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Theorem hgmapfnN 41891
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 7393 . . 3 (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥))) ∈ V
2 eqid 2736 . . 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 6710 . 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 2736 . . . 4 (Base‘𝑈) = (Base‘𝑈)
7 eqid 2736 . . . 4 ( ·𝑠𝑈) = ( ·𝑠𝑈)
8 hgmapfn.r . . . 4 𝑅 = (Scalar‘𝑈)
9 hgmapfn.b . . . 4 𝐵 = (Base‘𝑅)
10 eqid 2736 . . . 4 ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
11 eqid 2736 . . . 4 ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊)) = ( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))
12 eqid 2736 . . . 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 41889 . . 3 (𝜑𝐺 = (𝑘𝐵 ↦ (𝑗𝐵𝑥 ∈ (Base‘𝑈)(((HDMap‘𝐾)‘𝑊)‘(𝑘( ·𝑠𝑈)𝑥)) = (𝑗( ·𝑠 ‘((LCDual‘𝐾)‘𝑊))(((HDMap‘𝐾)‘𝑊)‘𝑥)))))
1615fneq1d 6660 . 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 258 1 (𝜑𝐺 Fn 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  cmpt 5224   Fn wfn 6555  cfv 6560  crio 7388  (class class class)co 7432  Basecbs 17248  Scalarcsca 17301   ·𝑠 cvsca 17302  HLchlt 39352  LHypclh 39987  DVecHcdvh 41081  LCDualclcd 41589  HDMapchdma 41795  HGMapchg 41886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-hgmap 41887
This theorem is referenced by:  hgmaprnlem1N  41899  hgmaprnN  41904  hgmapf1oN  41906
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