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Theorem prrngorngo 36513
Description: A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
prrngorngo (𝑅 ∈ PrRing β†’ 𝑅 ∈ RingOps)

Proof of Theorem prrngorngo
StepHypRef Expression
1 eqid 2737 . . 3 (1st β€˜π‘…) = (1st β€˜π‘…)
2 eqid 2737 . . 3 (GIdβ€˜(1st β€˜π‘…)) = (GIdβ€˜(1st β€˜π‘…))
31, 2isprrngo 36512 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GIdβ€˜(1st β€˜π‘…))} ∈ (PrIdlβ€˜π‘…)))
43simplbi 499 1 (𝑅 ∈ PrRing β†’ 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  {csn 4587  β€˜cfv 6497  1st c1st 7920  GIdcgi 29435  RingOpscrngo 36356  PrIdlcpridl 36470  PrRingcprrng 36508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-prrngo 36510
This theorem is referenced by:  isdmn2  36517
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