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Theorem hgmapfnN 32081
Description: Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hgmapfn.h  |-  H  =  ( LHyp `  K
)
hgmapfn.u  |-  U  =  ( ( DVecH `  K
) `  W )
hgmapfn.r  |-  R  =  (Scalar `  U )
hgmapfn.b  |-  B  =  ( Base `  R
)
hgmapfn.g  |-  G  =  ( (HGMap `  K
) `  W )
hgmapfn.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
hgmapfnN  |-  ( ph  ->  G  Fn  B )

Proof of Theorem hgmapfnN
Dummy variables  j 
k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6308 . . 3  |-  ( iota_ j  e.  B A. x  e.  ( Base `  U
) ( ( (HDMap `  K ) `  W
) `  ( k
( .s `  U
) x ) )  =  ( j ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) ) )  e. 
_V
2 eqid 2283 . . 3  |-  ( k  e.  B  |->  ( iota_ j  e.  B A. x  e.  ( Base `  U
) ( ( (HDMap `  K ) `  W
) `  ( k
( .s `  U
) x ) )  =  ( j ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) ) ) )  =  ( k  e.  B  |->  ( iota_ j  e.  B A. x  e.  ( Base `  U
) ( ( (HDMap `  K ) `  W
) `  ( k
( .s `  U
) x ) )  =  ( j ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) ) ) )
31, 2fnmpti 5372 . 2  |-  ( k  e.  B  |->  ( iota_ j  e.  B A. x  e.  ( Base `  U
) ( ( (HDMap `  K ) `  W
) `  ( k
( .s `  U
) x ) )  =  ( j ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) ) ) )  Fn  B
4 hgmapfn.h . . . 4  |-  H  =  ( LHyp `  K
)
5 hgmapfn.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
6 eqid 2283 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
7 eqid 2283 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
8 hgmapfn.r . . . 4  |-  R  =  (Scalar `  U )
9 hgmapfn.b . . . 4  |-  B  =  ( Base `  R
)
10 eqid 2283 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
11 eqid 2283 . . . 4  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
12 eqid 2283 . . . 4  |-  ( (HDMap `  K ) `  W
)  =  ( (HDMap `  K ) `  W
)
13 hgmapfn.g . . . 4  |-  G  =  ( (HGMap `  K
) `  W )
14 hgmapfn.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
154, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hgmapfval 32079 . . 3  |-  ( ph  ->  G  =  ( k  e.  B  |->  ( iota_ j  e.  B A. x  e.  ( Base `  U
) ( ( (HDMap `  K ) `  W
) `  ( k
( .s `  U
) x ) )  =  ( j ( .s `  ( (LCDual `  K ) `  W
) ) ( ( (HDMap `  K ) `  W ) `  x
) ) ) ) )
1615fneq1d 5335 . 2  |-  ( ph  ->  ( G  Fn  B  <->  ( k  e.  B  |->  (
iota_ j  e.  B A. x  e.  ( Base `  U ) ( ( (HDMap `  K
) `  W ) `  ( k ( .s
`  U ) x ) )  =  ( j ( .s `  ( (LCDual `  K ) `  W ) ) ( ( (HDMap `  K
) `  W ) `  x ) ) ) )  Fn  B ) )
173, 16mpbiri 224 1  |-  ( ph  ->  G  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    e. cmpt 4077    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268  LCDualclcd 31776  HDMapchdma 31983  HGMapchg 32076
This theorem is referenced by:  hgmaprnlem1N  32089  hgmaprnN  32094  hgmapf1oN  32096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-hgmap 32077
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