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Theorem vczcl 28361
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 28359 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
3 vczcl.2 . . 3 𝑋 = ran 𝐺
4 vczcl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grpoidcl 28303 . 2 (𝐺 ∈ GrpOp → 𝑍𝑋)
62, 5syl 17 1 (𝑊 ∈ CVecOLD𝑍𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  ran crn 5544  cfv 6344  1st c1st 7683  GrpOpcgr 28278  GIdcgi 28279  CVecOLDcvc 28347
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fo 6350  df-fv 6352  df-riota 7108  df-ov 7153  df-1st 7685  df-2nd 7686  df-grpo 28282  df-gid 28283  df-ablo 28334  df-vc 28348
This theorem is referenced by:  vc0  28363  vcz  28364
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