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Theorem hvmapvalvalN 36068
 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 36067 . . 3 (𝜑 → (𝑀𝑋) = (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))))
1413fveq1d 6105 . 2 (𝜑 → ((𝑀𝑋)‘𝑌) = ((𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌))
15 hvmapval.y . . 3 (𝜑𝑌𝑉)
16 riotaex 6515 . . 3 (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V
17 eqeq1 2614 . . . . . 6 (𝑦 = 𝑌 → (𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ 𝑌 = (𝑡 + (𝑗 · 𝑋))))
1817rexbidv 3034 . . . . 5 (𝑦 = 𝑌 → (∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
1918riotabidv 6513 . . . 4 (𝑦 = 𝑌 → (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
20 eqid 2610 . . . 4 (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))) = (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))
2119, 20fvmptg 6189 . . 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 693 . 2 (𝜑 → ((𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
2314, 22eqtrd 2644 1 (𝜑 → ((𝑀𝑋)‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537  {csn 4125   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  LHypclh 34288  DVecHcdvh 35385  ocHcoch 35654  HVMapchvm 36063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-hvmap 36064 This theorem is referenced by: (None)
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