Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hgmapval Structured version   Unicode version

Theorem hgmapval 34890
 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 34885. (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 34889 . . 3
1312fveq1d 5850 . 2
14 hgmapval.x . . 3
15 riotaex 6243 . . 3
16 oveq1 6284 . . . . . . . 8
1716fveq2d 5852 . . . . . . 7
1817eqeq1d 2404 . . . . . 6
1918ralbidv 2842 . . . . 5
2019riotabidv 6241 . . . 4
21 eqid 2402 . . . 4
2220, 21fvmptg 5929 . . 3
2314, 15, 22sylancl 660 . 2
2413, 23eqtrd 2443 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wceq 1405   wcel 1842  wral 2753  cvv 3058   cmpt 4452  cfv 5568  crio 6238  (class class class)co 6277  cbs 14839  Scalarcsca 14910  cvsca 14911  clh 32981  cdvh 34078  LCDualclcd 34586  HDMapchdma 34793  HGMapchg 34886 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-hgmap 34887 This theorem is referenced by:  hgmapcl  34892  hgmapvs  34894
 Copyright terms: Public domain W3C validator