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Theorem prrngorngo 38036
Description: A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
prrngorngo (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Proof of Theorem prrngorngo
StepHypRef Expression
1 eqid 2736 . . 3 (1st𝑅) = (1st𝑅)
2 eqid 2736 . . 3 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
31, 2isprrngo 38035 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st𝑅))} ∈ (PrIdl‘𝑅)))
43simplbi 497 1 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  {csn 4624  cfv 6559  1st c1st 8008  GIdcgi 30499  RingOpscrngo 37879  PrIdlcpridl 37993  PrRingcprrng 38031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-prrngo 38033
This theorem is referenced by:  isdmn2  38040
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