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Theorem rngo0lid 35259
 Description: The additive identity of a ring is a left 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
rngo0lid ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)

Proof of Theorem rngo0lid
StepHypRef Expression
1 ring0cl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 35248 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ring0cl.2 . . 3 𝑋 = ran 𝐺
4 ring0cl.3 . . 3 𝑍 = (GId‘𝐺)
53, 4grpolid 28288 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)
62, 5sylan 583 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ran crn 5537  ‘cfv 6336  (class class class)co 7138  1st c1st 7670  GrpOpcgr 28261  GIdcgi 28262  RingOpscrngo 35232 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 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 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 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fo 6342  df-fv 6344  df-riota 7096  df-ov 7141  df-1st 7672  df-2nd 7673  df-grpo 28265  df-gid 28266  df-ablo 28317  df-rngo 35233 This theorem is referenced by: (None)
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