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Theorem hdmapval 32727
 Description: Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector to be (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom (see dvheveccl 32008). is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 32665 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our that the ranges over. The middle term provides isolation to allow and to assume the same value without conflict. Closure is shown by hdmapcl 32729. If a separate auxiliary vector is known, hdmapval2 32731 provides a version without quantification. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmapval.h
hdmapfval.e
hdmapfval.u
hdmapfval.v
hdmapfval.n
hdmapfval.c LCDual
hdmapfval.d
hdmapfval.j HVMap
hdmapfval.i HDMap1
hdmapfval.s HDMap
hdmapfval.k
hdmapval.t
Assertion
Ref Expression
hdmapval
Distinct variable groups:   ,,   ,   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   ()   (,)   (,)   (,)   (,)

Proof of Theorem hdmapval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 hdmapval.h . . . 4
2 hdmapfval.e . . . 4
3 hdmapfval.u . . . 4
4 hdmapfval.v . . . 4
5 hdmapfval.n . . . 4
6 hdmapfval.c . . . 4 LCDual
7 hdmapfval.d . . . 4
8 hdmapfval.j . . . 4 HVMap
9 hdmapfval.i . . . 4 HDMap1
10 hdmapfval.s . . . 4 HDMap
11 hdmapfval.k . . . 4
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hdmapfval 32726 . . 3
1312fveq1d 5759 . 2
14 hdmapval.t . . 3
15 riotaex 6582 . . 3
16 sneq 3849 . . . . . . . . . . 11
1716fveq2d 5761 . . . . . . . . . 10
1817uneq2d 3487 . . . . . . . . 9
1918eleq2d 2509 . . . . . . . 8
2019notbid 287 . . . . . . 7
21 oteq3 4019 . . . . . . . . 9
2221fveq2d 5761 . . . . . . . 8
2322eqeq2d 2453 . . . . . . 7
2420, 23imbi12d 313 . . . . . 6
2524ralbidv 2731 . . . . 5
2625riotabidv 6580 . . . 4
27 eqid 2442 . . . 4
2826, 27fvmptg 5833 . . 3
2914, 15, 28sylancl 645 . 2
3013, 29eqtrd 2474 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   wceq 1653   wcel 1727  wral 2711  cvv 2962   cun 3304  csn 3838  cop 3841  cotp 3842   cmpt 4291   cid 4522   cres 4909  cfv 5483  crio 6571  cbs 13500  clspn 16078  clh 30879  cltrn 30996  cdvh 31974  LCDualclcd 32482  HVMapchvm 32652  HDMap1chdma1 32688  HDMapchdma 32689 This theorem is referenced by:  hdmapcl  32729  hdmapval2lem  32730 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 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432 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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-ot 3848  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-riota 6578  df-hdmap 32691
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