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Theorem hgmapval 37496
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 37491. (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 37495 . . 3 (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
1312fveq1d 6231 . 2 (𝜑 → (𝐼𝑋) = ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋))
14 hgmapval.x . . 3 (𝜑𝑋𝐵)
15 riotaex 6655 . . 3 (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))) ∈ V
16 oveq1 6697 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 · 𝑣) = (𝑋 · 𝑣))
1716fveq2d 6233 . . . . . . 7 (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣)))
1817eqeq1d 2653 . . . . . 6 (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
1918ralbidv 3015 . . . . 5 (𝑥 = 𝑋 → (∀𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)) ↔ ∀𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
2019riotabidv 6653 . . . 4 (𝑥 = 𝑋 → (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
21 eqid 2651 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))) = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
2220, 21fvmptg 6319 . . 3 ((𝑋𝐵 ∧ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))) ∈ V) → ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
2314, 15, 22sylancl 695 . 2 (𝜑 → ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
2413, 23eqtrd 2685 1 (𝜑 → (𝐼𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  LHypclh 35588  DVecHcdvh 36684  LCDualclcd 37192  HDMapchdma 37399  HGMapchg 37492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-hgmap 37493
This theorem is referenced by:  hgmapcl  37498  hgmapvs  37500
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