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Theorem vczcl 30395
Description: The zero vector is a vector. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vczcl.1 𝐺 = (1st𝑊)
vczcl.2 𝑋 = ran 𝐺
vczcl.3 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
vczcl (𝑊 ∈ CVecOLD𝑍𝑋)

Proof of Theorem vczcl
StepHypRef Expression
1 vczcl.1 . . 3 𝐺 = (1st𝑊)
21vcgrp 30393 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
3 vczcl.2 . . 3 𝑋 = ran 𝐺
4 vczcl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grpoidcl 30337 . 2 (𝐺 ∈ GrpOp → 𝑍𝑋)
62, 5syl 17 1 (𝑊 ∈ CVecOLD𝑍𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  ran crn 5679  cfv 6548  1st c1st 7991  GrpOpcgr 30312  GIdcgi 30313  CVecOLDcvc 30381
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 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
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 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554  df-fv 6556  df-riota 7376  df-ov 7423  df-1st 7993  df-2nd 7994  df-grpo 30316  df-gid 30317  df-ablo 30368  df-vc 30382
This theorem is referenced by:  vc0  30397  vcz  30398
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