Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inidl Structured version   Unicode version

Theorem inidl 26654
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
StepHypRef Expression
1 intprg 4086 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  |^| { I ,  J }  =  ( I  i^i  J ) )
213adant1 976 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  J  e.  ( Idl `  R ) )  ->  |^| { I ,  J }  =  ( I  i^i  J ) )
3 prnzg 3926 . . . . . 6  |-  ( I  e.  ( Idl `  R
)  ->  { I ,  J }  =/=  (/) )
43adantr 453 . . . . 5  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  { I ,  J }  =/=  (/) )
5 prssi 3956 . . . . 5  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  { I ,  J }  C_  ( Idl `  R ) )
64, 5jca 520 . . . 4  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  ( { I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) ) )
7 intidl 26653 . . . . 5  |-  ( ( R  e.  RingOps  /\  {
I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
873expb 1155 . . . 4  |-  ( ( R  e.  RingOps  /\  ( { I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
96, 8sylan2 462 . . 3  |-  ( ( R  e.  RingOps  /\  (
I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
1093impb 1150 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  J  e.  ( Idl `  R ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
112, 10eqeltrrd 2513 1  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  J  e.  ( Idl `  R ) )  ->  ( I  i^i  J )  e.  ( Idl `  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   {cpr 3817   |^|cint 4052   ` cfv 5457   RingOpscrngo 21968   Idlcidl 26631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-idl 26634
  Copyright terms: Public domain W3C validator