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Theorem hvmapfval 40625
Description: Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
Hypotheses
Ref Expression
hvmapval.h 𝐻 = (LHypβ€˜πΎ)
hvmapval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hvmapval.o 𝑂 = ((ocHβ€˜πΎ)β€˜π‘Š)
hvmapval.v 𝑉 = (Baseβ€˜π‘ˆ)
hvmapval.p + = (+gβ€˜π‘ˆ)
hvmapval.t Β· = ( ·𝑠 β€˜π‘ˆ)
hvmapval.z 0 = (0gβ€˜π‘ˆ)
hvmapval.s 𝑆 = (Scalarβ€˜π‘ˆ)
hvmapval.r 𝑅 = (Baseβ€˜π‘†)
hvmapval.m 𝑀 = ((HVMapβ€˜πΎ)β€˜π‘Š)
hvmapval.k (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
Assertion
Ref Expression
hvmapfval (πœ‘ β†’ 𝑀 = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))))
Distinct variable groups:   𝑑,𝑗,𝑣,π‘₯,𝐾   𝑑,π‘Š   𝑑,𝑂   𝑅,𝑗   π‘₯,𝑉   𝑗,π‘Š,𝑣,π‘₯   π‘₯, 0
Allowed substitution hints:   πœ‘(π‘₯,𝑣,𝑑,𝑗)   𝐴(π‘₯,𝑣,𝑑,𝑗)   + (π‘₯,𝑣,𝑑,𝑗)   𝑅(π‘₯,𝑣,𝑑)   𝑆(π‘₯,𝑣,𝑑,𝑗)   Β· (π‘₯,𝑣,𝑑,𝑗)   π‘ˆ(π‘₯,𝑣,𝑑,𝑗)   𝐻(π‘₯,𝑣,𝑑,𝑗)   𝑀(π‘₯,𝑣,𝑑,𝑗)   𝑂(π‘₯,𝑣,𝑗)   𝑉(𝑣,𝑑,𝑗)   0 (𝑣,𝑑,𝑗)

Proof of Theorem hvmapfval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 hvmapval.k . 2 (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
2 hvmapval.m . . . 4 𝑀 = ((HVMapβ€˜πΎ)β€˜π‘Š)
3 hvmapval.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
43hvmapffval 40624 . . . . 5 (𝐾 ∈ 𝐴 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‘₯)))))))
54fveq1d 6893 . . . 4 (𝐾 ∈ 𝐴 β†’ ((HVMapβ€˜πΎ)β€˜π‘Š) = ((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‘₯))))))β€˜π‘Š))
62, 5eqtrid 2784 . . 3 (𝐾 ∈ 𝐴 β†’ 𝑀 = ((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‘₯))))))β€˜π‘Š))
7 fveq2 6891 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = ((DVecHβ€˜πΎ)β€˜π‘Š))
8 hvmapval.u . . . . . . . . 9 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
97, 8eqtr4di 2790 . . . . . . . 8 (𝑀 = π‘Š β†’ ((DVecHβ€˜πΎ)β€˜π‘€) = π‘ˆ)
109fveq2d 6895 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (Baseβ€˜π‘ˆ))
11 hvmapval.v . . . . . . 7 𝑉 = (Baseβ€˜π‘ˆ)
1210, 11eqtr4di 2790 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝑉)
139fveq2d 6895 . . . . . . . 8 (𝑀 = π‘Š β†’ (0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (0gβ€˜π‘ˆ))
14 hvmapval.z . . . . . . . 8 0 = (0gβ€˜π‘ˆ)
1513, 14eqtr4di 2790 . . . . . . 7 (𝑀 = π‘Š β†’ (0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 0 )
1615sneqd 4640 . . . . . 6 (𝑀 = π‘Š β†’ {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))} = { 0 })
1712, 16difeq12d 4123 . . . . 5 (𝑀 = π‘Š β†’ ((Baseβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) βˆ– {(0gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))}) = (𝑉 βˆ– { 0 }))
189fveq2d 6895 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (Scalarβ€˜π‘ˆ))
19 hvmapval.s . . . . . . . . . 10 𝑆 = (Scalarβ€˜π‘ˆ)
2018, 19eqtr4di 2790 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = 𝑆)
2120fveq2d 6895 . . . . . . . 8 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€))) = (Baseβ€˜π‘†))
22 hvmapval.r . . . . . . . 8 𝑅 = (Baseβ€˜π‘†)
2321, 22eqtr4di 2790 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜(Scalarβ€˜((DVecHβ€˜πΎ)β€˜π‘€))) = 𝑅)
