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Theorem hgmapfnN 41488
Description: Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hgmapfn.h 𝐻 = (LHyp‘𝐾)
hgmapfn.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hgmapfn.r 𝑅 = (Scalar‘𝑈)
hgmapfn.b 𝐵 = (Base‘𝑅)
hgmapfn.g 𝐺 = ((HGMap‘𝐾)‘𝑊)
hgmapfn.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
Assertion
Ref Expression
hgmapfnN (𝜑𝐺 Fn 𝐵)

Proof of Theorem hgmapfnN
Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7379 . . 3 (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ V
2 eqid 2725 . . 3 (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) = (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))))
31, 2fnmpti 6699 . 2 (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) Fn 𝐵
4 hgmapfn.h . . . 4 𝐻 = (LHyp‘𝐾)
5 hgmapfn.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
6 eqid 2725 . . . 4 (Base‘𝑈) = (Base‘𝑈)
7 eqid 2725 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
8 hgmapfn.r . . . 4 𝑅 = (Scalar‘𝑈)
9 hgmapfn.b . . . 4 𝐵 = (Base‘𝑅)
10 eqid 2725 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
11 eqid 2725 . . . 4 ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))
12 eqid 2725 . . . 4 ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊)
13 hgmapfn.g . . . 4 𝐺 = ((HGMap‘𝐾)‘𝑊)
14 hgmapfn.k . . . 4 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hgmapfval 41486 . . 3 (𝜑𝐺 = (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))))
1615fneq1d 6648 . 2 (𝜑 → (𝐺 Fn 𝐵 ↔ (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))) Fn 𝐵))
173, 16mpbiri 257 1 (𝜑𝐺 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  cmpt 5232   Fn wfn 6544  cfv 6549  crio 7374  (class class class)co 7419  Basecbs 17183  Scalarcsca 17239   ·𝑠 cvsca 17240  HLchlt 38949  LHypclh 39584  DVecHcdvh 40678  LCDualclcd 41186  HDMapchdma 41392  HGMapchg 41483
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 41484
This theorem is referenced by:  hgmaprnlem1N  41496  hgmaprnN  41501  hgmapf1oN  41503
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