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Theorem idlsubcl 26647
Description: An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
idlsubcl.1  |-  G  =  ( 1st `  R
)
idlsubcl.2  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
idlsubcl  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  e.  I )

Proof of Theorem idlsubcl
StepHypRef Expression
1 idlsubcl.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 eqid 2438 . . . . 5  |-  ran  G  =  ran  G
31, 2idlcl 26641 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  A  e.  ran  G
)
41, 2idlcl 26641 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  B  e.  ran  G
)
53, 4anim12da 26426 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  ran  G  /\  B  e.  ran  G ) )
6 eqid 2438 . . . . . 6  |-  ( inv `  G )  =  ( inv `  G )
7 idlsubcl.2 . . . . . 6  |-  D  =  (  /g  `  G
)
81, 2, 6, 7rngosub 26578 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G  /\  B  e.  ran  G )  -> 
( A D B )  =  ( A G ( ( inv `  G ) `  B
) ) )
983expb 1155 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  ran  G  /\  B  e.  ran  G ) )  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
109adantlr 697 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  ran  G  /\  B  e.  ran  G ) )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
115, 10syldan 458 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
12 simprl 734 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  A  e.  I )
131, 6idlnegcl 26646 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  ( ( inv `  G
) `  B )  e.  I )
1413adantrl 698 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  (
( inv `  G
) `  B )  e.  I )
1512, 14jca 520 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  I  /\  ( ( inv `  G
) `  B )  e.  I ) )
161idladdcl 26643 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  ( ( inv `  G
) `  B )  e.  I ) )  -> 
( A G ( ( inv `  G
) `  B )
)  e.  I )
1715, 16syldan 458 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A G ( ( inv `  G ) `  B
) )  e.  I
)
1811, 17eqeltrd 2512 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   ran crn 4882   ` cfv 5457  (class class class)co 6084   1stc1st 6350   invcgn 21781    /g cgs 21782   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-rep 4323  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-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  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-iun 4097  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-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-ass 21906  df-exid 21908  df-mgm 21912  df-sgr 21924  df-mndo 21931  df-rngo 21969  df-idl 26634
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