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Theorem vczcl 27982
 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 27980 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
3 vczcl.2 . . 3 𝑋 = ran 𝐺
4 vczcl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grpoidcl 27924 . 2 (𝐺 ∈ GrpOp → 𝑍𝑋)
62, 5syl 17 1 (𝑊 ∈ CVecOLD𝑍𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ran crn 5343  ‘cfv 6123  1st c1st 7426  GrpOpcgr 27899  GIdcgi 27900  CVecOLDcvc 27968 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fo 6129  df-fv 6131  df-riota 6866  df-ov 6908  df-1st 7428  df-2nd 7429  df-grpo 27903  df-gid 27904  df-ablo 27955  df-vc 27969 This theorem is referenced by:  vc0  27984  vcz  27985
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