Theorem prrngorngo 35489

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

Step	Hyp	Ref	Expression
1		eqid 2798	. . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		eqid 2798	. . 3 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅))
3	1, 2	isprrngo 35488	. 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st ‘𝑅))} ∈ (PrIdl‘𝑅)))
4	3	simplbi 501	1 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2111  {csn 4525  ‘cfv 6324  1^st c1st 7669  GIdcgi 28273  RingOpscrngo 35332  PrIdlcpridl 35446  PrRingcprrng 35484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-prrngo 35486

This theorem is referenced by:  isdmn2  35493