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Theorem isprrngo 37004
Description: The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isprrng.1 𝐺 = (1st β€˜π‘…)
isprrng.2 𝑍 = (GIdβ€˜πΊ)
Assertion
Ref Expression
isprrngo (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…)))

Proof of Theorem isprrngo
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = (1st β€˜π‘…))
2 isprrng.1 . . . . . . 7 𝐺 = (1st β€˜π‘…)
31, 2eqtr4di 2790 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = 𝐺)
43fveq2d 6895 . . . . 5 (π‘Ÿ = 𝑅 β†’ (GIdβ€˜(1st β€˜π‘Ÿ)) = (GIdβ€˜πΊ))
5 isprrng.2 . . . . 5 𝑍 = (GIdβ€˜πΊ)
64, 5eqtr4di 2790 . . . 4 (π‘Ÿ = 𝑅 β†’ (GIdβ€˜(1st β€˜π‘Ÿ)) = 𝑍)
76sneqd 4640 . . 3 (π‘Ÿ = 𝑅 β†’ {(GIdβ€˜(1st β€˜π‘Ÿ))} = {𝑍})
8 fveq2 6891 . . 3 (π‘Ÿ = 𝑅 β†’ (PrIdlβ€˜π‘Ÿ) = (PrIdlβ€˜π‘…))
97, 8eleq12d 2827 . 2 (π‘Ÿ = 𝑅 β†’ ({(GIdβ€˜(1st β€˜π‘Ÿ))} ∈ (PrIdlβ€˜π‘Ÿ) ↔ {𝑍} ∈ (PrIdlβ€˜π‘…)))
10 df-prrngo 37002 . 2 PrRing = {π‘Ÿ ∈ RingOps ∣ {(GIdβ€˜(1st β€˜π‘Ÿ))} ∈ (PrIdlβ€˜π‘Ÿ)}
119, 10elrab2 3686 1 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {csn 4628  β€˜cfv 6543  1st c1st 7975  GIdcgi 29781  RingOpscrngo 36848  PrIdlcpridl 36962  PrRingcprrng 37000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-prrngo 37002
This theorem is referenced by:  prrngorngo  37005  smprngopr  37006  isdmn3  37028
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