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Theorem inidl 25978
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 3977 . . 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 3822 . . . . . 6  |-  ( I  e.  ( Idl `  R
)  ->  { I ,  J }  =/=  (/) )
43adantr 451 . . . . 5  |-  ( ( I  e.  ( Idl `  R )  /\  J  e.  ( Idl `  R
) )  ->  { I ,  J }  =/=  (/) )
5 prssi 3850 . . . . 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 25977 . . . . 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 2433 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 1642    e. wcel 1710    =/= wne 2521    i^i cin 3227    C_ wss 3228   (/)c0 3531   {cpr 3717   |^|cint 3943   ` cfv 5337   RingOpscrngo 21154   Idlcidl 25955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-idl 25958
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