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Theorem vcgrp 28508
Description: Vector addition is a group operation. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
vcabl.1 𝐺 = (1st𝑊)
Assertion
Ref Expression
vcgrp (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)

Proof of Theorem vcgrp
StepHypRef Expression
1 vcabl.1 . . 3 𝐺 = (1st𝑊)
21vcablo 28507 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)
3 ablogrpo 28485 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cfv 6340  1st c1st 7715  GrpOpcgr 28427  AbelOpcablo 28482  CVecOLDcvc 28496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7176  df-1st 7717  df-2nd 7718  df-ablo 28483  df-vc 28497
This theorem is referenced by:  vclcan  28509  vczcl  28510  vc0rid  28511  vcm  28514
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