Theorem vcgrp 30373

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 30372	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)
3		ablogrpo 30350	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ‘cfv 6542  1^st c1st 7985  GrpOpcgr 30292  AbelOpcablo 30347  CVec_OLDcvc 30361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-1st 7987  df-2nd 7988  df-ablo 30348  df-vc 30362

This theorem is referenced by:  vclcan  30374  vczcl  30375  vc0rid  30376  vcm  30379