Theorem isprmrng 49068

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 6881	. . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
2	1	sneqd 4600	. . . 4 ⊢ (𝑟 = 𝑅 → {(0g‘𝑟)} = {(0g‘𝑅)})
3		fveq2 6881	. . . 4 ⊢ (𝑟 = 𝑅 → (PrmIdeal‘𝑟) = (PrmIdeal‘𝑅))
4	2, 3	eleq12d 2855	. . 3 ⊢ (𝑟 = 𝑅 → ({(0g‘𝑟)} ∈ (PrmIdeal‘𝑟) ↔ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅)))
5		df-prmring 49067	. . 3 ⊢ PrmRing = {𝑟 ∈ Ring ∣ {(0g‘𝑟)} ∈ (PrmIdeal‘𝑟)}
6	4, 5	elrab2 3653	. 2 ⊢ (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅)))
7		isprmrng.z	. . . . . 6 ⊢ 0 = (0g‘𝑅)
8	7	sneqi 4599	. . . . 5 ⊢ { 0 } = {(0g‘𝑅)}
9		isprmrng.p	. . . . 5 ⊢ 𝑃 = (PrmIdeal‘𝑅)
10	8, 9	eleq12i 2854	. . . 4 ⊢ ({ 0 } ∈ 𝑃 ↔ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅))
11	10	bicomi 227	. . 3 ⊢ ({(0g‘𝑅)} ∈ (PrmIdeal‘𝑅) ↔ { 0 } ∈ 𝑃)
12	11	anbi2i 634	. 2 ⊢ ((𝑅 ∈ Ring ∧ {(0g‘𝑅)} ∈ (PrmIdeal‘𝑅)) ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))
13	6, 12	bitri 278	1 ⊢ (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  {csn 4588  ‘cfv 6536  0_gc0g 17491  Ringcrg 20314  PrmIdealcprmidl 21439  PrmRingcprmrng 49066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-prmring 49067

This theorem is referenced by:  prmringnzring  49069  smprngprmrng  49071  crngprmringidom  49073  isidom3  49077