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Theorem hgmapffval 32623
 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 2956 . 2
2 fveq2 5720 . . . . 5
3 hgmapval.h . . . . 5
42, 3syl6eqr 2485 . . . 4
5 fveq2 5720 . . . . . . . 8
65fveq1d 5722 . . . . . . 7
7 dfsbcq 3155 . . . . . . 7 Scalar HDMap LCDual Scalar HDMap LCDual
86, 7syl 16 . . . . . 6 Scalar HDMap LCDual Scalar HDMap LCDual
9 fveq2 5720 . . . . . . . . . . 11 HDMap HDMap
109fveq1d 5722 . . . . . . . . . 10 HDMap HDMap
11 dfsbcq 3155 . . . . . . . . . 10 HDMap HDMap HDMap LCDual HDMap LCDual
1210, 11syl 16 . . . . . . . . 9 HDMap LCDual HDMap LCDual
13 fveq2 5720 . . . . . . . . . . . . . . . . . 18 LCDual LCDual
1413fveq1d 5722 . . . . . . . . . . . . . . . . 17 LCDual LCDual
1514fveq2d 5724 . . . . . . . . . . . . . . . 16 LCDual LCDual
1615oveqd 6090 . . . . . . . . . . . . . . 15 LCDual LCDual
1716eqeq2d 2446 . . . . . . . . . . . . . 14 LCDual LCDual
1817ralbidv 2717 . . . . . . . . . . . . 13 LCDual LCDual
1918riotabidv 6543 . . . . . . . . . . . 12 LCDual LCDual
2019mpteq2dv 4288 . . . . . . . . . . 11 LCDual LCDual
2120eleq2d 2502 . . . . . . . . . 10 LCDual LCDual
2221sbcbidv 3207 . . . . . . . . 9 HDMap LCDual HDMap LCDual
2312, 22bitrd 245 . . . . . . . 8 HDMap LCDual HDMap LCDual
2423sbcbidv 3207 . . . . . . 7 Scalar HDMap LCDual Scalar HDMap LCDual
2524sbcbidv 3207 . . . . . 6 Scalar HDMap LCDual Scalar HDMap LCDual
268, 25bitrd 245 . . . . 5 Scalar HDMap LCDual Scalar HDMap LCDual
2726abbidv 2549 . . . 4 Scalar HDMap LCDual Scalar HDMap LCDual
284, 27mpteq12dv 4279 . . 3 Scalar HDMap LCDual Scalar HDMap LCDual
29 df-hgmap 32622 . . 3 HGMap Scalar HDMap LCDual
30 fvex 5734 . . . . 5
313, 30eqeltri 2505 . . . 4
3231mptex 5958 . . 3 Scalar HDMap LCDual
3328, 29, 32fvmpt 5798 . 2 HGMap Scalar HDMap LCDual
341, 33syl 16 1 HGMap Scalar HDMap LCDual
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652   wcel 1725  cab 2421  wral 2697  cvv 2948  wsbc 3153   cmpt 4258  cfv 5446  (class class class)co 6073  crio 6534  cbs 13461  Scalarcsca 13524  cvsca 13525  clh 30718  cdvh 31813  LCDualclcd 32321  HDMapchdma 32528  HGMapchg 32621 This theorem is referenced by:  hgmapfval  32624 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-riota 6541  df-hgmap 32622
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