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Theorem vcgrp 30599
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 30598 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)
3 ablogrpo 30576 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  1st c1st 8011  GrpOpcgr 30518  AbelOpcablo 30573  CVecOLDcvc 30587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014  df-ablo 30574  df-vc 30588
This theorem is referenced by:  vclcan  30600  vczcl  30601  vc0rid  30602  vcm  30605
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