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Theorem isprrngo 36249
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 6800 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = (1st β€˜π‘…))
2 isprrng.1 . . . . . . 7 𝐺 = (1st β€˜π‘…)
31, 2eqtr4di 2794 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (1st β€˜π‘Ÿ) = 𝐺)
43fveq2d 6804 . . . . 5 (π‘Ÿ = 𝑅 β†’ (GIdβ€˜(1st β€˜π‘Ÿ)) = (GIdβ€˜πΊ))
5 isprrng.2 . . . . 5 𝑍 = (GIdβ€˜πΊ)
64, 5eqtr4di 2794 . . . 4 (π‘Ÿ = 𝑅 β†’ (GIdβ€˜(1st β€˜π‘Ÿ)) = 𝑍)
76sneqd 4577 . . 3 (π‘Ÿ = 𝑅 β†’ {(GIdβ€˜(1st β€˜π‘Ÿ))} = {𝑍})
8 fveq2 6800 . . 3 (π‘Ÿ = 𝑅 β†’ (PrIdlβ€˜π‘Ÿ) = (PrIdlβ€˜π‘…))
97, 8eleq12d 2831 . 2 (π‘Ÿ = 𝑅 β†’ ({(GIdβ€˜(1st β€˜π‘Ÿ))} ∈ (PrIdlβ€˜π‘Ÿ) ↔ {𝑍} ∈ (PrIdlβ€˜π‘…)))
10 df-prrngo 36247 . 2 PrRing = {π‘Ÿ ∈ RingOps ∣ {(GIdβ€˜(1st β€˜π‘Ÿ))} ∈ (PrIdlβ€˜π‘Ÿ)}
119, 10elrab2 3632 1 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  {csn 4565  β€˜cfv 6454  1st c1st 7857  GIdcgi 28893  RingOpscrngo 36093  PrIdlcpridl 36207  PrRingcprrng 36245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6406  df-fv 6462  df-prrngo 36247
This theorem is referenced by:  prrngorngo  36250  smprngopr  36251  isdmn3  36273
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