Theorem prrngorngo 38683

Description: Obsolete theorem, use prmrngring 49086 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 38682	. 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st ‘𝑅))} ∈ (PrIdl‘𝑅)))
4	3	simplbi 501	1 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2143  {csn 4590  ‘cfv 6538  1^st c1st 7985  GIdcgi 30823  RingOpscrngo 38526  PrIdlcpridl 38640  PrRingcprrng 38678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-prrngo 38680

This theorem is referenced by:  isdmn2  38687