Theorem prrngorngo 38730

Description: Obsolete theorem, use prmrngring 49131 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 2762	. . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		eqid 2762	. . 3 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅))
3	1, 2	isprrngo 38729	. 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st ‘𝑅))} ∈ (PrIdl‘𝑅)))
4	3	simplbi 501	1 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  {csn 4588  ‘cfv 6536  1^st c1st 7982  GIdcgi 30853  RingOpscrngo 38573  PrIdlcpridl 38687  PrRingcprrng 38725

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-prrngo 38727

This theorem is used by:  isdmn2  38734