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Theorem hvmapval 40273
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 = (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 }))
Assertion
Ref Expression
hvmapval (πœ‘ β†’ (π‘€β€˜π‘‹) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋)))))
Distinct variable groups:   𝑑,𝑗,𝑣,𝐾   𝑑,π‘Š   𝑑,𝑂   𝑅,𝑗   𝑗,π‘Š,𝑣   𝑣,𝑉   𝑗,𝑋,𝑑,𝑣
Allowed substitution hints:   πœ‘(𝑣,𝑑,𝑗)   𝐴(𝑣,𝑑,𝑗)   + (𝑣,𝑑,𝑗)   𝑅(𝑣,𝑑)   𝑆(𝑣,𝑑,𝑗)   Β· (𝑣,𝑑,𝑗)   π‘ˆ(𝑣,𝑑,𝑗)   𝐻(𝑣,𝑑,𝑗)   𝑀(𝑣,𝑑,𝑗)   𝑂(𝑣,𝑗)   𝑉(𝑑,𝑗)   0 (𝑣,𝑑,𝑗)

Proof of Theorem hvmapval
Dummy variable π‘₯ 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))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hvmapfval 40272 . . 3 (πœ‘ β†’ 𝑀 = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))))
1312fveq1d 6848 . 2 (πœ‘ β†’ (π‘€β€˜π‘‹) = ((π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯)))))β€˜π‘‹))
14 hvmapval.x . . 3 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
154fvexi 6860 . . . 4 𝑉 ∈ V
1615mptex 7177 . . 3 (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋)))) ∈ V
17 sneq 4600 . . . . . . . 8 (π‘₯ = 𝑋 β†’ {π‘₯} = {𝑋})
1817fveq2d 6850 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘‚β€˜{π‘₯}) = (π‘‚β€˜{𝑋}))
19 oveq2 7369 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (𝑗 Β· π‘₯) = (𝑗 Β· 𝑋))
2019oveq2d 7377 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (𝑑 + (𝑗 Β· π‘₯)) = (𝑑 + (𝑗 Β· 𝑋)))
2120eqeq2d 2744 . . . . . . 7 (π‘₯ = 𝑋 β†’ (𝑣 = (𝑑 + (𝑗 Β· π‘₯)) ↔ 𝑣 = (𝑑 + (𝑗 Β· 𝑋))))
2218, 21rexeqbidv 3319 . . . . . 6 (π‘₯ = 𝑋 β†’ (βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯)) ↔ βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋))))
2322riotabidv 7319 . . . . 5 (π‘₯ = 𝑋 β†’ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))) = (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋))))
2423mpteq2dv 5211 . . . 4 (π‘₯ = 𝑋 β†’ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋)))))
25 eqid 2733 . . . 4 (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯))))) = (π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯)))))
2624, 25fvmptg 6950 . . 3 ((𝑋 ∈ (𝑉 βˆ– { 0 }) ∧ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋)))) ∈ V) β†’ ((π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯)))))β€˜π‘‹) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋)))))
2714, 16, 26sylancl 587 . 2 (πœ‘ β†’ ((π‘₯ ∈ (𝑉 βˆ– { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{π‘₯})𝑣 = (𝑑 + (𝑗 Β· π‘₯)))))β€˜π‘‹) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋)))))
2813, 27eqtrd 2773 1 (πœ‘ β†’ (π‘€β€˜π‘‹) = (𝑣 ∈ 𝑉 ↦ (℩𝑗 ∈ 𝑅 βˆƒπ‘‘ ∈ (π‘‚β€˜{𝑋})𝑣 = (𝑑 + (𝑗 Β· 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070  Vcvv 3447   βˆ– cdif 3911  {csn 4590   ↦ cmpt 5192  β€˜cfv 6500  β„©crio 7316  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  Scalarcsca 17144   ·𝑠 cvsca 17145  0gc0g 17329  LHypclh 38497  DVecHcdvh 39591  ocHcoch 39860  HVMapchvm 40269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-hvmap 40270
This theorem is referenced by:  hvmapvalvalN  40274  hvmapidN  40275  hdmapevec2  40349
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