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Theorem isprrngo 35197
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 6666 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2 isprrng.1 . . . . . . 7 𝐺 = (1st𝑅)
31, 2syl6eqr 2878 . . . . . 6 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
43fveq2d 6670 . . . . 5 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = (GId‘𝐺))
5 isprrng.2 . . . . 5 𝑍 = (GId‘𝐺)
64, 5syl6eqr 2878 . . . 4 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = 𝑍)
76sneqd 4575 . . 3 (𝑟 = 𝑅 → {(GId‘(1st𝑟))} = {𝑍})
8 fveq2 6666 . . 3 (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅))
97, 8eleq12d 2911 . 2 (𝑟 = 𝑅 → ({(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅)))
10 df-prrngo 35195 . 2 PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
119, 10elrab2 3686 1 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2106  {csn 4563  cfv 6351  1st c1st 7681  GIdcgi 28182  RingOpscrngo 35041  PrIdlcpridl 35155  PrRingcprrng 35193
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-prrngo 35195
This theorem is referenced by:  prrngorngo  35198  smprngopr  35199  isdmn3  35221
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