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Theorem hvmapval 36958
 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 36957 . . 3
1312fveq1d 5874 . 2
14 hvmapval.x . . 3
15 fvex 5882 . . . . 5
164, 15eqeltri 2551 . . . 4
1716mptex 6142 . . 3
18 sneq 4043 . . . . . . . 8
1918fveq2d 5876 . . . . . . 7
20 oveq2 6303 . . . . . . . . 9
2120oveq2d 6311 . . . . . . . 8
2221eqeq2d 2481 . . . . . . 7
2319, 22rexeqbidv 3078 . . . . . 6
2423riotabidv 6258 . . . . 5
2524mpteq2dv 4540 . . . 4
26 eqid 2467 . . . 4
2725, 26fvmptg 5955 . . 3
2814, 17, 27sylancl 662 . 2
2913, 28eqtrd 2508 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  wrex 2818  cvv 3118   cdif 3478  csn 4033   cmpt 4511  cfv 5594  crio 6255  (class class class)co 6295  cbs 14507   cplusg 14572  Scalarcsca 14575  cvsca 14576  c0g 14712  clh 35181  cdvh 36276  coch 36545  HVMapchvm 36954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-hvmap 36955 This theorem is referenced by:  hvmapvalvalN  36959  hvmapidN  36960  hdmapevec2  37037
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