Theorem prrngorngo 39251

Description: Obsolete theorem, use prmrngring 49766 instead. A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
prrngorngo	⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Proof of Theorem prrngorngo

Step	Hyp	Ref	Expression
1		eqid 2763	. . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		eqid 2763	. . 3 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅))
3	1, 2	isprrngo 39250	. 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st ‘𝑅))} ∈ (PrIdl‘𝑅)))
4	3	simplbi 502	1 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  {csn 4588  ‘cfv 6566  1^st c1st 8033  GIdcgi 31149  RingOpscrngo 39094  PrIdlcpridl 39208  PrRingcprrng 39246

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-prrngo 39248

This theorem is used by:  isdmn2  39255