24 fveq2 6891 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ((ocHβ€˜πΎ)β€˜π‘€) = ((ocHβ€˜πΎ)β€˜π‘Š))
25 hvmapval.o . . . . . . . . . 10 𝑂 = ((ocHβ€˜πΎ)β€˜π‘Š)
2624, 25eqtr4di 2790 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((ocHβ€˜πΎ)β€˜π‘€) = 𝑂)
2726fveq1d 6893 . . . . . . . 8 (𝑀 = π‘Š β†’ (((ocHβ€˜πΎ)β€˜π‘€)β€˜{π‘₯}) = (π‘‚β€˜{π‘₯}))
289fveq2d 6895 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = (+gβ€˜π‘ˆ))
29 hvmapval.p . . . . . . . . . . 11 + = (+gβ€˜π‘ˆ)
3028, 29eqtr4di 2790 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€)) = + )
31 eqidd 2733 . . . . . . . . . 10 (𝑀 = π‘Š β†’ 𝑑 = 𝑑)
329fveq2d 6895 . . . . . . . . . . . 12 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€)) = ( ·𝑠 β€˜π‘ˆ))
33 hvmapval.t . . . . . . . . . . . 12 Β· = ( ·𝑠 β€˜π‘ˆ)
3432, 33eqtr4di 2790 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€)) = Β· )
3534oveqd 7425 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯) = (𝑗 Β· π‘₯))
3630, 31, 35oveq123d 7429 . . . . . . . . 9 (𝑀 = π‘Š β†’ (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)) = (𝑑 + (𝑗 Β· π‘₯)))
3736eqeq2d 2743 . . . . . . . 8 (𝑀 = π‘Š β†’ (𝑣 = (𝑑(+gβ€˜((DVecHβ€˜πΎ)β€˜π‘€))(𝑗( ·𝑠 β€˜((DVecHβ€˜πΎ)β€˜π‘€))π‘₯)) ↔ 𝑣 = (𝑑 + (𝑗 Β· π‘₯))))
3827, 37rexeqbidv 3343 . . . . . . 7 (𝑀 = π‘Š β†’ (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‘₯)) ↔ βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))
3923, 38riotaeqbidv 7367 . . . . . 6 (𝑀 = π‘Š β†’ (℩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‘₯))) = (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))
4012, 39mpteq12dv 5239 . . . . 5 (𝑀 = π‘Š β†’ (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‘₯)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯)))))
4117, 40mpteq12dv 5239 . . . 4 (𝑀 = π‘Š β†’ (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‘₯))))) = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))))
42 eqid 2732 . . . 4 (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‘₯)))))) = (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‘₯))))))
4311fvexi 6905 . . . . . 6 𝑉 ∈ V
4443difexi 5328 . . . . 5 (𝑉 βˆ– { 0 }) ∈ V
4544mptex 7224 . . . 4 (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))) ∈ V
4641, 42, 45fvmpt 6998 . . 3 (π‘Š ∈ 𝐻 β†’ ((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‘₯))))))β€˜π‘Š) = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))))
476, 46sylan9eq 2792 . 2 ((𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻) β†’ 𝑀 = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))))
481, 47syl 17 1 (πœ‘ β†’ 𝑀 = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βˆ– cdif 3945  {csn 4628   ↦ cmpt 5231  β€˜cfv 6543  β„©crio 7363  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  Scalarcsca 17199   ·𝑠 cvsca 17200  0gc0g 17384  LHypclh 38850  DVecHcdvh 39944  ocHcoch 40213  HVMapchvm 40622
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 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-hvmap 40623
This theorem is referenced by:  hvmapval  40626  hvmap1o  40629  hvmaplkr  40634
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