . 7 ((𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹) β†’ (𝑑 ∈ (π‘Œ ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m π‘Œ)))
40393adant3 1132 . . . . . 6 ((𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦))) β†’ (𝑑 ∈ (π‘Œ ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m π‘Œ)))
4140ssopab2i 5550 . . . . 5 {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))} βŠ† {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑 ∈ (π‘Œ ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m π‘Œ))}
42 df-xp 5682 . . . . 5 ((π‘Œ ↑m 𝑋) Γ— (𝑋 ↑m π‘Œ)) = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑 ∈ (π‘Œ ↑m 𝑋) ∧ 𝑠 ∈ (𝑋 ↑m π‘Œ))}
4341, 42sseqtrri 4019 . . . 4 {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))} βŠ† ((π‘Œ ↑m 𝑋) Γ— (𝑋 ↑m π‘Œ))
4433, 43ssexi 5322 . . 3 {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))} ∈ V
4515, 29, 30, 44ovmpo 7567 . 2 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (π‘ˆadjπ‘Š) = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))})
461, 45eqtrid 2784 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝐴 = {βŸ¨π‘‘, π‘ βŸ© ∣ (𝑑:π‘‹βŸΆπ‘Œ ∧ 𝑠:π‘ŒβŸΆπ‘‹ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ ((π‘‘β€˜π‘₯)𝑄𝑦) = (π‘₯𝑃(π‘ β€˜π‘¦)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {copab 5210   Γ— cxp 5674  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑m cmap 8819  NrmCVeccnv 29832  BaseSetcba 29834  Β·π‘–OLDcdip 29948  adjcaj 29996
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-aj 29998
This theorem is referenced by:  ajfuni  30107  ajval  30109
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