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Theorem hdmap1val2 31991
 Description: Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero . (Contributed by NM, 16-May-2015.)
Hypotheses
Ref Expression
hdmap1val2.h
hdmap1val2.u
hdmap1val2.v
hdmap1val2.s
hdmap1val2.o
hdmap1val2.n
hdmap1val2.c LCDual
hdmap1val2.d
hdmap1val2.r
hdmap1val2.l
hdmap1val2.m mapd
hdmap1val2.i HDMap1
hdmap1val2.k
hdmap1val2.x
hdmap1val2.f
hdmap1val2.y
Assertion
Ref Expression
hdmap1val2
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()   ()   ()   ()   ()

Proof of Theorem hdmap1val2
StepHypRef Expression
1 hdmap1val2.h . . 3
2 hdmap1val2.u . . 3
3 hdmap1val2.v . . 3
4 hdmap1val2.s . . 3
5 hdmap1val2.o . . 3
6 hdmap1val2.n . . 3
7 hdmap1val2.c . . 3 LCDual
8 hdmap1val2.d . . 3
9 hdmap1val2.r . . 3
10 eqid 2283 . . 3
11 hdmap1val2.l . . 3
12 hdmap1val2.m . . 3 mapd
13 hdmap1val2.i . . 3 HDMap1
14 hdmap1val2.k . . 3
15 hdmap1val2.x . . 3
16 hdmap1val2.f . . 3
17 hdmap1val2.y . . . 4
18 eldifi 3298 . . . 4
1917, 18syl 15 . . 3
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19hdmap1val 31989 . 2
21 eldifsni 3750 . . . 4
2221neneqd 2462 . . 3
23 iffalse 3572 . . 3
2417, 22, 233syl 18 . 2
2520, 24eqtrd 2315 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   wceq 1623   wcel 1684   cdif 3149  cif 3565  csn 3640  cotp 3644  cfv 5255  (class class class)co 5858  crio 6297  cbs 13148  c0g 13400  csg 14365  clspn 15728  chlt 29540  clh 30173  cdvh 31268  LCDualclcd 31776  mapdcmpd 31814  HDMap1chdma1 31982 This theorem is referenced by:  hdmap1eq  31992 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-hdmap1 31984
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