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Metamath Proof Explorer

Theorem pridlc

Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011)

Ref Expression
Hypotheses ispridlc.1 
|- G = ( 1st ` R )
ispridlc.2 
|- H = ( 2nd ` R )
ispridlc.3 
|- X = ran G
Assertion pridlc 
|- ( ( ( R e. CRingOps /\ P e. ( PrIdl ` R ) ) /\ ( A e. X /\ B e. X /\ ( A H B ) e. P ) ) -> ( A e. P \/ B e. P ) )

Proof

Step Hyp Ref Expression
1 ispridlc.1 
 |-  G = ( 1st ` R )
2 ispridlc.2 
 |-  H = ( 2nd ` R )
3 ispridlc.3 
 |-  X = ran G
4 1 2 3 ispridlc 
 |-  ( R e. CRingOps -> ( P e. ( PrIdl ` R ) <-> ( P e. ( Idl ` R ) /\ P =/= X /\ A. a e. X A. b e. X ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) ) ) )
5 4 biimpa 
 |-  ( ( R e. CRingOps /\ P e. ( PrIdl ` R ) ) -> ( P e. ( Idl ` R ) /\ P =/= X /\ A. a e. X A. b e. X ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) ) )
6 5 simp3d 
 |-  ( ( R e. CRingOps /\ P e. ( PrIdl ` R ) ) -> A. a e. X A. b e. X ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) )
7 oveq1 
 |-  ( a = A -> ( a H b ) = ( A H b ) )
8 7 eleq1d 
 |-  ( a = A -> ( ( a H b ) e. P <-> ( A H b ) e. P ) )
9 eleq1 
 |-  ( a = A -> ( a e. P <-> A e. P ) )
10 9 orbi1d 
 |-  ( a = A -> ( ( a e. P \/ b e. P ) <-> ( A e. P \/ b e. P ) ) )
11 8 10 imbi12d 
 |-  ( a = A -> ( ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) <-> ( ( A H b ) e. P -> ( A e. P \/ b e. P ) ) ) )
12 oveq2 
 |-  ( b = B -> ( A H b ) = ( A H B ) )
13 12 eleq1d 
 |-  ( b = B -> ( ( A H b ) e. P <-> ( A H B ) e. P ) )
14 eleq1 
 |-  ( b = B -> ( b e. P <-> B e. P ) )
15 14 orbi2d 
 |-  ( b = B -> ( ( A e. P \/ b e. P ) <-> ( A e. P \/ B e. P ) ) )
16 13 15 imbi12d 
 |-  ( b = B -> ( ( ( A H b ) e. P -> ( A e. P \/ b e. P ) ) <-> ( ( A H B ) e. P -> ( A e. P \/ B e. P ) ) ) )
17 11 16 rspc2v 
 |-  ( ( A e. X /\ B e. X ) -> ( A. a e. X A. b e. X ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) -> ( ( A H B ) e. P -> ( A e. P \/ B e. P ) ) ) )
18 17 com12 
 |-  ( A. a e. X A. b e. X ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) -> ( ( A e. X /\ B e. X ) -> ( ( A H B ) e. P -> ( A e. P \/ B e. P ) ) ) )
19 18 expd 
 |-  ( A. a e. X A. b e. X ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) -> ( A e. X -> ( B e. X -> ( ( A H B ) e. P -> ( A e. P \/ B e. P ) ) ) ) )
20 19 3imp2 
 |-  ( ( A. a e. X A. b e. X ( ( a H b ) e. P -> ( a e. P \/ b e. P ) ) /\ ( A e. X /\ B e. X /\ ( A H B ) e. P ) ) -> ( A e. P \/ B e. P ) )
21 6 20 sylan 
 |-  ( ( ( R e. CRingOps /\ P e. ( PrIdl ` R ) ) /\ ( A e. X /\ B e. X /\ ( A H B ) e. P ) ) -> ( A e. P \/ B e. P ) )