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Theorem hvmapvalvalN 35745
 Description: Value of value of map (i.e. functional value) from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.) (New usage is discouraged.)
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
hvmapval.y
Assertion
Ref Expression
hvmapvalvalN
Distinct variable groups:   ,,   ,   ,   ,   ,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   ()   (,)   (,)   (,)   (,)   (,)   ()   (,)   (,)

Proof of Theorem hvmapvalvalN
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
12 hvmapval.x . . . 4
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12hvmapval 35744 . . 3
1413fveq1d 5802 . 2
15 hvmapval.y . . 3
16 riotaex 6166 . . 3
17 eqeq1 2458 . . . . . 6
1817rexbidv 2868 . . . . 5
1918riotabidv 6164 . . . 4
20 eqid 2454 . . . 4
2119, 20fvmptg 5882 . . 3
2215, 16, 21sylancl 662 . 2
2314, 22eqtrd 2495 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1370   wcel 1758  wrex 2800  cvv 3078   cdif 3434  csn 3986   cmpt 4459  cfv 5527  crio 6161  (class class class)co 6201  cbs 14293   cplusg 14358  Scalarcsca 14361  cvsca 14362  c0g 14498  clh 33967  cdvh 35062  coch 35331  HVMapchvm 35740 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pr 4640 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-hvmap 35741 This theorem is referenced by: (None)
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