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Theorem hgmapfnN 36688
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 6247 . . 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 2467 . . 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 5707 . 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 2467 . . . 4  |-  ( Base `  U )  =  (
Base `  U )
7 eqid 2467 . . . 4  |-  ( .s
`  U )  =  ( .s `  U
)
8 hgmapfn.r . . . 4  |-  R  =  (Scalar `  U )
9 hgmapfn.b . . . 4  |-  B  =  ( Base `  R
)
10 eqid 2467 . . . 4  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
11 eqid 2467 . . . 4  |-  ( .s
`  ( (LCDual `  K ) `  W
) )  =  ( .s `  ( (LCDual `  K ) `  W
) )
12 eqid 2467 . . . 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 36686 . . 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 5669 . 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 369    = wceq 1379    e. wcel 1767   A.wral 2814    |-> cmpt 4505    Fn wfn 5581   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14486  Scalarcsca 14554   .scvsca 14555   HLchlt 34147   LHypclh 34780   DVecHcdvh 35875  LCDualclcd 36383  HDMapchdma 36590  HGMapchg 36683
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 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-hgmap 36684
This theorem is referenced by:  hgmaprnlem1N  36696  hgmaprnN  36701  hgmapf1oN  36703
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