Theorem pridlidl 33834

Description: A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

Assertion

Ref	Expression
pridlidl	|- ( ( R e. RingOps /\ P e. ( PrIdl ` R ) ) -> P e. ( Idl ` R ) )

Proof of Theorem pridlidl

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( 1st ` R ) = ( 1st ` R )
2		eqid 2622	. . . 4 |- ( 2nd ` R ) = ( 2nd ` R )
3		eqid 2622	. . . 4 |- ran ( 1st ` R ) = ran ( 1st ` R )
4	1, 2, 3	ispridl 33833	. . 3 |- ( R e. RingOps -> ( P e. ( PrIdl ` R ) <-> ( P e. ( Idl ` R ) /\ P =/= ran ( 1st ` R ) /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x ( 2nd ` R ) y ) e. P -> ( a C_ P \/ b C_ P ) ) ) ) )
5		3anass 1042	. . 3 |- ( ( P e. ( Idl ` R ) /\ P =/= ran ( 1st ` R ) /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x ( 2nd ` R ) y ) e. P -> ( a C_ P \/ b C_ P ) ) ) <-> ( P e. ( Idl ` R ) /\ ( P =/= ran ( 1st ` R ) /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x ( 2nd ` R ) y ) e. P -> ( a C_ P \/ b C_ P ) ) ) ) )
6	4, 5	syl6bb 276	. 2 |- ( R e. RingOps -> ( P e. ( PrIdl ` R ) <-> ( P e. ( Idl ` R ) /\ ( P =/= ran ( 1st ` R ) /\ A. a e. ( Idl ` R ) A. b e. ( Idl ` R ) ( A. x e. a A. y e. b ( x ( 2nd ` R ) y ) e. P -> ( a C_ P \/ b C_ P ) ) ) ) ) )
7	6	simprbda 653	1 |- ( ( R e. RingOps /\ P e. ( PrIdl ` R ) ) -> P e. ( Idl ` R ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RingOpscrngo 33693   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-pridl 33810

This theorem is referenced by: (None)