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Theorem vcgrp 29554
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 29553 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ AbelOp)
3 ablogrpo 29531 . 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 2107  cfv 6497  1st c1st 7920  GrpOpcgr 29473  AbelOpcablo 29528  CVecOLDcvc 29542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-1st 7922  df-2nd 7923  df-ablo 29529  df-vc 29543
This theorem is referenced by:  vclcan  29555  vczcl  29556  vc0rid  29557  vcm  29560
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