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Theorem hgmapfnN 38015
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 6236 . . 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 2454 . . 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 5691 . 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 2454 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
7 eqid 2454 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
8 hgmapfn.r . . . 4  |-  R  =  (Scalar `  U )
9 hgmapfn.b . . . 4  |-  B  =  ( Base `  R
)
10 eqid 2454 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
11 eqid 2454 . . . 4  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
12 eqid 2454 . . . 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 38013 . . 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 5653 . 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 233 1  |-  ( ph  ->  G  Fn  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    |-> cmpt 4497    Fn wfn 5565   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716  Scalarcsca 14787   .scvsca 14788   HLchlt 35472   LHypclh 36105   DVecHcdvh 37202  LCDualclcd 37710  HDMapchdma 37917  HGMapchg 38010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-hgmap 38011
This theorem is referenced by:  hgmaprnlem1N  38023  hgmaprnN  38028  hgmapf1oN  38030
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