Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isprrngo Structured version   Visualization version   GIF version

Theorem isprrngo 37651
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 6896 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2 isprrng.1 . . . . . . 7 𝐺 = (1st𝑅)
31, 2eqtr4di 2783 . . . . . 6 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
43fveq2d 6900 . . . . 5 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = (GId‘𝐺))
5 isprrng.2 . . . . 5 𝑍 = (GId‘𝐺)
64, 5eqtr4di 2783 . . . 4 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = 𝑍)
76sneqd 4642 . . 3 (𝑟 = 𝑅 → {(GId‘(1st𝑟))} = {𝑍})
8 fveq2 6896 . . 3 (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅))
97, 8eleq12d 2819 . 2 (𝑟 = 𝑅 → ({(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅)))
10 df-prrngo 37649 . 2 PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
119, 10elrab2 3682 1 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  {csn 4630  cfv 6549  1st c1st 7992  GIdcgi 30372  RingOpscrngo 37495  PrIdlcpridl 37609  PrRingcprrng 37647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-prrngo 37649
This theorem is referenced by:  prrngorngo  37652  smprngopr  37653  isdmn3  37675
  Copyright terms: Public domain W3C validator