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Theorem isprrngo 34162
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 6352 . . . . . . 7 (𝑟 = 𝑅 → (1st𝑟) = (1st𝑅))
2 isprrng.1 . . . . . . 7 𝐺 = (1st𝑅)
31, 2syl6eqr 2812 . . . . . 6 (𝑟 = 𝑅 → (1st𝑟) = 𝐺)
43fveq2d 6356 . . . . 5 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = (GId‘𝐺))
5 isprrng.2 . . . . 5 𝑍 = (GId‘𝐺)
64, 5syl6eqr 2812 . . . 4 (𝑟 = 𝑅 → (GId‘(1st𝑟)) = 𝑍)
76sneqd 4333 . . 3 (𝑟 = 𝑅 → {(GId‘(1st𝑟))} = {𝑍})
8 fveq2 6352 . . 3 (𝑟 = 𝑅 → (PrIdl‘𝑟) = (PrIdl‘𝑅))
97, 8eleq12d 2833 . 2 (𝑟 = 𝑅 → ({(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟) ↔ {𝑍} ∈ (PrIdl‘𝑅)))
10 df-prrngo 34160 . 2 PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st𝑟))} ∈ (PrIdl‘𝑟)}
119, 10elrab2 3507 1 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  {csn 4321  cfv 6049  1st c1st 7331  GIdcgi 27653  RingOpscrngo 34006  PrIdlcpridl 34120  PrRingcprrng 34158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-prrngo 34160
This theorem is referenced by:  prrngorngo  34163  smprngopr  34164  isdmn3  34186
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