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Theorem vcgrp 25113
Description: Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
vcabl.1  |-  G  =  ( 1st `  W
)
Assertion
Ref Expression
vcgrp  |-  ( W  e.  CVecOLD  ->  G  e.  GrpOp )

Proof of Theorem vcgrp
StepHypRef Expression
1 vcabl.1 . . 3  |-  G  =  ( 1st `  W
)
21vcablo 25112 . 2  |-  ( W  e.  CVecOLD  ->  G  e.  AbelOp )
3 ablogrpo 24948 . 2  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
42, 3syl 16 1  |-  ( W  e.  CVecOLD  ->  G  e.  GrpOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   ` cfv 5579   1stc1st 6772   GrpOpcgr 24850   AbelOpcablo 24945   CVecOLDcvc 25100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-1st 6774  df-2nd 6775  df-ablo 24946  df-vc 25101
This theorem is referenced by:  vcgcl  25114  vcaass  25116  vcrcan  25119  vclcan  25120  vczcl  25121  vc0rid  25122  vc0lid  25123  vcm  25126  vcrinv  25127  vclinv  25128  vcoprne  25134
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