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Theorem hgmapffval 40351
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 3464 . 2 (𝐾 ∈ 𝑋 β†’ 𝐾 ∈ V)
2 fveq2 6843 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 hgmapval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2795 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6843 . . . . . . 7 (π‘˜ = 𝐾 β†’ (DVecHβ€˜π‘˜) = (DVecHβ€˜πΎ))
65fveq1d 6845 . . . . . 6 (π‘˜ = 𝐾 β†’ ((DVecHβ€˜π‘˜)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘€))
7 fveq2 6843 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (HDMapβ€˜π‘˜) = (HDMapβ€˜πΎ))
87fveq1d 6845 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((HDMapβ€˜π‘˜)β€˜π‘€) = ((HDMapβ€˜πΎ)β€˜π‘€))
9 fveq2 6843 . . . . . . . . . . . . . . . 16 (π‘˜ = 𝐾 β†’ (LCDualβ€˜π‘˜) = (LCDualβ€˜πΎ))
109fveq1d 6845 . . . . . . . . . . . . . . 15 (π‘˜ = 𝐾 β†’ ((LCDualβ€˜π‘˜)β€˜π‘€) = ((LCDualβ€˜πΎ)β€˜π‘€))
1110fveq2d 6847 . . . . . . . . . . . . . 14 (π‘˜ = 𝐾 β†’ ( ·𝑠 β€˜((LCDualβ€˜π‘˜)β€˜π‘€)) = ( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘€)))
1211oveqd 7375 . . . . . . . . . . . . 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 2748 . . . . . . . . . . . 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 3175 . . . . . . . . . . 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 7316 . . . . . . . . . 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 5208 . . . . . . . . 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 2824 . . . . . . . 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 3747 . . . . . . 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 3799 . . . . . 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 3747 . . . . 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 2806 . . . 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 5197 . . 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 40350 . . 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 7179 . 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 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  Vcvv 3446  [wsbc 3740   ↦ cmpt 5189  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  Basecbs 17084  Scalarcsca 17137   ·𝑠 cvsca 17138  LHypclh 38450  DVecHcdvh 39544  LCDualclcd 40052  HDMapchdma 40258  HGMapchg 40349
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-hgmap 40350
This theorem is referenced by:  hgmapfval  40352
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