Theorem vcgrp 31051

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 31050	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)
3		ablogrpo 31028	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 18	1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  1^st c1st 7984  GrpOpcgr 30970  AbelOpcablo 31025  CVec_OLDcvc 31039

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-1st 7986  df-2nd 7987  df-ablo 31026  df-vc 31040

This theorem is used by:  vclcan  31052  vczcl  31053  vc0rid  31054  vcm  31057