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Theorem hgmapfnN 40354
Description: Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hgmapfn.h 𝐻 = (LHypβ€˜πΎ)
hgmapfn.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hgmapfn.r 𝑅 = (Scalarβ€˜π‘ˆ)
hgmapfn.b 𝐡 = (Baseβ€˜π‘…)
hgmapfn.g 𝐺 = ((HGMapβ€˜πΎ)β€˜π‘Š)
hgmapfn.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
Assertion
Ref Expression
hgmapfnN (πœ‘ β†’ 𝐺 Fn 𝐡)

Proof of Theorem hgmapfnN
Dummy variables 𝑗 π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 7318 . . 3 (℩𝑗 ∈ 𝐡 βˆ€π‘₯ ∈ (Baseβ€˜π‘ˆ)(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜(π‘˜( ·𝑠 β€˜π‘ˆ)π‘₯)) = (𝑗( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š))(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜π‘₯))) ∈ V
2 eqid 2737 . . 3 (π‘˜ ∈ 𝐡 ↦ (℩𝑗 ∈ 𝐡 βˆ€π‘₯ ∈ (Baseβ€˜π‘ˆ)(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜(π‘˜( ·𝑠 β€˜π‘ˆ)π‘₯)) = (𝑗( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š))(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜π‘₯)))) = (π‘˜ ∈ 𝐡 ↦ (℩𝑗 ∈ 𝐡 βˆ€π‘₯ ∈ (Baseβ€˜π‘ˆ)(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜(π‘˜( ·𝑠 β€˜π‘ˆ)π‘₯)) = (𝑗( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š))(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜π‘₯))))
31, 2fnmpti 6645 . 2 (π‘˜ ∈ 𝐡 ↦ (℩𝑗 ∈ 𝐡 βˆ€π‘₯ ∈ (Baseβ€˜π‘ˆ)(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜(π‘˜( ·𝑠 β€˜π‘ˆ)π‘₯)) = (𝑗( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š))(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜π‘₯)))) Fn 𝐡
4 hgmapfn.h . . . 4 𝐻 = (LHypβ€˜πΎ)
5 hgmapfn.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
6 eqid 2737 . . . 4 (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘ˆ)
7 eqid 2737 . . . 4 ( ·𝑠 β€˜π‘ˆ) = ( ·𝑠 β€˜π‘ˆ)
8 hgmapfn.r . . . 4 𝑅 = (Scalarβ€˜π‘ˆ)
9 hgmapfn.b . . . 4 𝐡 = (Baseβ€˜π‘…)
10 eqid 2737 . . . 4 ((LCDualβ€˜πΎ)β€˜π‘Š) = ((LCDualβ€˜πΎ)β€˜π‘Š)
11 eqid 2737 . . . 4 ( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š)) = ( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š))
12 eqid 2737 . . . 4 ((HDMapβ€˜πΎ)β€˜π‘Š) = ((HDMapβ€˜πΎ)β€˜π‘Š)
13 hgmapfn.g . . . 4 𝐺 = ((HGMapβ€˜πΎ)β€˜π‘Š)
14 hgmapfn.k . . . 4 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hgmapfval 40352 . . 3 (πœ‘ β†’ 𝐺 = (π‘˜ ∈ 𝐡 ↦ (℩𝑗 ∈ 𝐡 βˆ€π‘₯ ∈ (Baseβ€˜π‘ˆ)(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜(π‘˜( ·𝑠 β€˜π‘ˆ)π‘₯)) = (𝑗( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š))(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜π‘₯)))))
1615fneq1d 6596 . 2 (πœ‘ β†’ (𝐺 Fn 𝐡 ↔ (π‘˜ ∈ 𝐡 ↦ (℩𝑗 ∈ 𝐡 βˆ€π‘₯ ∈ (Baseβ€˜π‘ˆ)(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜(π‘˜( ·𝑠 β€˜π‘ˆ)π‘₯)) = (𝑗( ·𝑠 β€˜((LCDualβ€˜πΎ)β€˜π‘Š))(((HDMapβ€˜πΎ)β€˜π‘Š)β€˜π‘₯)))) Fn 𝐡))
173, 16mpbiri 258 1 (πœ‘ β†’ 𝐺 Fn 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065   ↦ cmpt 5189   Fn wfn 6492  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  Basecbs 17084  Scalarcsca 17137   ·𝑠 cvsca 17138  HLchlt 37815  LHypclh 38450  DVecHcdvh 39544  LCDualclcd 40052  HDMapchdma 40258  HGMapchg 40349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-hgmap 40350
This theorem is referenced by:  hgmaprnlem1N  40362  hgmaprnN  40367  hgmapf1oN  40369
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