Theorem inidl 33829

Description: The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)

Assertion

Ref	Expression
inidl	|- ( ( R e. RingOps /\ I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> ( I i^i J ) e. ( Idl ` R ) )

Proof of Theorem inidl

Step	Hyp	Ref	Expression
1		intprg 4511	. . 3 |- ( ( I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> |^| { I , J } = ( I i^i J ) )
2	1	3adant1 1079	. 2 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> |^| { I , J } = ( I i^i J ) )
3		prnzg 4311	. . . . . 6 |- ( I e. ( Idl ` R ) -> { I , J } =/= (/) )
4	3	adantr 481	. . . . 5 |- ( ( I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> { I , J } =/= (/) )
5		prssi 4353	. . . . 5 |- ( ( I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> { I , J } C_ ( Idl ` R ) )
6	4, 5	jca 554	. . . 4 |- ( ( I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> ( { I , J } =/= (/) /\ { I , J } C_ ( Idl ` R ) ) )
7		intidl 33828	. . . . 5 |- ( ( R e. RingOps /\ { I , J } =/= (/) /\ { I , J } C_ ( Idl ` R ) ) -> |^| { I , J } e. ( Idl ` R ) )
8	7	3expb 1266	. . . 4 |- ( ( R e. RingOps /\ ( { I , J } =/= (/) /\ { I , J } C_ ( Idl ` R ) ) ) -> |^| { I , J } e. ( Idl ` R ) )
9	6, 8	sylan2 491	. . 3 |- ( ( R e. RingOps /\ ( I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) ) -> |^| { I , J } e. ( Idl ` R ) )
10	9	3impb 1260	. 2 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> |^| { I , J } e. ( Idl ` R ) )
11	2, 10	eqeltrrd 2702	1 |- ( ( R e. RingOps /\ I e. ( Idl ` R ) /\ J e. ( Idl ` R ) ) -> ( I i^i J ) e. ( Idl ` R ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   {cpr 4179   |^|cint 4475   ` cfv 5888   RingOpscrngo 33693   Idlcidl 33806

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-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-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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  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-idl 33809

This theorem is referenced by: (None)