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Theorem vcgrp 25745
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 25744 . 2  |-  ( W  e.  CVecOLD  ->  G  e.  AbelOp )
3 ablogrpo 25580 . 2  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
42, 3syl 17 1  |-  ( W  e.  CVecOLD  ->  G  e.  GrpOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   ` cfv 5525   1stc1st 6736   GrpOpcgr 25482   AbelOpcablo 25577   CVecOLDcvc 25732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-1st 6738  df-2nd 6739  df-ablo 25578  df-vc 25733
This theorem is referenced by:  vcgcl  25746  vcaass  25748  vcrcan  25751  vclcan  25752  vczcl  25753  vc0rid  25754  vc0lid  25755  vcm  25758  vcrinv  25759  vclinv  25760  vcoprne  25766
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