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Theorem hvmapval 37551
Description: Value of 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))
hvmapval.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
Assertion
Ref Expression
hvmapval (𝜑 → (𝑀𝑋) = (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
Distinct variable groups:   𝑡,𝑗,𝑣,𝐾   𝑡,𝑊   𝑡,𝑂   𝑅,𝑗   𝑗,𝑊,𝑣   𝑣,𝑉   𝑗,𝑋,𝑡,𝑣
Allowed substitution hints:   𝜑(𝑣,𝑡,𝑗)   𝐴(𝑣,𝑡,𝑗)   + (𝑣,𝑡,𝑗)   𝑅(𝑣,𝑡)   𝑆(𝑣,𝑡,𝑗)   · (𝑣,𝑡,𝑗)   𝑈(𝑣,𝑡,𝑗)   𝐻(𝑣,𝑡,𝑗)   𝑀(𝑣,𝑡,𝑗)   𝑂(𝑣,𝑗)   𝑉(𝑡,𝑗)   0 (𝑣,𝑡,𝑗)

Proof of Theorem hvmapval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hvmapval.h . . . 4 𝐻 = (LHyp‘𝐾)
2 hvmapval.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 hvmapval.o . . . 4 𝑂 = ((ocH‘𝐾)‘𝑊)
4 hvmapval.v . . . 4 𝑉 = (Base‘𝑈)
5 hvmapval.p . . . 4 + = (+g𝑈)
6 hvmapval.t . . . 4 · = ( ·𝑠𝑈)
7 hvmapval.z . . . 4 0 = (0g𝑈)
8 hvmapval.s . . . 4 𝑆 = (Scalar‘𝑈)
9 hvmapval.r . . . 4 𝑅 = (Base‘𝑆)
10 hvmapval.m . . . 4 𝑀 = ((HVMap‘𝐾)‘𝑊)
11 hvmapval.k . . . 4 (𝜑 → (𝐾𝐴𝑊𝐻))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hvmapfval 37550 . . 3 (𝜑𝑀 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))))
1312fveq1d 6354 . 2 (𝜑 → (𝑀𝑋) = ((𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))‘𝑋))
14 hvmapval.x . . 3 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
15 fvex 6362 . . . . 5 (Base‘𝑈) ∈ V
164, 15eqeltri 2835 . . . 4 𝑉 ∈ V
1716mptex 6650 . . 3 (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) ∈ V
18 sneq 4331 . . . . . . . 8 (𝑥 = 𝑋 → {𝑥} = {𝑋})
1918fveq2d 6356 . . . . . . 7 (𝑥 = 𝑋 → (𝑂‘{𝑥}) = (𝑂‘{𝑋}))
20 oveq2 6821 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑗 · 𝑥) = (𝑗 · 𝑋))
2120oveq2d 6829 . . . . . . . 8 (𝑥 = 𝑋 → (𝑡 + (𝑗 · 𝑥)) = (𝑡 + (𝑗 · 𝑋)))
2221eqeq2d 2770 . . . . . . 7 (𝑥 = 𝑋 → (𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ 𝑣 = (𝑡 + (𝑗 · 𝑋))))
2319, 22rexeqbidv 3292 . . . . . 6 (𝑥 = 𝑋 → (∃𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))
2423riotabidv 6776 . . . . 5 (𝑥 = 𝑋 → (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋))))
2524mpteq2dv 4897 . . . 4 (𝑥 = 𝑋 → (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))) = (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
26 eqid 2760 . . . 4 (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥))))) = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))
2725, 26fvmptg 6442 . . 3 ((𝑋 ∈ (𝑉 ∖ { 0 }) ∧ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))) ∈ V) → ((𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))‘𝑋) = (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
2814, 17, 27sylancl 697 . 2 (𝜑 → ((𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑥})𝑣 = (𝑡 + (𝑗 · 𝑥)))))‘𝑋) = (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
2913, 28eqtrd 2794 1 (𝜑 → (𝑀𝑋) = (𝑣𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑣 = (𝑡 + (𝑗 · 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  cdif 3712  {csn 4321  cmpt 4881  cfv 6049  crio 6773  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Scalarcsca 16146   ·𝑠 cvsca 16147  0gc0g 16302  LHypclh 35773  DVecHcdvh 36869  ocHcoch 37138  HVMapchvm 37547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-hvmap 37548
This theorem is referenced by:  hvmapvalvalN  37552  hvmapidN  37553  hdmapevec2  37630
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