Theorem vczcl 30592

Description: The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vczcl.1	⊢ 𝐺 = (1st ‘𝑊)
vczcl.2	⊢ 𝑋 = ran 𝐺
vczcl.3	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
vczcl	⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋)

Proof of Theorem vczcl

Step	Hyp	Ref	Expression
1		vczcl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2	1	vcgrp 30590	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3		vczcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		vczcl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 30534	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	2, 5	syl 17	1 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ran crn 5685  ‘cfv 6560  1^st c1st 8013  GrpOpcgr 30509  GIdcgi 30510  CVec_OLDcvc 30578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-riota 7389  df-ov 7435  df-1st 8015  df-2nd 8016  df-grpo 30513  df-gid 30514  df-ablo 30565  df-vc 30579

This theorem is referenced by:  vc0  30594  vcz  30595