Theorem vczcl 29812

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 29810	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3		vczcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		vczcl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grpoidcl 29754	. 2 ⊢ (𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋)
6	2, 5	syl 17	1 ⊢ (𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ran crn 5676  ‘cfv 6540  1^st c1st 7969  GrpOpcgr 29729  GIdcgi 29730  CVec_OLDcvc 29798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-riota 7361  df-ov 7408  df-1st 7971  df-2nd 7972  df-grpo 29733  df-gid 29734  df-ablo 29785  df-vc 29799

This theorem is referenced by:  vc0  29814  vcz  29815