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Theorem hgmapfnN 32689
 Description: Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hgmapfn.h
hgmapfn.u
hgmapfn.r Scalar
hgmapfn.b
hgmapfn.g HGMap
hgmapfn.k
Assertion
Ref Expression
hgmapfnN

Proof of Theorem hgmapfnN
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6553 . . 3 HDMap LCDualHDMap
2 eqid 2436 . . 3 HDMap LCDualHDMap HDMap LCDualHDMap
31, 2fnmpti 5573 . 2 HDMap LCDualHDMap
4 hgmapfn.h . . . 4
5 hgmapfn.u . . . 4
6 eqid 2436 . . . 4
7 eqid 2436 . . . 4
8 hgmapfn.r . . . 4 Scalar
9 hgmapfn.b . . . 4
10 eqid 2436 . . . 4 LCDual LCDual
11 eqid 2436 . . . 4 LCDual LCDual
12 eqid 2436 . . . 4 HDMap HDMap
13 hgmapfn.g . . . 4 HGMap
14 hgmapfn.k . . . 4
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hgmapfval 32687 . . 3 HDMap LCDualHDMap
1615fneq1d 5536 . 2 HDMap LCDualHDMap
173, 16mpbiri 225 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2705   cmpt 4266   wfn 5449  cfv 5454  (class class class)co 6081  crio 6542  cbs 13469  Scalarcsca 13532  cvsca 13533  chlt 30148  clh 30781  cdvh 31876  LCDualclcd 32384  HDMapchdma 32591  HGMapchg 32684 This theorem is referenced by:  hgmaprnlem1N  32697  hgmaprnN  32702  hgmapf1oN  32704 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-hgmap 32685
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