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Theorem nvgf 25284
Description: Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvgf.1  |-  X  =  ( BaseSet `  U )
nvgf.2  |-  G  =  ( +v `  U
)
Assertion
Ref Expression
nvgf  |-  ( U  e.  NrmCVec  ->  G : ( X  X.  X ) --> X )

Proof of Theorem nvgf
StepHypRef Expression
1 nvgf.2 . . 3  |-  G  =  ( +v `  U
)
21nvgrp 25283 . 2  |-  ( U  e.  NrmCVec  ->  G  e.  GrpOp )
3 nvgf.1 . . . 4  |-  X  =  ( BaseSet `  U )
43, 1bafval 25270 . . 3  |-  X  =  ran  G
54grpofo 24974 . 2  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
6 fof 5795 . 2  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
72, 5, 63syl 20 1  |-  ( U  e.  NrmCVec  ->  G : ( X  X.  X ) --> X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    X. cxp 4997   -->wf 5584   -onto->wfo 5586   ` cfv 5588   GrpOpcgr 24961   NrmCVeccnv 25250   +vcpv 25251   BaseSetcba 25252
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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6577
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-1st 6785  df-2nd 6786  df-grpo 24966  df-ablo 25057  df-vc 25212  df-nv 25258  df-va 25261  df-ba 25262  df-sm 25263  df-0v 25264  df-nmcv 25266
This theorem is referenced by:  vacn  25377  sspg  25414  hladdf  25588
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