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Theorem inidl 26534
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 4048 . . 3  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  |^| { I ,  J }  =  ( I  i^i  J ) )
213adant1 975 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  J  e.  ( Idl `  R ) )  ->  |^| { I ,  J }  =  ( I  i^i  J ) )
3 prnzg 3888 . . . . . 6  |-  ( I  e.  ( Idl `  R
)  ->  { I ,  J }  =/=  (/) )
43adantr 452 . . . . 5  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  { I ,  J }  =/=  (/) )
5 prssi 3918 . . . . 5  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  { I ,  J }  C_  ( Idl `  R ) )
64, 5jca 519 . . . 4  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  ( { I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) ) )
7 intidl 26533 . . . . 5  |-  ( ( R  e.  RingOps  /\  {
I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
873expb 1154 . . . 4  |-  ( ( R  e.  RingOps  /\  ( { I ,  J }  =/=  (/)  /\  { I ,  J }  C_  ( Idl `  R ) ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
96, 8sylan2 461 . . 3  |-  ( ( R  e.  RingOps  /\  (
I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
1093impb 1149 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
)  /\  J  e.  ( Idl `  R ) )  ->  |^| { I ,  J }  e.  ( Idl `  R ) )
112, 10eqeltrrd 2483 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571    i^i cin 3283    C_ wss 3284   (/)c0 3592   {cpr 3779   |^|cint 4014   ` cfv 5417   RingOpscrngo 21920   Idlcidl 26511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-idl 26514
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