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Theorem vc0rid 29344
Description: The zero vector is a right identity element. (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
vc0rid ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Proof of Theorem vc0rid
StepHypRef Expression
1 vczcl.1 . . 3 𝐺 = (1st𝑊)
21vcgrp 29341 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
3 vczcl.2 . . 3 𝑋 = ran 𝐺
4 vczcl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grporid 29288 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
62, 5sylan 580 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ran crn 5632  cfv 6493  (class class class)co 7351  1st c1st 7911  GrpOpcgr 29260  GIdcgi 29261  CVecOLDcvc 29329
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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-riota 7307  df-ov 7354  df-1st 7913  df-2nd 7914  df-grpo 29264  df-gid 29265  df-ablo 29316  df-vc 29330
This theorem is referenced by:  vc0  29345
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