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Theorem hdmap1ffval 35129
 Description: Preliminary map from vectors to functionals in the closed kernel dual space. (Contributed by NM, 14-May-2015.)
Hypothesis
Ref Expression
hdmap1val.h
Assertion
Ref Expression
hdmap1ffval HDMap1 LCDual mapd
Distinct variable groups:   ,   ,,,,,,,,,   ,,,,,,,,,,
Allowed substitution hints:   (,,,,,,,,,)   (,)   (,,,,,,,,,,)

Proof of Theorem hdmap1ffval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2979 . 2
2 fveq2 5688 . . . . 5
3 hdmap1val.h . . . . 5
42, 3syl6eqr 2491 . . . 4
5 fveq2 5688 . . . . . . . 8
65fveq1d 5690 . . . . . . 7
7 dfsbcq 3185 . . . . . . 7 LCDual mapd LCDual mapd
86, 7syl 16 . . . . . 6 LCDual mapd LCDual mapd
9 fveq2 5688 . . . . . . . . . . . 12 LCDual LCDual
109fveq1d 5690 . . . . . . . . . . 11 LCDual LCDual
11 dfsbcq 3185 . . . . . . . . . . 11 LCDual LCDual LCDual mapd LCDual mapd
1210, 11syl 16 . . . . . . . . . 10 LCDual mapd LCDual mapd
13 fveq2 5688 . . . . . . . . . . . . . . 15 mapd mapd
1413fveq1d 5690 . . . . . . . . . . . . . 14 mapd mapd
15 dfsbcq 3185 . . . . . . . . . . . . . 14 mapd mapd mapd mapd
1614, 15syl 16 . . . . . . . . . . . . 13 mapd mapd
1716sbcbidv 3242 . . . . . . . . . . . 12 mapd mapd
1817sbcbidv 3242 . . . . . . . . . . 11 mapd mapd
1918sbcbidv 3242 . . . . . . . . . 10 LCDual mapd LCDual mapd
2012, 19bitrd 253 . . . . . . . . 9 LCDual mapd LCDual mapd
2120sbcbidv 3242 . . . . . . . 8 LCDual mapd LCDual mapd
2221sbcbidv 3242 . . . . . . 7 LCDual mapd LCDual mapd
2322sbcbidv 3242 . . . . . 6 LCDual mapd LCDual mapd
248, 23bitrd 253 . . . . 5 LCDual mapd LCDual mapd
2524abbidv 2555 . . . 4 LCDual mapd LCDual mapd
264, 25mpteq12dv 4367 . . 3 LCDual mapd LCDual mapd
27 df-hdmap1 35127 . . 3 HDMap1 LCDual mapd
28 fvex 5698 . . . . 5
293, 28eqeltri 2511 . . . 4
3029mptex 5945 . . 3 LCDual mapd
3126, 27, 30fvmpt 5771 . 2 HDMap1 LCDual mapd
321, 31syl 16 1 HDMap1 LCDual mapd
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1364   wcel 1761  cab 2427  cvv 2970  wsbc 3183  cif 3788  csn 3874   cmpt 4347   cxp 4834  cfv 5415  crio 6048  (class class class)co 6090  c1st 6574  c2nd 6575  cbs 14170  c0g 14374  csg 15409  clspn 17030  clh 33316  cdvh 34411  LCDualclcd 34919  mapdcmpd 34957  HDMap1chdma1 35125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-hdmap1 35127 This theorem is referenced by:  hdmap1fval  35130
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