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Theorem hgmapval 39005
Description: Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 39000. (Contributed by NM, 25-Mar-2015.)
Hypotheses
Ref Expression
hgmapval.h 𝐻 = (LHyp‘𝐾)
hgmapfval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hgmapfval.v 𝑉 = (Base‘𝑈)
hgmapfval.t · = ( ·𝑠𝑈)
hgmapfval.r 𝑅 = (Scalar‘𝑈)
hgmapfval.b 𝐵 = (Base‘𝑅)
hgmapfval.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hgmapfval.s = ( ·𝑠𝐶)
hgmapfval.m 𝑀 = ((HDMap‘𝐾)‘𝑊)
hgmapfval.i 𝐼 = ((HGMap‘𝐾)‘𝑊)
hgmapfval.k (𝜑 → (𝐾𝑌𝑊𝐻))
hgmapval.x (𝜑𝑋𝐵)
Assertion
Ref Expression
hgmapval (𝜑 → (𝐼𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
Distinct variable groups:   𝑦,𝑣,𝐾   𝑣,𝐵,𝑦   𝑣,𝑀,𝑦   𝑣,𝑈,𝑦   𝑣,𝑉   𝑣,𝑊,𝑦   𝑣,𝑋,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑣)   𝐶(𝑦,𝑣)   𝑅(𝑦,𝑣)   (𝑦,𝑣)   · (𝑦,𝑣)   𝐻(𝑦,𝑣)   𝐼(𝑦,𝑣)   𝑉(𝑦)   𝑌(𝑦,𝑣)

Proof of Theorem hgmapval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hgmapval.h . . . 4 𝐻 = (LHyp‘𝐾)
2 hgmapfval.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 hgmapfval.v . . . 4 𝑉 = (Base‘𝑈)
4 hgmapfval.t . . . 4 · = ( ·𝑠𝑈)
5 hgmapfval.r . . . 4 𝑅 = (Scalar‘𝑈)
6 hgmapfval.b . . . 4 𝐵 = (Base‘𝑅)
7 hgmapfval.c . . . 4 𝐶 = ((LCDual‘𝐾)‘𝑊)
8 hgmapfval.s . . . 4 = ( ·𝑠𝐶)
9 hgmapfval.m . . . 4 𝑀 = ((HDMap‘𝐾)‘𝑊)
10 hgmapfval.i . . . 4 𝐼 = ((HGMap‘𝐾)‘𝑊)
11 hgmapfval.k . . . 4 (𝜑 → (𝐾𝑌𝑊𝐻))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hgmapfval 39004 . . 3 (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
1312fveq1d 6665 . 2 (𝜑 → (𝐼𝑋) = ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋))
14 hgmapval.x . . 3 (𝜑𝑋𝐵)
15 riotaex 7110 . . 3 (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))) ∈ V
16 fvoveq1 7171 . . . . . . 7 (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣)))
1716eqeq1d 2821 . . . . . 6 (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
1817ralbidv 3195 . . . . 5 (𝑥 = 𝑋 → (∀𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)) ↔ ∀𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
1918riotabidv 7108 . . . 4 (𝑥 = 𝑋 → (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
20 eqid 2819 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))) = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
2119, 20fvmptg 6759 . . 3 ((𝑋𝐵 ∧ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))) ∈ V) → ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
2214, 15, 21sylancl 588 . 2 (𝜑 → ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
2313, 22eqtrd 2854 1 (𝜑 → (𝐼𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  wral 3136  Vcvv 3493  cmpt 5137  cfv 6348  crio 7105  (class class class)co 7148  Basecbs 16475  Scalarcsca 16560   ·𝑠 cvsca 16561  LHypclh 37102  DVecHcdvh 38196  LCDualclcd 38704  HDMapchdma 38910  HGMapchg 39001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-hgmap 39002
This theorem is referenced by:  hgmapcl  39007  hgmapvs  39009
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