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Theorem hvmapffval 40933
Description: Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
Hypothesis
Ref Expression
hvmapval.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
hvmapffval (𝐾 ∈ 𝑋 β†’ (HVMapβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)))))))
Distinct variable groups:   𝑀,𝐻   𝑑,𝑗,𝑣,π‘₯,𝑀,𝐾
Allowed substitution hints:   𝐻(π‘₯,𝑣,𝑑,𝑗)   𝑋(π‘₯,𝑀,𝑣,𝑑,𝑗)

Proof of Theorem hvmapffval
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3492 . 2 (𝐾 ∈ 𝑋 β†’ 𝐾 ∈ V)
2 fveq2 6892 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 hvmapval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2789 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6892 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (DVecHβ€˜π‘˜) = (DVecHβ€˜πΎ))
65fveq1d 6894 . . . . . . 7 (π‘˜ = 𝐾 β†’ ((DVecHβ€˜π‘˜)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘€))
76fveq2d 6896 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) = (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))
86fveq2d 6896 . . . . . . 7 (π‘˜ = 𝐾 β†’ (0gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) = (0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))
98sneqd 4641 . . . . . 6 (π‘˜ = 𝐾 β†’ {(0gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))} = {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))})
107, 9difeq12d 4124 . . . . 5 (π‘˜ = 𝐾 β†’ ((Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))}) = ((Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))}))
116fveq2d 6896 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (Scalarβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) = (Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))
1211fveq2d 6896 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))) = (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€))))
13 fveq2 6892 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (ocHβ€˜π‘˜) = (ocHβ€˜πΎ))
1413fveq1d 6894 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ ((ocHβ€˜π‘˜)β€˜π‘€) = ((ocHβ€˜πΎ)β€˜π‘€))
1514fveq1d 6894 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜{π‘₯}) = (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯}))
166fveq2d 6896 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) = (+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))
17 eqidd 2732 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ 𝑑 = 𝑑)
186fveq2d 6896 . . . . . . . . . . 11 (π‘˜ = 𝐾 β†’ ( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€)) = ( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€)))
1918oveqd 7429 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯) = (𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯))
2016, 17, 19oveq123d 7433 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯)) = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)))
2120eqeq2d 2742 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (𝑣 = (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯)) ↔ 𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯))))
2215, 21rexeqbidv 3342 . . . . . . 7 (π‘˜ = 𝐾 β†’ (βˆƒπ‘‘ ∈ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯)) ↔ βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯))))
2312, 22riotaeqbidv 7371 . . . . . 6 (π‘˜ = 𝐾 β†’ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯))) = (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯))))
247, 23mpteq12dv 5240 . . . . 5 (π‘˜ = 𝐾 β†’ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯)))) = (𝑣 ∈ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)))))
2510, 24mpteq12dv 5240 . . . 4 (π‘˜ = 𝐾 β†’ (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯))))) = (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯))))))
264, 25mpteq12dv 5240 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯)))))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)))))))
27 df-hvmap 40932 . . 3 HVMap = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜π‘˜)β€˜π‘€))π‘₯)))))))
2826, 27, 3mptfvmpt 7233 . 2 (𝐾 ∈ V β†’ (HVMapβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)))))))
291, 28syl 17 1 (𝐾 ∈ 𝑋 β†’ (HVMapβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ ((Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))}) ↦ (𝑣 ∈ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) ↦ (℩𝑗 ∈ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)))βˆƒπ‘‘ ∈ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯})𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1540   ∈ wcel 2105  βˆƒwrex 3069  Vcvv 3473   βˆ– cdif 3946  {csn 4629   ↦ cmpt 5232  β€˜cfv 6544  β„©crio 7367  (class class class)co 7412  Basecbs 17149  +gcplusg 17202  Scalarcsca 17205   ·𝑠 cvsca 17206  0gc0g 17390  LHypclh 39159  DVecHcdvh 40253  ocHcoch 40522  HVMapchvm 40931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-hvmap 40932
This theorem is referenced by:  hvmapfval  40934
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