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Theorem hdmapfnN 36629
 Description: Functionality of map from vectors to functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hdmapfn.h
hdmapfn.u
hdmapfn.v
hdmapfn.s HDMap
hdmapfn.k
Assertion
Ref Expression
hdmapfnN

Proof of Theorem hdmapfnN
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6247 . . 3 LCDual HDMap1 HDMap1 HVMap
2 eqid 2467 . . 3 LCDual HDMap1 HDMap1 HVMap LCDual HDMap1 HDMap1 HVMap
31, 2fnmpti 5707 . 2 LCDual HDMap1 HDMap1 HVMap
4 hdmapfn.h . . . 4
5 eqid 2467 . . . 4
6 hdmapfn.u . . . 4
7 hdmapfn.v . . . 4
8 eqid 2467 . . . 4
9 eqid 2467 . . . 4 LCDual LCDual
10 eqid 2467 . . . 4 LCDual LCDual
11 eqid 2467 . . . 4 HVMap HVMap
12 eqid 2467 . . . 4 HDMap1 HDMap1
13 hdmapfn.s . . . 4 HDMap
14 hdmapfn.k . . . 4
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hdmapfval 36627 . . 3 LCDual HDMap1 HDMap1 HVMap
1615fneq1d 5669 . 2 LCDual HDMap1 HDMap1 HVMap
173, 16mpbiri 233 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 369   wceq 1379   wcel 1767  wral 2814   cun 3474  csn 4027  cop 4033  cotp 4035   cmpt 4505   cid 4790   cres 5001   wfn 5581  cfv 5586  crio 6242  cbs 14486  clspn 17400  chlt 34147  clh 34780  cltrn 34897  cdvh 35875  LCDualclcd 36383  HVMapchvm 36553  HDMap1chdma1 36589  HDMapchdma 36590 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 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686 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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-ot 4036  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-hdmap 36592 This theorem is referenced by:  hdmaprnlem11N  36660  hdmaprnlem17N  36663  hdmaprnN  36664  hdmapf1oN  36665  hgmaprnlem4N  36699
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