Metamath Proof Explorer


Theorem isprmidl

Description: The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by Thierry Arnoux, 12-Jan-2024)

Ref Expression
Hypotheses prmidlval.1 𝐵 = ( Base ‘ 𝑅 )
prmidlval.2 · = ( .r𝑅 )
Assertion isprmidl ( 𝑅 ∈ Ring → ( 𝑃 ∈ ( PrmIdeal ‘ 𝑅 ) ↔ ( 𝑃 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑃𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 prmidlval.1 𝐵 = ( Base ‘ 𝑅 )
2 prmidlval.2 · = ( .r𝑅 )
3 1 2 prmidlval ( 𝑅 ∈ Ring → ( PrmIdeal ‘ 𝑅 ) = { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎𝑖𝑏𝑖 ) ) ) } )
4 3 eleq2d ( 𝑅 ∈ Ring → ( 𝑃 ∈ ( PrmIdeal ‘ 𝑅 ) ↔ 𝑃 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎𝑖𝑏𝑖 ) ) ) } ) )
5 neeq1 ( 𝑖 = 𝑃 → ( 𝑖𝐵𝑃𝐵 ) )
6 eleq2 ( 𝑖 = 𝑃 → ( ( 𝑥 · 𝑦 ) ∈ 𝑖 ↔ ( 𝑥 · 𝑦 ) ∈ 𝑃 ) )
7 6 2ralbidv ( 𝑖 = 𝑃 → ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 ↔ ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 ) )
8 sseq2 ( 𝑖 = 𝑃 → ( 𝑎𝑖𝑎𝑃 ) )
9 sseq2 ( 𝑖 = 𝑃 → ( 𝑏𝑖𝑏𝑃 ) )
10 8 9 orbi12d ( 𝑖 = 𝑃 → ( ( 𝑎𝑖𝑏𝑖 ) ↔ ( 𝑎𝑃𝑏𝑃 ) ) )
11 7 10 imbi12d ( 𝑖 = 𝑃 → ( ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎𝑖𝑏𝑖 ) ) ↔ ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) )
12 11 2ralbidv ( 𝑖 = 𝑃 → ( ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎𝑖𝑏𝑖 ) ) ↔ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) )
13 5 12 anbi12d ( 𝑖 = 𝑃 → ( ( 𝑖𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎𝑖𝑏𝑖 ) ) ) ↔ ( 𝑃𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) )
14 13 elrab ( 𝑃 ∈ { 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ∣ ( 𝑖𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑖 → ( 𝑎𝑖𝑏𝑖 ) ) ) } ↔ ( 𝑃 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑃𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) )
15 4 14 bitrdi ( 𝑅 ∈ Ring → ( 𝑃 ∈ ( PrmIdeal ‘ 𝑅 ) ↔ ( 𝑃 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑃𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) ) )
16 3anass ( ( 𝑃 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑃𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ↔ ( 𝑃 ∈ ( LIdeal ‘ 𝑅 ) ∧ ( 𝑃𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) )
17 15 16 bitr4di ( 𝑅 ∈ Ring → ( 𝑃 ∈ ( PrmIdeal ‘ 𝑅 ) ↔ ( 𝑃 ∈ ( LIdeal ‘ 𝑅 ) ∧ 𝑃𝐵 ∧ ∀ 𝑎 ∈ ( LIdeal ‘ 𝑅 ) ∀ 𝑏 ∈ ( LIdeal ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 · 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) )