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Theorem hgmapval 32080
 Description: Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 32075. (Contributed by NM, 25-Mar-2015.)
Hypotheses
Ref Expression
hgmapval.h
hgmapfval.u
hgmapfval.v
hgmapfval.t
hgmapfval.r Scalar
hgmapfval.b
hgmapfval.c LCDual
hgmapfval.s
hgmapfval.m HDMap
hgmapfval.i HGMap
hgmapfval.k
hgmapval.x
Assertion
Ref Expression
hgmapval
Distinct variable groups:   ,,   ,,   ,,   ,,   ,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)   (,)   (,)   (,)   ()   (,)

Proof of Theorem hgmapval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 hgmapval.h . . . 4
2 hgmapfval.u . . . 4
3 hgmapfval.v . . . 4
4 hgmapfval.t . . . 4
5 hgmapfval.r . . . 4 Scalar
6 hgmapfval.b . . . 4
7 hgmapfval.c . . . 4 LCDual
8 hgmapfval.s . . . 4
9 hgmapfval.m . . . 4 HDMap
10 hgmapfval.i . . . 4 HGMap
11 hgmapfval.k . . . 4
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hgmapfval 32079 . . 3
1312fveq1d 5527 . 2
14 hgmapval.x . . 3
15 riotaex 6308 . . 3
16 oveq1 5865 . . . . . . . 8
1716fveq2d 5529 . . . . . . 7
1817eqeq1d 2291 . . . . . 6
1918ralbidv 2563 . . . . 5
2019riotabidv 6306 . . . 4
21 eqid 2283 . . . 4
2220, 21fvmptg 5600 . . 3
2314, 15, 22sylancl 643 . 2
2413, 23eqtrd 2315 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1623   wcel 1684  wral 2543  cvv 2788   cmpt 4077  cfv 5255  (class class class)co 5858  crio 6297  cbs 13148  Scalarcsca 13211  cvsca 13212  clh 30173  cdvh 31268  LCDualclcd 31776  HDMapchdma 31983  HGMapchg 32076 This theorem is referenced by:  hgmapcl  32082  hgmapvs  32084 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-hgmap 32077
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