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Theorem hgmapval 41490
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 41485. (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 41489 . . 3 (𝜑𝐼 = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))))
1312fveq1d 6898 . 2 (𝜑 → (𝐼𝑋) = ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋))
14 hgmapval.x . . 3 (𝜑𝑋𝐵)
15 riotaex 7379 . . 3 (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))) ∈ V
16 fvoveq1 7442 . . . . . . 7 (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣)))
1716eqeq1d 2727 . . . . . 6 (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
1817ralbidv 3167 . . . . 5 (𝑥 = 𝑋 → (∀𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)) ↔ ∀𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
1918riotabidv 7377 . . . 4 (𝑥 = 𝑋 → (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
20 eqid 2725 . . . 4 (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣)))) = (𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))
2119, 20fvmptg 7002 . . 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 584 . 2 (𝜑 → ((𝑥𝐵 ↦ (𝑦𝐵𝑣𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 (𝑀𝑣))))‘𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
2313, 22eqtrd 2765 1 (𝜑 → (𝐼𝑋) = (𝑦𝐵𝑣𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 (𝑀𝑣))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  cmpt 5232  cfv 6549  crio 7374  (class class class)co 7419  Basecbs 17183  Scalarcsca 17239   ·𝑠 cvsca 17240  LHypclh 39587  DVecHcdvh 40681  LCDualclcd 41189  HDMapchdma 41395  HGMapchg 41486
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-hgmap 41487
This theorem is referenced by:  hgmapcl  41492  hgmapvs  41494
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