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Theorem hvmapfval 36567
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 = (LHyp‘𝐾)
hvmapval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hvmapval.o 𝑂 = ((ocH‘𝐾)‘𝑊)
hvmapval.v 𝑉 = (Base‘𝑈)
hvmapval.p + = (+g𝑈)
hvmapval.t · = ( ·𝑠𝑈)
hvmapval.z 0 = (0g𝑈)
hvmapval.s 𝑆 = (Scalar‘𝑈)
hvmapval.r 𝑅 = (Base‘𝑆)
hvmapval.m 𝑀 = ((HVMap‘𝐾)‘𝑊)
hvmapval.k (𝜑 → (𝐾𝐴𝑊𝐻))
Assertion
Ref Expression
hvmapfval (𝜑𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
Distinct variable groups:   𝑡,𝑗,𝑣,𝑥,𝐾   𝑡,𝑊   𝑡,𝑂   𝑅,𝑗   𝑥,𝑉   𝑗,𝑊,𝑣,𝑥   𝑥, 0
Allowed substitution hints:   𝜑(𝑥,𝑣,𝑡,𝑗)   𝐴(𝑥,𝑣,𝑡,𝑗)   + (𝑥,𝑣,𝑡,𝑗)   𝑅(𝑥,𝑣,𝑡)   𝑆(𝑥,𝑣,𝑡,𝑗)   · (𝑥,𝑣,𝑡,𝑗)   𝑈(𝑥,𝑣,𝑡,𝑗)   𝐻(𝑥,𝑣,𝑡,𝑗)   𝑀(𝑥,𝑣,𝑡,𝑗)   𝑂(𝑥,𝑣,𝑗)   𝑉(𝑣,𝑡,𝑗)   0 (𝑣,𝑡,𝑗)

Proof of Theorem hvmapfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 hvmapval.k . 2 (𝜑 → (𝐾𝐴𝑊𝐻))
2 hvmapval.m . . . 4 𝑀 = ((HVMap‘𝐾)‘𝑊)
3 hvmapval.h . . . . . 6 𝐻 = (LHyp‘𝐾)
43hvmapffval 36566 . . . . 5 (𝐾𝐴 → (HVMap‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))))
54fveq1d 6160 . . . 4 (𝐾𝐴 → ((HVMap‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))‘𝑊))
62, 5syl5eq 2667 . . 3 (𝐾𝐴𝑀 = ((𝑤𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))‘𝑊))
7 fveq2 6158 . . . . . . . . 9 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 hvmapval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8syl6eqr 2673 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
109fveq2d 6162 . . . . . . 7 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
11 hvmapval.v . . . . . . 7 𝑉 = (Base‘𝑈)
1210, 11syl6eqr 2673 . . . . . 6 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
139fveq2d 6162 . . . . . . . 8 (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = (0g𝑈))
14 hvmapval.z . . . . . . . 8 0 = (0g𝑈)
1513, 14syl6eqr 2673 . . . . . . 7 (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = 0 )
1615sneqd 4167 . . . . . 6 (𝑤 = 𝑊 → {(0g‘((DVecH‘𝐾)‘𝑤))} = { 0 })
1712, 16difeq12d 3713 . . . . 5 (𝑤 = 𝑊 → ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) = (𝑉 ∖ { 0 }))
189fveq2d 6162 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = (Scalar‘𝑈))
19 hvmapval.s . . . . . . . . . 10 𝑆 = (Scalar‘𝑈)
2018, 19syl6eqr 2673 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
2120fveq2d 6162 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = (Base‘𝑆))
22 hvmapval.r . . . . . . . 8 𝑅 = (Base‘𝑆)
2321, 22syl6eqr 2673 . . . . . . 7 (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = 𝑅)
24 fveq2 6158 . . . . . . . . . 10 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
25 hvmapval.o . . . . . . . . . 10 𝑂 = ((ocH‘𝐾)‘𝑊)
2624, 25syl6eqr 2673 . . . . . . . . 9 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = 𝑂)
2726fveq1d 6160 . . . . . . . 8 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘{𝑥}) = (𝑂‘{𝑥}))
289fveq2d 6162 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = (+g𝑈))
29 hvmapval.p . . . . . . . . . . 11 + = (+g𝑈)
3028, 29syl6eqr 2673 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = + )
31 eqidd 2622 . . . . . . . . . 10 (𝑤 = 𝑊𝑡 = 𝑡)
329fveq2d 6162 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = ( ·𝑠𝑈))
33 hvmapval.t . . . . . . . . . . . 12 · = ( ·𝑠𝑈)
3432, 33syl6eqr 2673 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = · )
3534oveqd 6632 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥) = (𝑗 · 𝑥))
3630, 31, 35oveq123d 6636 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) = (𝑡 + (𝑗 · 𝑥)))
3736eqeq2d 2631 . . . . . . . 8 (𝑤 = 𝑊 → (𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑥))))
3827, 37rexeqbidv 3146 . . . . . . 7 (𝑤 = 𝑊 → (∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) ↔ ∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))
3923, 38riotaeqbidv 6579 . . . . . 6 (𝑤 = 𝑊 → (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))
4012, 39mpteq12dv 4703 . . . . 5 (𝑤 = 𝑊 → (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))) = (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))
4117, 40mpteq12dv 4703 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
42 eqid 2621 . . . 4 (𝑤𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))) = (𝑤𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))
43 fvex 6168 . . . . . . 7 (Base‘𝑈) ∈ V
4411, 43eqeltri 2694 . . . . . 6 𝑉 ∈ V
45 difexg 4778 . . . . . 6 (𝑉 ∈ V → (𝑉 ∖ { 0 }) ∈ V)
4644, 45ax-mp 5 . . . . 5 (𝑉 ∖ { 0 }) ∈ V
4746mptex 6451 . . . 4 (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))) ∈ V
4841, 42, 47fvmpt 6249 . . 3 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))‘𝑊) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
496, 48sylan9eq 2675 . 2 ((𝐾𝐴𝑊𝐻) → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
501, 49syl 17 1 (𝜑𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  Vcvv 3190  cdif 3557  {csn 4155  cmpt 4683  cfv 5857  crio 6575  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  Scalarcsca 15884   ·𝑠 cvsca 15885  0gc0g 16040  LHypclh 34789  DVecHcdvh 35886  ocHcoch 36155  HVMapchvm 36564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-hvmap 36565
This theorem is referenced by:  hvmapval  36568  hvmap1o  36571  hvmaplkr  36576
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