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Theorem inidl 26655
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 3896 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  |^| { I ,  J }  =  ( I  i^i  J ) )
213adant1 973 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  J  e.  ( Idl `  R ) )  ->  |^| { I ,  J }  =  ( I  i^i  J ) )
3 prnzg 3746 . . . . . 6  |-  ( I  e.  ( Idl `  R
)  ->  { I ,  J }  =/=  (/) )
43adantr 451 . . . . 5  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  { I ,  J }  =/=  (/) )
5 prssi 3771 . . . . 5  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  { I ,  J }  C_  ( Idl `  R ) )
64, 5jca 518 . . . 4  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  ( { I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) ) )
7 intidl 26654 . . . . 5  |-  ( ( R  e.  RingOps  /\  {
I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
873expb 1152 . . . 4  |-  ( ( R  e.  RingOps  /\  ( { I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
96, 8sylan2 460 . . 3  |-  ( ( R  e.  RingOps  /\  (
I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
1093impb 1147 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  J  e.  ( Idl `  R ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
112, 10eqeltrrd 2358 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   {cpr 3641   |^|cint 3862   ` cfv 5255   RingOpscrngo 21042   Idlcidl 26632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-idl 26635
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