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Theorem hvmapfval 37647
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 37646 . . . . 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 6377 . . . 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 2811 . . 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 6375 . . . . . . . . 9 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
8 hvmapval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
97, 8syl6eqr 2817 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
109fveq2d 6379 . . . . . . 7 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘𝑈))
11 hvmapval.v . . . . . . 7 𝑉 = (Base‘𝑈)
1210, 11syl6eqr 2817 . . . . . 6 (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
139fveq2d 6379 . . . . . . . 8 (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = (0g𝑈))
14 hvmapval.z . . . . . . . 8 0 = (0g𝑈)
1513, 14syl6eqr 2817 . . . . . . 7 (𝑤 = 𝑊 → (0g‘((DVecH‘𝐾)‘𝑤)) = 0 )
1615sneqd 4346 . . . . . 6 (𝑤 = 𝑊 → {(0g‘((DVecH‘𝐾)‘𝑤))} = { 0 })
1712, 16difeq12d 3891 . . . . 5 (𝑤 = 𝑊 → ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) = (𝑉 ∖ { 0 }))
189fveq2d 6379 . . . . . . . . . 10 (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = (Scalar‘𝑈))
19 hvmapval.s . . . . . . . . . 10 𝑆 = (Scalar‘𝑈)
2018, 19syl6eqr 2817 . . . . . . . . 9 (𝑤 = 𝑊 → (Scalar‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
2120fveq2d 6379 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = (Base‘𝑆))
22 hvmapval.r . . . . . . . 8 𝑅 = (Base‘𝑆)
2321, 22syl6eqr 2817 . . . . . . 7 (𝑤 = 𝑊 → (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))) = 𝑅)
24 fveq2 6375 . . . . . . . . . 10 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
25 hvmapval.o . . . . . . . . . 10 𝑂 = ((ocH‘𝐾)‘𝑊)
2624, 25syl6eqr 2817 . . . . . . . . 9 (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = 𝑂)
2726fveq1d 6377 . . . . . . . 8 (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘{𝑥}) = (𝑂‘{𝑥}))
289fveq2d 6379 . . . . . . . . . . 11 (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = (+g𝑈))
29 hvmapval.p . . . . . . . . . . 11 + = (+g𝑈)
3028, 29syl6eqr 2817 . . . . . . . . . 10 (𝑤 = 𝑊 → (+g‘((DVecH‘𝐾)‘𝑤)) = + )
31 eqidd 2766 . . . . . . . . . 10 (𝑤 = 𝑊𝑡 = 𝑡)
329fveq2d 6379 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = ( ·𝑠𝑈))
33 hvmapval.t . . . . . . . . . . . 12 · = ( ·𝑠𝑈)
3432, 33syl6eqr 2817 . . . . . . . . . . 11 (𝑤 = 𝑊 → ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)) = · )
3534oveqd 6859 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥) = (𝑗 · 𝑥))
3630, 31, 35oveq123d 6863 . . . . . . . . 9 (𝑤 = 𝑊 → (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) = (𝑡 + (𝑗 · 𝑥)))
3736eqeq2d 2775 . . . . . . . 8 (𝑤 = 𝑊 → (𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑥))))
3827, 37rexeqbidv 3301 . . . . . . 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 6806 . . . . . 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 4892 . . . . 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 4892 . . . 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 2765 . . . 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))))))
4311fvexi 6389 . . . . . 6 𝑉 ∈ V
44 difexg 4969 . . . . . 6 (𝑉 ∈ V → (𝑉 ∖ { 0 }) ∈ V)
4543, 44ax-mp 5 . . . . 5 (𝑉 ∖ { 0 }) ∈ V
4645mptex 6679 . . . 4 (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))) ∈ V
4741, 42, 46fvmpt 6471 . . 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))))))
486, 47sylan9eq 2819 . 2 ((𝐾𝐴𝑊𝐻) → 𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
491, 48syl 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 1652  wcel 2155  wrex 3056  Vcvv 3350  cdif 3729  {csn 4334  cmpt 4888  cfv 6068  crio 6802  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Scalarcsca 16219   ·𝑠 cvsca 16220  0gc0g 16368  LHypclh 35872  DVecHcdvh 36966  ocHcoch 37235  HVMapchvm 37644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-hvmap 37645
This theorem is referenced by:  hvmapval  37648  hvmap1o  37651  hvmaplkr  37656
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