Theorem prrngorngo 36209

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 2738	. . 3 ⊢ (1st ‘𝑅) = (1st ‘𝑅)
2		eqid 2738	. . 3 ⊢ (GId‘(1st ‘𝑅)) = (GId‘(1st ‘𝑅))
3	1, 2	isprrngo 36208	. 2 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st ‘𝑅))} ∈ (PrIdl‘𝑅)))
4	3	simplbi 498	1 ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)

Syntax hints:   → wi 4   ∈ wcel 2106  {csn 4561  ‘cfv 6433  1^st c1st 7829  GIdcgi 28852  RingOpscrngo 36052  PrIdlcpridl 36166  PrRingcprrng 36204

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-prrngo 36206

This theorem is referenced by:  isdmn2  36213