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Theorem hgmapfnN 36198
 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 6515 . . 3 (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ V
2 eqid 2610 . . 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 5935 . 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 2610 . . . 4 (Base‘𝑈) = (Base‘𝑈)
7 eqid 2610 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
8 hgmapfn.r . . . 4 𝑅 = (Scalar‘𝑈)
9 hgmapfn.b . . . 4 𝐵 = (Base‘𝑅)
10 eqid 2610 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
11 eqid 2610 . . . 4 ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))
12 eqid 2610 . . . 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 36196 . . 3 (𝜑𝐺 = (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))))
1615fneq1d 5895 . 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 247 1 (𝜑𝐺 Fn 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  HLchlt 33655  LHypclh 34288  DVecHcdvh 35385  LCDualclcd 35893  HDMapchdma 36100  HGMapchg 36193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-hgmap 36194 This theorem is referenced by:  hgmaprnlem1N  36206  hgmaprnN  36211  hgmapf1oN  36213
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