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Theorem vc0rid 28354
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 28351 . 2 (𝑊 ∈ CVecOLD𝐺 ∈ GrpOp)
3 vczcl.2 . . 3 𝑋 = ran 𝐺
4 vczcl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grporid 28298 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
62, 5sylan 583 1 ((𝑊 ∈ CVecOLD𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  ran crn 5533  cfv 6334  (class class class)co 7140  1st c1st 7673  GrpOpcgr 28270  GIdcgi 28271  CVecOLDcvc 28339
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fo 6340  df-fv 6342  df-riota 7098  df-ov 7143  df-1st 7675  df-2nd 7676  df-grpo 28274  df-gid 28275  df-ablo 28326  df-vc 28340
This theorem is referenced by:  vc0  28355
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