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Theorem hvmapvalvalN 41158
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 = (LHypβ€˜πΎ)
hvmapval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hvmapval.o 𝑂 = ((ocHβ€˜πΎ)β€˜π‘Š)
hvmapval.v 𝑉 = (Baseβ€˜π‘ˆ)
hvmapval.p + = (+gβ€˜π‘ˆ)
hvmapval.t Β· = ( ·𝑠 β€˜π‘ˆ)
hvmapval.z 0 = (0gβ€˜π‘ˆ)
hvmapval.s 𝑆 = (Scalarβ€˜π‘ˆ)
hvmapval.r 𝑅 = (Baseβ€˜π‘†)
hvmapval.m 𝑀 = ((HVMapβ€˜πΎ)β€˜π‘Š)
hvmapval.k (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
hvmapval.x (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
hvmapval.y (πœ‘ β†’ π‘Œ ∈ 𝑉)
Assertion
Ref Expression
hvmapvalvalN (πœ‘ β†’ ((π‘€β€˜π‘‹)β€˜π‘Œ) = (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))))
Distinct variable groups:   𝑑,𝑗,𝐾   𝑑,π‘Š   𝑑,𝑂   𝑅,𝑗   𝑗,π‘Š   𝑗,𝑋,𝑑   𝑗,π‘Œ,𝑑
Allowed substitution hints:   πœ‘(𝑑,𝑗)   𝐴(𝑑,𝑗)   + (𝑑,𝑗)   𝑅(𝑑)   𝑆(𝑑,𝑗)   Β· (𝑑,𝑗)   π‘ˆ(𝑑,𝑗)   𝐻(𝑑,𝑗)   𝑀(𝑑,𝑗)   𝑂(𝑗)   𝑉(𝑑,𝑗)   0 (𝑑,𝑗)

Proof of Theorem hvmapvalvalN
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 hvmapval.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 hvmapval.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
3 hvmapval.o . . . 4 𝑂 = ((ocHβ€˜πΎ)β€˜π‘Š)
4 hvmapval.v . . . 4 𝑉 = (Baseβ€˜π‘ˆ)
5 hvmapval.p . . . 4 + = (+gβ€˜π‘ˆ)
6 hvmapval.t . . . 4 Β· = ( ·𝑠 β€˜π‘ˆ)
7 hvmapval.z . . . 4 0 = (0gβ€˜π‘ˆ)
8 hvmapval.s . . . 4 𝑆 = (Scalarβ€˜π‘ˆ)
9 hvmapval.r . . . 4 𝑅 = (Baseβ€˜π‘†)
10 hvmapval.m . . . 4 𝑀 = ((HVMapβ€˜πΎ)β€˜π‘Š)
11 hvmapval.k . . . 4 (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
12 hvmapval.x . . . 4 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12hvmapval 41157 . . 3 (πœ‘ β†’ (π‘€β€˜π‘‹) = (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋)))))
1413fveq1d 6893 . 2 (πœ‘ β†’ ((π‘€β€˜π‘‹)β€˜π‘Œ) = ((𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋))))β€˜π‘Œ))
15 hvmapval.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝑉)
16 riotaex 7374 . . 3 (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))) ∈ V
17 eqeq1 2731 . . . . . 6 (𝑦 = π‘Œ β†’ (𝑦 = (𝑑 + (𝑗 Β· 𝑋)) ↔ π‘Œ = (𝑑 + (𝑗 Β· 𝑋))))
1817rexbidv 3173 . . . . 5 (𝑦 = π‘Œ β†’ (βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋)) ↔ βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))))
1918riotabidv 7372 . . . 4 (𝑦 = π‘Œ β†’ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋))) = (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))))
20 eqid 2727 . . . 4 (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋)))) = (𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋))))
2119, 20fvmptg 6997 . . 3 ((π‘Œ ∈ 𝑉 ∧ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))) ∈ V) β†’ ((𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋))))β€˜π‘Œ) = (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))))
2215, 16, 21sylancl 585 . 2 (πœ‘ β†’ ((𝑦 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑦 = (𝑑 + (𝑗 Β· 𝑋))))β€˜π‘Œ) = (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))))
2314, 22eqtrd 2767 1 (πœ‘ β†’ ((π‘€β€˜π‘‹)β€˜π‘Œ) = (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})π‘Œ = (𝑑 + (𝑗 Β· 𝑋))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3065  Vcvv 3469   βˆ– cdif 3941  {csn 4624   ↦ cmpt 5225  β€˜cfv 6542  β„©crio 7369  (class class class)co 7414  Basecbs 17165  +gcplusg 17218  Scalarcsca 17221   ·𝑠 cvsca 17222  0gc0g 17406  LHypclh 39381  DVecHcdvh 40475  ocHcoch 40744  HVMapchvm 41153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-hvmap 41154
This theorem is referenced by: (None)
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