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Theorem rngo0rid 37621
Description: The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ring0cl.1 𝐺 = (1st𝑅)
ring0cl.2 𝑋 = ran 𝐺
ring0cl.3 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
rngo0rid ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Proof of Theorem rngo0rid
StepHypRef Expression
1 ring0cl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 37611 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ring0cl.2 . . 3 𝑋 = ran 𝐺
4 ring0cl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grporid 30450 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
62, 5sylan 578 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  ran crn 5683  cfv 6554  (class class class)co 7424  1st c1st 8001  GrpOpcgr 30422  GIdcgi 30423  RingOpscrngo 37595
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-rngo 37596
This theorem is referenced by:  0idl  37726
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