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Theorem hgmapfnN 32689
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 6553 . . 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 2436 . . 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 5573 . 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 2436 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
7 eqid 2436 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
8 hgmapfn.r . . . 4  |-  R  =  (Scalar `  U )
9 hgmapfn.b . . . 4  |-  B  =  ( Base `  R
)
10 eqid 2436 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
11 eqid 2436 . . . 4  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
12 eqid 2436 . . . 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 32687 . . 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 5536 . 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 225 1  |-  ( ph  ->  G  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    e. cmpt 4266    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469  Scalarcsca 13532   .scvsca 13533   HLchlt 30148   LHypclh 30781   DVecHcdvh 31876  LCDualclcd 32384  HDMapchdma 32591  HGMapchg 32684
This theorem is referenced by:  hgmaprnlem1N  32697  hgmaprnN  32702  hgmapf1oN  32704
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-hgmap 32685
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