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Theorem hvmapval 32632
 Description: Value of map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)
Hypotheses
Ref Expression
hvmapval.h
hvmapval.u
hvmapval.o
hvmapval.v
hvmapval.p
hvmapval.t
hvmapval.z
hvmapval.s Scalar
hvmapval.r
hvmapval.m HVMap
hvmapval.k
hvmapval.x
Assertion
Ref Expression
hvmapval
Distinct variable groups:   ,,,   ,   ,   ,   ,,   ,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,)   (,,)   (,,)   (,,)   (,,)   (,,)   (,)   (,)   (,,)

Proof of Theorem hvmapval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 hvmapval.h . . . 4
2 hvmapval.u . . . 4
3 hvmapval.o . . . 4
4 hvmapval.v . . . 4
5 hvmapval.p . . . 4
6 hvmapval.t . . . 4
7 hvmapval.z . . . 4
8 hvmapval.s . . . 4 Scalar
9 hvmapval.r . . . 4
10 hvmapval.m . . . 4 HVMap
11 hvmapval.k . . . 4
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hvmapfval 32631 . . 3
1312fveq1d 5733 . 2
14 hvmapval.x . . 3
15 fvex 5745 . . . . 5
164, 15eqeltri 2508 . . . 4
1716mptex 5969 . . 3
18 sneq 3827 . . . . . . . 8
1918fveq2d 5735 . . . . . . 7
20 oveq2 6092 . . . . . . . . 9
2120oveq2d 6100 . . . . . . . 8
2221eqeq2d 2449 . . . . . . 7
2319, 22rexeqbidv 2919 . . . . . 6
2423riotabidv 6554 . . . . 5
2524mpteq2dv 4299 . . . 4
26 eqid 2438 . . . 4
2725, 26fvmptg 5807 . . 3
2814, 17, 27sylancl 645 . 2
2913, 28eqtrd 2470 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  wrex 2708  cvv 2958   cdif 3319  csn 3816   cmpt 4269  cfv 5457  (class class class)co 6084  crio 6545  cbs 13474   cplusg 13534  Scalarcsca 13537  cvsca 13538  c0g 13728  clh 30855  cdvh 31950  coch 32219  HVMapchvm 32628 This theorem is referenced by:  hvmapvalvalN  32633  hvmapidN  32634  hdmapevec2  32711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-riota 6552  df-hvmap 32629
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