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Theorem isprrngo 38035
Description: The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isprrng.1 𝐺 = (1st𝑅)
isprrng.2 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
isprrngo (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))

Proof of Theorem isprrngo
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6904 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2 isprrng.1 . . . . . . 7 𝐺 = (1st𝑅)
31, 2eqtr4di 2794 . . . . . 6 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
43fveq2d 6908 . . . . 5 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = (GId‘𝐺))
5 isprrng.2 . . . . 5 𝑍 = (GId‘𝐺)
64, 5eqtr4di 2794 . . . 4 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = 𝑍)
76sneqd 4636 . . 3 (𝑟 = 𝑅 → {(GId‘(1st𝑟))} = {𝑍})
8 fveq2 6904 . . 3 (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅))
97, 8eleq12d 2834 . 2 (𝑟 = 𝑅 → ({(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅)))
10 df-prrngo 38033 . 2 PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
119, 10elrab2 3694 1 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {csn 4624  cfv 6559  1st c1st 8008  GIdcgi 30499  RingOpscrngo 37879  PrIdlcpridl 37993  PrRingcprrng 38031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-prrngo 38033
This theorem is referenced by:  prrngorngo  38036  smprngopr  38037  isdmn3  38059
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