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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
StepHypRef Expression
1 eqid 2738 . . 3 (1st𝑅) = (1st𝑅)
2 eqid 2738 . . 3 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
31, 2isprrngo 36208 . 2 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {(GId‘(1st𝑅))} ∈ (PrIdl‘𝑅)))
43simplbi 498 1 (𝑅 ∈ PrRing → 𝑅 ∈ RingOps)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  {csn 4561  cfv 6433  1st 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
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