Theorem isprmrng 49355

Description: The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.)

Hypotheses

Ref	Expression
isprmrng.z	⊢ 0 = (0g‘𝑅)
isprmrng.p	⊢ 𝑃 = (PrmIdeal‘𝑅)

Assertion

Ref	Expression
isprmrng	⊢ (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))

Proof of Theorem isprmrng

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6873	. . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
2	1	sneqd 4595	. . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = {(0g‘𝑅)})
3		fveq2 6873	. . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
4	2, 3	eleq12d 2854	. . 3 ⊢ (𝑟 = 𝑅 → ({(0g‘𝑟)} ∈ (PrmIdeal‘𝑟) ↔ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅)))
5		df-prmring 49354	. . 3 ⊢ PrmRing = {𝑟 ∈ Ring ∣ {(0g‘𝑟)} ∈ (PrmIdeal‘𝑟)}
6	4, 5	elrab2 3648	. 2 ⊢ (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅)))
7		isprmrng.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
8	7	sneqi 4594	. . . . 5 ⊢ { 0 } = {(0g‘𝑅)}
9		isprmrng.p	. . . . 5 ⊢ 𝑃 = (PrmIdeal‘𝑅)
10	8, 9	eleq12i 2853	. . . 4 ⊢ ({ 0 } ∈ 𝑃 ↔ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅))
11	10	bicomi 227	. . 3 ⊢ ({(0g‘𝑅)} ∈ (PrmIdeal‘𝑅) ↔ { 0 } ∈ 𝑃)
12	11	anbi2i 635	. 2 ⊢ ((𝑅 ∈ Ring ∧ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅)) ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))
13	6, 12	bitri 278	1 ⊢ (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {csn 4583  ‘cfv 6527  0_gc0g 17572  Ringcrg 20421  PrmIdealcprmidl 21578  PrmRingcprmrng 49353

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-prmring 49354

This theorem is used by:  prmringnzring  49356  smprngprmrng  49358  crngprmringidom  49360  isidom3  49364