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Theorem prrngorngo 34329
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 2797 . . 3 (1st𝑅) = (1st𝑅)
2 eqid 2797 . . 3 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
31, 2isprrngo 34328 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st𝑅))} ∈ (PrIdl‘𝑅)))
43simplbi 492 1 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  {csn 4366  cfv 6099  1st c1st 7397  GIdcgi 27862  RingOpscrngo 34172  PrIdlcpridl 34286  PrRingcprrng 34324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-iota 6062  df-fv 6107  df-prrngo 34326
This theorem is referenced by:  isdmn2  34333
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