Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1ffval Structured version   Visualization version   Unicode version

Theorem hdmap1ffval 35376
 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 3056 . 2
2 fveq2 5870 . . . . 5
3 hdmap1val.h . . . . 5
42, 3syl6eqr 2505 . . . 4
5 fveq2 5870 . . . . . . 7
65fveq1d 5872 . . . . . 6
7 fveq2 5870 . . . . . . . . . 10 LCDual LCDual
87fveq1d 5872 . . . . . . . . 9 LCDual LCDual
9 fveq2 5870 . . . . . . . . . . . . 13 mapd mapd
109fveq1d 5872 . . . . . . . . . . . 12 mapd mapd
1110sbceq1d 3274 . . . . . . . . . . 11 mapd mapd
1211sbcbidv 3324 . . . . . . . . . 10 mapd mapd
1312sbcbidv 3324 . . . . . . . . 9 mapd mapd
148, 13sbceqbid 3276 . . . . . . . 8 LCDual mapd LCDual mapd
1514sbcbidv 3324 . . . . . . 7 LCDual mapd LCDual mapd
1615sbcbidv 3324 . . . . . 6 LCDual mapd LCDual mapd
176, 16sbceqbid 3276 . . . . 5 LCDual mapd LCDual mapd
1817abbidv 2571 . . . 4 LCDual mapd LCDual mapd
194, 18mpteq12dv 4484 . . 3 LCDual mapd LCDual mapd
20 df-hdmap1 35374 . . 3 HDMap1 LCDual mapd
21 fvex 5880 . . . . 5
223, 21eqeltri 2527 . . . 4
2322mptex 6141 . . 3 LCDual mapd
2419, 20, 23fvmpt 5953 . 2 HDMap1 LCDual mapd
251, 24syl 17 1 HDMap1 LCDual mapd
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   wceq 1446   wcel 1889  cab 2439  cvv 3047  wsbc 3269  cif 3883  csn 3970   cmpt 4464   cxp 4835  cfv 5585  crio 6256  (class class class)co 6295  c1st 6796  c2nd 6797  cbs 15133  c0g 15350  csg 16683  clspn 18206  clh 33561  cdvh 34658  LCDualclcd 35166  mapdcmpd 35204  HDMap1chdma1 35372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pr 4642 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-hdmap1 35374 This theorem is referenced by:  hdmap1fval  35377
 Copyright terms: Public domain W3C validator