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Theorem hvmapvalvalN 41096
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 41095 . . 3 (𝜑 → (𝑀𝑋) = (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))))
1413fveq1d 6893 . 2 (𝜑 → ((𝑀𝑋)‘𝑌) = ((𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌))
15 hvmapval.y . . 3 (𝜑𝑌𝑉)
16 riotaex 7372 . . 3 (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))) ∈ V
17 eqeq1 2735 . . . . . 6 (𝑦 = 𝑌 → (𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ 𝑌 = (𝑡 + (𝑗 · 𝑋))))
1817rexbidv 3177 . . . . 5 (𝑦 = 𝑌 → (∃𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)) ↔ ∃𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
1918riotabidv 7370 . . . 4 (𝑦 = 𝑌 → (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
20 eqid 2731 . . . 4 (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋)))) = (𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))
2119, 20fvmptg 6996 . . 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 585 . 2 (𝜑 → ((𝑦𝑉 ↦ (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑦 = (𝑡 + (𝑗 · 𝑋))))‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
2314, 22eqtrd 2771 1 (𝜑 → ((𝑀𝑋)‘𝑌) = (𝑗𝑅𝑡 ∈ (𝑂‘{𝑋})𝑌 = (𝑡 + (𝑗 · 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wrex 3069  Vcvv 3473  cdif 3945  {csn 4628  cmpt 5231  cfv 6543  crio 7367  (class class class)co 7412  Basecbs 17151  +gcplusg 17204  Scalarcsca 17207   ·𝑠 cvsca 17208  0gc0g 17392  LHypclh 39319  DVecHcdvh 40413  ocHcoch 40682  HVMapchvm 41091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-hvmap 41092
This theorem is referenced by: (None)
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