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Theorem hgmapffval 40744
Description: Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
Hypothesis
Ref Expression
hgmapval.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
hgmapffval (𝐾 ∈ 𝑋 β†’ (HGMapβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))}))
Distinct variable groups:   𝑀,𝐻   π‘Ž,𝑏,π‘š,𝑒,𝑣,𝑀,π‘₯,𝑦,𝐾
Allowed substitution hints:   𝐻(π‘₯,𝑦,𝑣,𝑒,π‘š,π‘Ž,𝑏)   𝑋(π‘₯,𝑦,𝑀,𝑣,𝑒,π‘š,π‘Ž,𝑏)

Proof of Theorem hgmapffval
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝑋 β†’ 𝐾 ∈ V)
2 fveq2 6888 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 hgmapval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2790 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6888 . . . . . . 7 (π‘˜ = 𝐾 β†’ (DVecHβ€˜π‘˜) = (DVecHβ€˜πΎ))
65fveq1d 6890 . . . . . 6 (π‘˜ = 𝐾 β†’ ((DVecHβ€˜π‘˜)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘€))
7 fveq2 6888 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (HDMapβ€˜π‘˜) = (HDMapβ€˜πΎ))
87fveq1d 6890 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((HDMapβ€˜π‘˜)β€˜π‘€) = ((HDMapβ€˜πΎ)β€˜π‘€))
9 fveq2 6888 . . . . . . . . . . . . . . . 16 (π‘˜ = 𝐾 β†’ (LCDualβ€˜π‘˜) = (LCDualβ€˜πΎ))
109fveq1d 6890 . . . . . . . . . . . . . . 15 (π‘˜ = 𝐾 β†’ ((LCDualβ€˜π‘˜)β€˜π‘€) = ((LCDualβ€˜πΎ)β€˜π‘€))
1110fveq2d 6892 . . . . . . . . . . . . . 14 (π‘˜ = 𝐾 β†’ ( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€)) = ( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€)))
1211oveqd 7422 . . . . . . . . . . . . 13 (π‘˜ = 𝐾 β†’ (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£)))
1312eqeq2d 2743 . . . . . . . . . . . 12 (π‘˜ = 𝐾 β†’ ((π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)) ↔ (π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))
1413ralbidv 3177 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ (βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)) ↔ βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))
1514riotabidv 7363 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£))) = (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))
1615mpteq2dv 5249 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)))) = (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£)))))
1716eleq2d 2819 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)))) ↔ π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))))
188, 17sbceqbid 3783 . . . . . . 7 (π‘˜ = 𝐾 β†’ ([((HDMapβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)))) ↔ [((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))))
1918sbcbidv 3835 . . . . . 6 (π‘˜ = 𝐾 β†’ ([(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)))) ↔ [(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))))
206, 19sbceqbid 3783 . . . . 5 (π‘˜ = 𝐾 β†’ ([((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£)))) ↔ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))))
2120abbidv 2801 . . . 4 (π‘˜ = 𝐾 β†’ {π‘Ž ∣ [((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£))))} = {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))})
224, 21mpteq12dv 5238 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {π‘Ž ∣ [((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£))))}) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))}))
23 df-hgmap 40743 . . 3 HGMap = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {π‘Ž ∣ [((DVecHβ€˜π‘˜)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜π‘˜)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€))(π‘šβ€˜π‘£))))}))
2422, 23, 3mptfvmpt 7226 . 2 (𝐾 ∈ V β†’ (HGMapβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))}))
251, 24syl 17 1 (𝐾 ∈ 𝑋 β†’ (HGMapβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {π‘Ž ∣ [((DVecHβ€˜πΎ)β€˜π‘€) / 𝑒][(Baseβ€˜(Scalarβ€˜π‘’)) / 𝑏][((HDMapβ€˜πΎ)β€˜π‘€) / π‘š]π‘Ž ∈ (π‘₯ ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 βˆ€π‘£ ∈ (Baseβ€˜π‘’)(π‘šβ€˜(π‘₯( ·𝑠 β€˜π‘’)𝑣)) = (𝑦( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€))(π‘šβ€˜π‘£))))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  Vcvv 3474  [wsbc 3776   ↦ cmpt 5230  β€˜cfv 6540  β„©crio 7360  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  LHypclh 38843  DVecHcdvh 39937  LCDualclcd 40445  HDMapchdma 40651  HGMapchg 40742
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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-hgmap 40743
This theorem is referenced by:  hgmapfval  40745
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