Theorem vc0rid 30506

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 = (1st ‘𝑊)
vczcl.2	⊢ 𝑋 = ran 𝐺
vczcl.3	⊢ 𝑍 = (GId‘𝐺)

Assertion

Ref	Expression
vc0rid	⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Proof of Theorem vc0rid

Step	Hyp	Ref	Expression
1		vczcl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2	1	vcgrp 30503	. 2 ⊢ (𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp)
3		vczcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		vczcl.3	. . 3 ⊢ 𝑍 = (GId‘𝐺)
5	3, 4	grporid 30450	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
6	2, 5	sylan 578	1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ran crn 5683  ‘cfv 6554  (class class class)co 7424  1^st c1st 8001  GrpOpcgr 30422  GIdcgi 30423  CVec_OLDcvc 30491

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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fo 6560  df-fv 6562  df-riota 7380  df-ov 7427  df-1st 8003  df-2nd 8004  df-grpo 30426  df-gid 30427  df-ablo 30478  df-vc 30492

This theorem is referenced by:  vc0  30507