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Theorem hdmapval 30710
 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 29991). is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 30648 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 30712. If a separate auxiliary vector is known, hdmapval2 30714 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
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 30709 . . 3
1312fveq1d 5379 . 2
14 hdmapval.t . . 3
15 riotaex 6194 . . 3
16 sneq 3555 . . . . . . . . . . 11
1716fveq2d 5381 . . . . . . . . . 10
1817uneq2d 3239 . . . . . . . . 9
1918eleq2d 2320 . . . . . . . 8
2019notbid 287 . . . . . . 7
21 oteq3 3707 . . . . . . . . 9
2221fveq2d 5381 . . . . . . . 8
2322eqeq2d 2264 . . . . . . 7
2420, 23imbi12d 313 . . . . . 6
2524ralbidv 2527 . . . . 5
2625riotabidv 6192 . . . 4
27 eqid 2253 . . . 4
2826, 27fvmptg 5452 . . 3
2914, 15, 28sylancl 646 . 2
3013, 29eqtrd 2285 1
 Colors of variables: wff set class Syntax hints:   wn 5   wi 6   wa 360   wceq 1619   wcel 1621  wral 2509  cvv 2727   cun 3076  csn 3544  cop 3547  cotp 3548   cmpt 3974   cid 4197   cres 4582  cfv 4592  crio 6181  cbs 13022  clspn 15563  clh 28862  cltrn 28979  cdvh 29957  LCDualclcd 30465  HVMapchvm 30635  HDMap1chdma1 30671  HDMapchdma 30672 This theorem is referenced by:  hdmapcl  30712  hdmapval2lem  30713 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-ot 3554  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-iota 6143  df-riota 6190  df-hdmap 30674
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