Theorem vcgrp 28331

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 28330	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp)
3		ablogrpo 28308	. 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ‘cfv 6341  1^st c1st 7673  GrpOpcgr 28250  AbelOpcablo 28305  CVec_OLDcvc 28319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-fv 6349  df-ov 7145  df-1st 7675  df-2nd 7676  df-ablo 28306  df-vc 28320

This theorem is referenced by:  vclcan  28332  vczcl  28333  vc0rid  28334  vcm  28337