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

Theorem hgmapffval 37086
 Description: Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
Hypothesis
Ref Expression
hgmapval.h
Assertion
Ref Expression
hgmapffval HGMap Scalar HDMap LCDual
Distinct variable groups:   ,   ,,,,,,,,
Allowed substitution hints:   (,,,,,,)   (,,,,,,,)

Proof of Theorem hgmapffval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3127 . 2
2 fveq2 5872 . . . . 5
3 hgmapval.h . . . . 5
42, 3syl6eqr 2526 . . . 4
5 fveq2 5872 . . . . . . . 8
65fveq1d 5874 . . . . . . 7
7 dfsbcq 3338 . . . . . . 7 Scalar HDMap LCDual Scalar HDMap LCDual
86, 7syl 16 . . . . . 6 Scalar HDMap LCDual Scalar HDMap LCDual
9 fveq2 5872 . . . . . . . . . . 11 HDMap HDMap
109fveq1d 5874 . . . . . . . . . 10 HDMap HDMap
11 dfsbcq 3338 . . . . . . . . . 10 HDMap HDMap HDMap LCDual HDMap LCDual
1210, 11syl 16 . . . . . . . . 9 HDMap LCDual HDMap LCDual
13 fveq2 5872 . . . . . . . . . . . . . . . . . 18 LCDual LCDual
1413fveq1d 5874 . . . . . . . . . . . . . . . . 17 LCDual LCDual
1514fveq2d 5876 . . . . . . . . . . . . . . . 16 LCDual LCDual
1615oveqd 6312 . . . . . . . . . . . . . . 15 LCDual LCDual
1716eqeq2d 2481 . . . . . . . . . . . . . 14 LCDual LCDual
1817ralbidv 2906 . . . . . . . . . . . . 13 LCDual LCDual
1918riotabidv 6258 . . . . . . . . . . . 12 LCDual LCDual
2019mpteq2dv 4540 . . . . . . . . . . 11 LCDual LCDual
2120eleq2d 2537 . . . . . . . . . 10 LCDual LCDual
2221sbcbidv 3395 . . . . . . . . 9 HDMap LCDual HDMap LCDual
2312, 22bitrd 253 . . . . . . . 8 HDMap LCDual HDMap LCDual
2423sbcbidv 3395 . . . . . . 7 Scalar HDMap LCDual Scalar HDMap LCDual
2524sbcbidv 3395 . . . . . 6 Scalar HDMap LCDual Scalar HDMap LCDual
268, 25bitrd 253 . . . . 5 Scalar HDMap LCDual Scalar HDMap LCDual
2726abbidv 2603 . . . 4 Scalar HDMap LCDual Scalar HDMap LCDual
284, 27mpteq12dv 4531 . . 3 Scalar HDMap LCDual Scalar HDMap LCDual
29 df-hgmap 37085 . . 3 HGMap Scalar HDMap LCDual
30 fvex 5882 . . . . 5
313, 30eqeltri 2551 . . . 4
3231mptex 6142 . . 3 Scalar HDMap LCDual
3328, 29, 32fvmpt 5957 . 2 HGMap Scalar HDMap LCDual
341, 33syl 16 1 HGMap Scalar HDMap LCDual
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wceq 1379   wcel 1767  cab 2452  wral 2817  cvv 3118  wsbc 3336   cmpt 4511  cfv 5594  crio 6255  (class class class)co 6295  cbs 14507  Scalarcsca 14575  cvsca 14576  clh 35181  cdvh 36276  LCDualclcd 36784  HDMapchdma 36991  HGMapchg 37084 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 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692 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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-hgmap 37085 This theorem is referenced by:  hgmapfval  37087
 Copyright terms: Public domain W3C validator