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Theorem hvmapvalvalN 41743
Description: Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)
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 }))
hvmapval.y (𝜑𝑌𝑉)
Assertion
Ref Expression
hvmapvalvalN (𝜑 → ((𝑀𝑋)‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
Distinct variable groups:   𝑡,𝑗,𝐾   𝑡,𝑊   𝑡,𝑂   𝑅,𝑗   𝑗,𝑊   𝑗,𝑋,𝑡   𝑗,𝑌,𝑡
Allowed substitution hints:   𝜑(𝑡,𝑗)   𝐴(𝑡,𝑗)   + (𝑡,𝑗)   𝑅(𝑡)   𝑆(𝑡,𝑗)   · (𝑡,𝑗)   𝑈(𝑡,𝑗)   𝐻(𝑡,𝑗)   𝑀(𝑡,𝑗)   𝑂(𝑗)   𝑉(𝑡,𝑗)   0 (𝑡,𝑗)

Proof of Theorem hvmapvalvalN
Dummy variable 𝑦 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))
12 hvmapval.x . . . 4 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12hvmapval 41742 . . 3 (𝜑 → (𝑀𝑋) = (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))))
1413fveq1d 6908 . 2 (𝜑 → ((𝑀𝑋)‘𝑌) = ((𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌))
15 hvmapval.y . . 3 (𝜑𝑌𝑉)
16 riotaex 7391 . . 3 (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V
17 eqeq1 2738 . . . . . 6 (𝑦 = 𝑌 → (𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ 𝑌 = (𝑡 + (𝑗 · 𝑋))))
1817rexbidv 3176 . . . . 5 (𝑦 = 𝑌 → (∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
1918riotabidv 7389 . . . 4 (𝑦 = 𝑌 → (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
20 eqid 2734 . . . 4 (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))) = (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))
2119, 20fvmptg 7013 . . 3 ((𝑌𝑉 ∧ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V) → ((𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
2215, 16, 21sylancl 586 . 2 (𝜑 → ((𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
2314, 22eqtrd 2774 1 (𝜑 → ((𝑀𝑋)‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wrex 3067  Vcvv 3477  cdif 3959  {csn 4630  cmpt 5230  cfv 6562  crio 7386  (class class class)co 7430  Basecbs 17244  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485  LHypclh 39966  DVecHcdvh 41060  ocHcoch 41329  HVMapchvm 41738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-hvmap 41739
This theorem is referenced by: (None)
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