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Theorem prrngorngo 35199
 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 2825 . . 3 (1st𝑅) = (1st𝑅)
2 eqid 2825 . . 3 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
31, 2isprrngo 35198 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st𝑅))} ∈ (PrIdl‘𝑅)))
43simplbi 498 1 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2107  {csn 4563  ‘cfv 6351  1st c1st 7681  GIdcgi 28183  RingOpscrngo 35042  PrIdlcpridl 35156  PrRingcprrng 35194 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-prrngo 35196 This theorem is referenced by:  isdmn2  35203
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