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Theorem ajfval 30062
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 6892 . . . . . . 7 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = (BaseSetβ€˜π‘ˆ))
3 ajfval.1 . . . . . . 7 𝑋 = (BaseSetβ€˜π‘ˆ)
42, 3eqtr4di 2791 . . . . . 6 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = 𝑋)
54feq2d 6704 . . . . 5 (𝑒 = π‘ˆ β†’ (𝑑:(BaseSetβ€˜π‘’)⟢(BaseSetβ€˜π‘€) ↔ 𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€)))
64feq3d 6705 . . . . 5 (𝑒 = π‘ˆ β†’ (𝑠:(BaseSetβ€˜π‘€)⟢(BaseSetβ€˜π‘’) ↔ 𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹))
7 fveq2 6892 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ (·𝑖OLDβ€˜π‘’) = (·𝑖OLDβ€˜π‘ˆ))
8 ajfval.3 . . . . . . . . . 10 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
97, 8eqtr4di 2791 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (·𝑖OLDβ€˜π‘’) = 𝑃)
109oveqd 7426 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (π‘₯(·𝑖OLDβ€˜π‘’)(π‘ β€˜π‘¦)) = (π‘₯𝑃(π‘ β€˜π‘¦)))
1110eqeq2d 2744 . . . . . . 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 3178 . . . . . 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 3343 . . . . 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 1437 . . . 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 5215 . . 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 6892 . . . . . . 7 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = (BaseSetβ€˜π‘Š))
17 ajfval.2 . . . . . . 7 π‘Œ = (BaseSetβ€˜π‘Š)
1816, 17eqtr4di 2791 . . . . . 6 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = π‘Œ)
1918feq3d 6705 . . . . 5 (𝑀 = π‘Š β†’ (𝑑:π‘‹βŸΆ(BaseSetβ€˜π‘€) ↔ 𝑑:π‘‹βŸΆπ‘Œ))
2018feq2d 6704 . . . . 5 (𝑀 = π‘Š β†’ (𝑠:(BaseSetβ€˜π‘€)βŸΆπ‘‹ ↔ 𝑠:π‘ŒβŸΆπ‘‹))
21 fveq2 6892 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (·𝑖OLDβ€˜π‘€) = (·𝑖OLDβ€˜π‘Š))
22 ajfval.4 . . . . . . . . . 10 𝑄 = (·𝑖OLDβ€˜π‘Š)
2321, 22eqtr4di 2791 . . . . . . . . 9 (𝑀 = π‘Š β†’ (·𝑖OLDβ€˜π‘€) = 𝑄)
2423oveqd 7426 . . . . . . . 8 (𝑀 = π‘Š β†’ ((π‘‘β€˜π‘₯)(·𝑖OLDβ€˜π‘€)𝑦) = ((π‘‘β€˜π‘₯)𝑄𝑦))
2524eqeq1d 2735 . . . . . . 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 3343 . . . . . 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 3178 . . . . 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 1437 . . . 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 5215 . . 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 30003 . . 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 7442 . . . . 5 (π‘Œ ↑m 𝑋) ∈ V
32 ovex 7442 . . . . 5 (𝑋 ↑m π‘Œ) ∈ V
3331, 32xpex 7740 . . . 4 ((π‘Œ ↑m 𝑋) Γ— (𝑋 ↑m π‘Œ)) ∈ V
3417fvexi 6906 . . . . . . . . . 10 π‘Œ ∈ V
353fvexi 6906 . . . . . . . . . 10 𝑋 ∈ V
3634, 35elmap 8865 . . . . . . . . 9 (𝑑 ∈ (π‘Œ ↑m 𝑋) ↔ 𝑑:π‘‹βŸΆπ‘Œ)
3735, 34elmap 8865 . . . . . . . . 9 (𝑠 ∈ (𝑋 ↑m π‘Œ) ↔ 𝑠:π‘ŒβŸΆπ‘‹)
3836, 37anbi12i 628 . . . . . . . 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 1133 . . . . . 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 5551 . . . . 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 5683 . . . . 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 4020 . . . 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 5323 . . 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 7568 . 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 2785 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {copab 5211   Γ— cxp 5675  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  NrmCVeccnv 29837  BaseSetcba 29839  Β·π‘–OLDcdip 29953  adjcaj 30001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-aj 30003
This theorem is referenced by:  ajfuni  30112  ajval  30114
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