Metamath Proof Explorer


Theorem pridl

Description: The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010)

Ref Expression
Hypothesis pridl.1 𝐻 = ( 2nd𝑅 )
Assertion pridl ( ( ( 𝑅 ∈ RingOps ∧ 𝑃 ∈ ( PrIdl ‘ 𝑅 ) ) ∧ ( 𝐴 ∈ ( Idl ‘ 𝑅 ) ∧ 𝐵 ∈ ( Idl ‘ 𝑅 ) ∧ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ) ) → ( 𝐴𝑃𝐵𝑃 ) )

Proof

Step Hyp Ref Expression
1 pridl.1 𝐻 = ( 2nd𝑅 )
2 eqid ( 1st𝑅 ) = ( 1st𝑅 )
3 eqid ran ( 1st𝑅 ) = ran ( 1st𝑅 )
4 2 1 3 ispridl ( 𝑅 ∈ RingOps → ( 𝑃 ∈ ( PrIdl ‘ 𝑅 ) ↔ ( 𝑃 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑃 ≠ ran ( 1st𝑅 ) ∧ ∀ 𝑎 ∈ ( Idl ‘ 𝑅 ) ∀ 𝑏 ∈ ( Idl ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) )
5 df-3an ( ( 𝑃 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑃 ≠ ran ( 1st𝑅 ) ∧ ∀ 𝑎 ∈ ( Idl ‘ 𝑅 ) ∀ 𝑏 ∈ ( Idl ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ↔ ( ( 𝑃 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑃 ≠ ran ( 1st𝑅 ) ) ∧ ∀ 𝑎 ∈ ( Idl ‘ 𝑅 ) ∀ 𝑏 ∈ ( Idl ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) )
6 4 5 bitrdi ( 𝑅 ∈ RingOps → ( 𝑃 ∈ ( PrIdl ‘ 𝑅 ) ↔ ( ( 𝑃 ∈ ( Idl ‘ 𝑅 ) ∧ 𝑃 ≠ ran ( 1st𝑅 ) ) ∧ ∀ 𝑎 ∈ ( Idl ‘ 𝑅 ) ∀ 𝑏 ∈ ( Idl ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ) ) )
7 6 simplbda ( ( 𝑅 ∈ RingOps ∧ 𝑃 ∈ ( PrIdl ‘ 𝑅 ) ) → ∀ 𝑎 ∈ ( Idl ‘ 𝑅 ) ∀ 𝑏 ∈ ( Idl ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) )
8 raleq ( 𝑎 = 𝐴 → ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ↔ ∀ 𝑥𝐴𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ) )
9 sseq1 ( 𝑎 = 𝐴 → ( 𝑎𝑃𝐴𝑃 ) )
10 9 orbi1d ( 𝑎 = 𝐴 → ( ( 𝑎𝑃𝑏𝑃 ) ↔ ( 𝐴𝑃𝑏𝑃 ) ) )
11 8 10 imbi12d ( 𝑎 = 𝐴 → ( ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) ↔ ( ∀ 𝑥𝐴𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝐴𝑃𝑏𝑃 ) ) ) )
12 raleq ( 𝑏 = 𝐵 → ( ∀ 𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ↔ ∀ 𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ) )
13 12 ralbidv ( 𝑏 = 𝐵 → ( ∀ 𝑥𝐴𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ↔ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ) )
14 sseq1 ( 𝑏 = 𝐵 → ( 𝑏𝑃𝐵𝑃 ) )
15 14 orbi2d ( 𝑏 = 𝐵 → ( ( 𝐴𝑃𝑏𝑃 ) ↔ ( 𝐴𝑃𝐵𝑃 ) ) )
16 13 15 imbi12d ( 𝑏 = 𝐵 → ( ( ∀ 𝑥𝐴𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝐴𝑃𝑏𝑃 ) ) ↔ ( ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝐴𝑃𝐵𝑃 ) ) ) )
17 11 16 rspc2v ( ( 𝐴 ∈ ( Idl ‘ 𝑅 ) ∧ 𝐵 ∈ ( Idl ‘ 𝑅 ) ) → ( ∀ 𝑎 ∈ ( Idl ‘ 𝑅 ) ∀ 𝑏 ∈ ( Idl ‘ 𝑅 ) ( ∀ 𝑥𝑎𝑦𝑏 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝑎𝑃𝑏𝑃 ) ) → ( ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝐴𝑃𝐵𝑃 ) ) ) )
18 7 17 syl5com ( ( 𝑅 ∈ RingOps ∧ 𝑃 ∈ ( PrIdl ‘ 𝑅 ) ) → ( ( 𝐴 ∈ ( Idl ‘ 𝑅 ) ∧ 𝐵 ∈ ( Idl ‘ 𝑅 ) ) → ( ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝐴𝑃𝐵𝑃 ) ) ) )
19 18 expd ( ( 𝑅 ∈ RingOps ∧ 𝑃 ∈ ( PrIdl ‘ 𝑅 ) ) → ( 𝐴 ∈ ( Idl ‘ 𝑅 ) → ( 𝐵 ∈ ( Idl ‘ 𝑅 ) → ( ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 → ( 𝐴𝑃𝐵𝑃 ) ) ) ) )
20 19 3imp2 ( ( ( 𝑅 ∈ RingOps ∧ 𝑃 ∈ ( PrIdl ‘ 𝑅 ) ) ∧ ( 𝐴 ∈ ( Idl ‘ 𝑅 ) ∧ 𝐵 ∈ ( Idl ‘ 𝑅 ) ∧ ∀ 𝑥𝐴𝑦𝐵 ( 𝑥 𝐻 𝑦 ) ∈ 𝑃 ) ) → ( 𝐴𝑃𝐵𝑃 ) )