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Theorem hgmapfnN 36999
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 6600 . . 3 (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ V
2 eqid 2620 . . 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 6009 . 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 2620 . . . 4 (Base‘𝑈) = (Base‘𝑈)
7 eqid 2620 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
8 hgmapfn.r . . . 4 𝑅 = (Scalar‘𝑈)
9 hgmapfn.b . . . 4 𝐵 = (Base‘𝑅)
10 eqid 2620 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
11 eqid 2620 . . . 4 ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))
12 eqid 2620 . . . 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 36997 . . 3 (𝜑𝐺 = (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))))
1615fneq1d 5969 . 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 248 1 (𝜑𝐺 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  cmpt 4720   Fn wfn 5871  cfv 5876  crio 6595  (class class class)co 6635  Basecbs 15838  Scalarcsca 15925   ·𝑠 cvsca 15926  HLchlt 34456  LHypclh 35089  DVecHcdvh 36186  LCDualclcd 36694  HDMapchdma 36901  HGMapchg 36994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-hgmap 36995
This theorem is referenced by:  hgmaprnlem1N  37007  hgmaprnN  37012  hgmapf1oN  37014
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