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Theorem idlsubcl 26625
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 2436 . . . . 5  |-  ran  G  =  ran  G
31, 2idlcl 26619 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  A  e.  ran  G
)
41, 2idlcl 26619 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  B  e.  ran  G
)
53, 4anim12da 26404 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  ran  G  /\  B  e.  ran  G ) )
6 eqid 2436 . . . . . 6  |-  ( inv `  G )  =  ( inv `  G )
7 idlsubcl.2 . . . . . 6  |-  D  =  (  /g  `  G
)
81, 2, 6, 7rngosub 26556 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G  /\  B  e.  ran  G )  -> 
( A D B )  =  ( A G ( ( inv `  G ) `  B
) ) )
983expb 1154 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  ran  G  /\  B  e.  ran  G ) )  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
109adantlr 696 . . 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 457 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
12 simprl 733 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  A  e.  I )
131, 6idlnegcl 26624 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  ( ( inv `  G
) `  B )  e.  I )
1413adantrl 697 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  (
( inv `  G
) `  B )  e.  I )
1512, 14jca 519 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  I  /\  ( ( inv `  G
) `  B )  e.  I ) )
161idladdcl 26621 . . 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 457 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A G ( ( inv `  G ) `  B
) )  e.  I
)
1811, 17eqeltrd 2510 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 359    = wceq 1652    e. wcel 1725   ran crn 4872   ` cfv 5447  (class class class)co 6074   1stc1st 6340   invcgn 21769    /g cgs 21770   RingOpscrngo 21956   Idlcidl 26609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-grpo 21772  df-gid 21773  df-ginv 21774  df-gdiv 21775  df-ablo 21863  df-ass 21894  df-exid 21896  df-mgm 21900  df-sgr 21912  df-mndo 21919  df-rngo 21957  df-idl 26612
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