Theorem prmidlval

Description: The class of prime ideals of a ring R . (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by Thierry Arnoux, 12-Jan-2024)

	Ref	Expression
Hypotheses	prmidlval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
	prmidlval.2	⊢ · = ( .r ‘ 𝑅 )
Assertion	prmidlval	⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) = { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) } )

Proof

Step	Hyp	Ref	Expression
1		prmidlval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		prmidlval.2	⊢ · = ( .r ‘ 𝑅 )
3		df-prmidl	⊢ PrmIdeal = ( 𝑟 ∈ Ring ↦ { 𝑖 ∈ ( LIdeal ‘ 𝑟 ) ∣ ( 𝑖 ≠ ( Base ‘ 𝑟 ) ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑟 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑟 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) } )
4		fveq2	⊢ ( 𝑟 = 𝑅 → ( LIdeal ‘ 𝑟 ) = ( LIdeal ‘ 𝑅 ) )
5		fveq2	⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) )
6	5 1	eqtr4di	⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = 𝐵 )
7	6	neeq2d	⊢ ( 𝑟 = 𝑅 → ( 𝑖 ≠ ( Base ‘ 𝑟 ) ↔ 𝑖 ≠ 𝐵 ) )
8		fveq2	⊢ ( 𝑟 = 𝑅 → ( .r ‘ 𝑟 ) = ( .r ‘ 𝑅 ) )
9	8 2	eqtr4di	⊢ ( 𝑟 = 𝑅 → ( .r ‘ 𝑟 ) = · )
10	9	oveqd	⊢ ( 𝑟 = 𝑅 → ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 𝑥 · 𝑦 ) )
11	10	eleq1d	⊢ ( 𝑟 = 𝑅 → ( ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 ↔ ( 𝑥 · 𝑦 ) ∈ 𝑖 ) )
12	11	2ralbidv	⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 ↔ ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 ) )
13	12	imbi1d	⊢ ( 𝑟 = 𝑅 → ( ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ↔ ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) )
14	4 13	raleqbidv	⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑏 ∈ ( LIdeal ‘ 𝑟 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ↔ ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) )
15	4 14	raleqbidv	⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑎 ∈ ( LIdeal ‘ 𝑟 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑟 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ↔ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) )
16	7 15	anbi12d	⊢ ( 𝑟 = 𝑅 → ( ( 𝑖 ≠ ( Base ‘ 𝑟 ) ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑟 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑟 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) ↔ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) ) )
17	4 16	rabeqbidv	⊢ ( 𝑟 = 𝑅 → { 𝑖 ∈ ( LIdeal ‘ 𝑟 ) ∣ ( 𝑖 ≠ ( Base ‘ 𝑟 ) ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑟 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑟 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) } = { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) } )
18		id	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Ring )
19		eqid	⊢ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) } = { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) }
20		fvexd	⊢ ( 𝑅 ∈ Ring → ( LIdeal ‘ 𝑅 ) ∈ V )
21	19 20	rabexd	⊢ ( 𝑅 ∈ Ring → { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) } ∈ V )
22	3 17 18 21	fvmptd3	⊢ ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) = { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖 ≠ 𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥 ∈ 𝑎 ∀ 𝑦 ∈ 𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖 ) ) ) } )