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Theorem prrngorngo 32823
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 2609 . . 3 (1st𝑅) = (1st𝑅)
2 eqid 2609 . . 3 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
31, 2isprrngo 32822 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st𝑅))} ∈ (PrIdl‘𝑅)))
43simplbi 474 1 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  {csn 4124  cfv 5790  1st c1st 7034  GIdcgi 26494  RingOpscrngo 32666  PrIdlcpridl 32780  PrRingcprrng 32818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-prrngo 32820
This theorem is referenced by:  isdmn2  32827
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