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Theorem isprrngo 34328
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 6409 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2 isprrng.1 . . . . . . 7 𝐺 = (1st𝑅)
31, 2syl6eqr 2849 . . . . . 6 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
43fveq2d 6413 . . . . 5 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = (GId‘𝐺))
5 isprrng.2 . . . . 5 𝑍 = (GId‘𝐺)
64, 5syl6eqr 2849 . . . 4 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = 𝑍)
76sneqd 4378 . . 3 (𝑟 = 𝑅 → {(GId‘(1st𝑟))} = {𝑍})
8 fveq2 6409 . . 3 (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅))
97, 8eleq12d 2870 . 2 (𝑟 = 𝑅 → ({(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅)))
10 df-prrngo 34326 . 2 PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
119, 10elrab2 3558 1 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wcel 2157  {csn 4366  cfv 6099  1st c1st 7397  GIdcgi 27862  RingOpscrngo 34172  PrIdlcpridl 34286  PrRingcprrng 34324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-iota 6062  df-fv 6107  df-prrngo 34326
This theorem is referenced by:  prrngorngo  34329  smprngopr  34330  isdmn3  34352
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