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Theorem hgmapfval 32687
 Description: Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (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
Assertion
Ref Expression
hgmapfval
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,,)   (,,)   (,,)   (,,)   (,)   (,,)

Proof of Theorem hgmapfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hgmapfval.k . 2
2 hgmapfval.i . . . 4 HGMap
3 hgmapval.h . . . . . 6
43hgmapffval 32686 . . . . 5 HGMap Scalar HDMap LCDual
54fveq1d 5730 . . . 4 HGMap Scalar HDMap LCDual
62, 5syl5eq 2480 . . 3 Scalar HDMap LCDual
7 fveq2 5728 . . . . . . . . 9
8 hgmapfval.u . . . . . . . . 9
97, 8syl6eqr 2486 . . . . . . . 8
10 dfsbcq 3163 . . . . . . . 8 Scalar HDMap LCDual Scalar HDMap LCDual
119, 10syl 16 . . . . . . 7 Scalar HDMap LCDual Scalar HDMap LCDual
12 fveq2 5728 . . . . . . . . . . . 12 HDMap HDMap
13 hgmapfval.m . . . . . . . . . . . 12 HDMap
1412, 13syl6eqr 2486 . . . . . . . . . . 11 HDMap
15 dfsbcq 3163 . . . . . . . . . . 11 HDMap HDMap LCDual LCDual
1614, 15syl 16 . . . . . . . . . 10 HDMap LCDual LCDual
17 fveq2 5728 . . . . . . . . . . . . . . . . . 18 LCDual LCDual
1817fveq2d 5732 . . . . . . . . . . . . . . . . 17 LCDual LCDual
1918oveqd 6098 . . . . . . . . . . . . . . . 16 LCDual LCDual
2019eqeq2d 2447 . . . . . . . . . . . . . . 15 LCDual LCDual
2120ralbidv 2725 . . . . . . . . . . . . . 14 LCDual LCDual
2221riotabidv 6551 . . . . . . . . . . . . 13 LCDual LCDual
2322mpteq2dv 4296 . . . . . . . . . . . 12 LCDual LCDual
2423eleq2d 2503 . . . . . . . . . . 11 LCDual LCDual
2524sbcbidv 3215 . . . . . . . . . 10 LCDual LCDual
2616, 25bitrd 245 . . . . . . . . 9 HDMap LCDual LCDual
2726sbcbidv 3215 . . . . . . . 8 Scalar HDMap LCDual Scalar LCDual
2827sbcbidv 3215 . . . . . . 7 Scalar HDMap LCDual Scalar LCDual
2911, 28bitrd 245 . . . . . 6 Scalar HDMap LCDual Scalar LCDual
30 fvex 5742 . . . . . . . 8
318, 30eqeltri 2506 . . . . . . 7
32 fvex 5742 . . . . . . 7 Scalar
33 fvex 5742 . . . . . . . 8 HDMap
3413, 33eqeltri 2506 . . . . . . 7
35 simp2 958 . . . . . . . . . 10 Scalar Scalar
36 simp1 957 . . . . . . . . . . . . 13 Scalar
3736fveq2d 5732 . . . . . . . . . . . 12 Scalar Scalar Scalar
38 hgmapfval.r . . . . . . . . . . . 12 Scalar
3937, 38syl6eqr 2486 . . . . . . . . . . 11 Scalar Scalar
4039fveq2d 5732 . . . . . . . . . 10 Scalar Scalar
4135, 40eqtrd 2468 . . . . . . . . 9 Scalar
42 hgmapfval.b . . . . . . . . 9
4341, 42syl6eqr 2486 . . . . . . . 8 Scalar
44 simp2 958 . . . . . . . . . 10
45 simp1 957 . . . . . . . . . . . . . 14
4645fveq2d 5732 . . . . . . . . . . . . 13
47 hgmapfval.v . . . . . . . . . . . . 13
4846, 47syl6eqr 2486 . . . . . . . . . . . 12
49 simp3 959 . . . . . . . . . . . . . 14
5045fveq2d 5732 . . . . . . . . . . . . . . . 16
51 hgmapfval.t . . . . . . . . . . . . . . . 16
5250, 51syl6eqr 2486 . . . . . . . . . . . . . . 15
5352oveqd 6098 . . . . . . . . . . . . . 14
5449, 53fveq12d 5734 . . . . . . . . . . . . 13
55 eqidd 2437 . . . . . . . . . . . . . . . . 17 LCDual LCDual
56 hgmapfval.c . . . . . . . . . . . . . . . . 17 LCDual
5755, 56syl6eqr 2486 . . . . . . . . . . . . . . . 16 LCDual
5857fveq2d 5732 . . . . . . . . . . . . . . 15 LCDual
59 hgmapfval.s . . . . . . . . . . . . . . 15
6058, 59syl6eqr 2486 . . . . . . . . . . . . . 14 LCDual
61 eqidd 2437 . . . . . . . . . . . . . 14
6249fveq1d 5730 . . . . . . . . . . . . . 14
6360, 61, 62oveq123d 6102 . . . . . . . . . . . . 13 LCDual
6454, 63eqeq12d 2450 . . . . . . . . . . . 12 LCDual
6548, 64raleqbidv 2916 . . . . . . . . . . 11 LCDual
6644, 65riotaeqbidv 6552 . . . . . . . . . 10 LCDual
6744, 66mpteq12dv 4287 . . . . . . . . 9 LCDual
6867eleq2d 2503 . . . . . . . 8 LCDual
6943, 68syld3an2 1231 . . . . . . 7 Scalar LCDual
7031, 32, 34, 69sbc3ie 3230 . . . . . 6 Scalar LCDual
7129, 70syl6bb 253 . . . . 5 Scalar HDMap LCDual
7271abbi1dv 2552 . . . 4 Scalar HDMap LCDual
73 eqid 2436 . . . 4 Scalar HDMap LCDual Scalar HDMap LCDual
74 fvex 5742 . . . . . 6
7542, 74eqeltri 2506 . . . . 5
7675mptex 5966 . . . 4
7772, 73, 76fvmpt 5806 . . 3 Scalar HDMap LCDual
786, 77sylan9eq 2488 . 2
791, 78syl 16 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  cab 2422  wral 2705  cvv 2956  wsbc 3161   cmpt 4266  cfv 5454  (class class class)co 6081  crio 6542  cbs 13469  Scalarcsca 13532  cvsca 13533  clh 30781  cdvh 31876  LCDualclcd 32384  HDMapchdma 32591  HGMapchg 32684 This theorem is referenced by:  hgmapval  32688  hgmapfnN  32689 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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