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Theorem hgmapfnN 39018
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 7112 . . 3 (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ V
2 eqid 2821 . . 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 6485 . 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 2821 . . . 4 (Base‘𝑈) = (Base‘𝑈)
7 eqid 2821 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
8 hgmapfn.r . . . 4 𝑅 = (Scalar‘𝑈)
9 hgmapfn.b . . . 4 𝐵 = (Base‘𝑅)
10 eqid 2821 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
11 eqid 2821 . . . 4 ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))
12 eqid 2821 . . . 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 39016 . . 3 (𝜑𝐺 = (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))))
1615fneq1d 6440 . 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 260 1 (𝜑𝐺 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  wral 3138  cmpt 5138   Fn wfn 6344  cfv 6349  crio 7107  (class class class)co 7150  Basecbs 16477  Scalarcsca 16562   ·𝑠 cvsca 16563  HLchlt 36480  LHypclh 37114  DVecHcdvh 38208  LCDualclcd 38716  HDMapchdma 38922  HGMapchg 39013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-hgmap 39014
This theorem is referenced by:  hgmaprnlem1N  39026  hgmaprnN  39031  hgmapf1oN  39033
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