Theorem pridlc 33870

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

Hypotheses

Ref	Expression
ispridlc.1	|- G = ( 1st ` R )
ispridlc.2	|- H = ( 2nd ` R )
ispridlc.3	|- X = ran G

Assertion

Ref	Expression
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 of Theorem pridlc

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispridlc.1	. . . . 5 |- G = ( 1st ` R )
2		ispridlc.2	. . . . 5 |- H = ( 2nd ` R )
3		ispridlc.3	. . . . 5 |- X = ran G
4	1, 2, 3	ispridlc 33869	. . . 4 |- ( 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 501	. . 3 |- ( ( 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 1075	. 2 |- ( ( 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 6657	. . . . . . . 8 |- ( a = A -> ( a H b ) = ( A H b ) )
8	7	eleq1d 2686	. . . . . . 7 |- ( a = A -> ( ( a H b ) e. P <-> ( A H b ) e. P ) )
9		eleq1 2689	. . . . . . . 8 |- ( a = A -> ( a e. P <-> A e. P ) )
10	9	orbi1d 739	. . . . . . 7 |- ( a = A -> ( ( a e. P \/ b e. P ) <-> ( A e. P \/ b e. P ) ) )
11	8, 10	imbi12d 334	. . . . . 6 |- ( 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 6658	. . . . . . . 8 |- ( b = B -> ( A H b ) = ( A H B ) )
13	12	eleq1d 2686	. . . . . . 7 |- ( b = B -> ( ( A H b ) e. P <-> ( A H B ) e. P ) )
14		eleq1 2689	. . . . . . . 8 |- ( b = B -> ( b e. P <-> B e. P ) )
15	14	orbi2d 738	. . . . . . 7 |- ( b = B -> ( ( A e. P \/ b e. P ) <-> ( A e. P \/ B e. P ) ) )
16	13, 15	imbi12d 334	. . . . . 6 |- ( 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 3322	. . . . 5 |- ( ( 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 32	. . . 4 |- ( 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 452	. . 3 |- ( 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 1282	. 2 |- ( ( 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 488	1 |- ( ( ( R e. CRingOps /\ P e. ( PrIdl ` R ) ) /\ ( A e. X /\ B e. X /\ ( A H B ) e. P ) ) -> ( A e. P \/ B e. P ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  CRingOpsccring 33792   Idlcidl 33806   PrIdlcpridl 33807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-com2 33789  df-crngo 33793  df-idl 33809  df-pridl 33810  df-igen 33859

This theorem is referenced by:  pridlc2  33871