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Theorem hgmapval 30769
 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 30764. (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
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 30768 . . 3
1312fveq1d 5379 . 2
14 hgmapval.x . . 3
15 riotaex 6194 . . 3
16 oveq1 5717 . . . . . . . 8
1716fveq2d 5381 . . . . . . 7
1817eqeq1d 2261 . . . . . 6
1918ralbidv 2527 . . . . 5
2019riotabidv 6192 . . . 4
21 eqid 2253 . . . 4
2220, 21fvmptg 5452 . . 3
2314, 15, 22sylancl 646 . 2
2413, 23eqtrd 2285 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   wceq 1619   wcel 1621  wral 2509  cvv 2727   cmpt 3974  cfv 4592  (class class class)co 5710  crio 6181  cbs 13022  Scalarcsca 13085  cvsca 13086  clh 28862  cdvh 29957  LCDualclcd 30465  HDMapchdma 30672  HGMapchg 30765 This theorem is referenced by:  hgmapcl  30771  hgmapvs  30773 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-iota 6143  df-riota 6190  df-hgmap 30766
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