Theorem vcgrp 30903

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 ∈ CVecOLD → 𝐺 ∈ GrpOp)

Proof of Theorem vcgrp

Step	Hyp	Ref	Expression
1		vcabl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2	1	vcablo 30902	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)
3		ablogrpo 30880	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 18	1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ‘cfv 6538  1^st c1st 7985  GrpOpcgr 30822  AbelOpcablo 30877  CVec_OLDcvc 30891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-1st 7987  df-2nd 7988  df-ablo 30878  df-vc 30892

This theorem is referenced by:  vclcan  30904  vczcl  30905  vc0rid  30906  vcm  30909