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Theorem idlsubcl 26648
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 2283 . . . . 5  |-  ran  G  =  ran  G
31, 2idlcl 26642 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  A  e.  ran  G
)
41, 2idlcl 26642 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  B  e.  ran  G
)
53, 4anim12da 26332 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  ran  G  /\  B  e.  ran  G ) )
6 eqid 2283 . . . . . 6  |-  ( inv `  G )  =  ( inv `  G )
7 idlsubcl.2 . . . . . 6  |-  D  =  (  /g  `  G
)
81, 2, 6, 7rngosub 26579 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G  /\  B  e.  ran  G )  -> 
( A D B )  =  ( A G ( ( inv `  G ) `  B
) ) )
983expb 1152 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  ran  G  /\  B  e.  ran  G ) )  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
109adantlr 695 . . 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 456 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
12 simprl 732 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  A  e.  I )
131, 6idlnegcl 26647 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  ( ( inv `  G
) `  B )  e.  I )
1413adantrl 696 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  (
( inv `  G
) `  B )  e.  I )
1512, 14jca 518 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  I  /\  ( ( inv `  G
) `  B )  e.  I ) )
161idladdcl 26644 . . 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 456 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A G ( ( inv `  G ) `  B
) )  e.  I
)
1811, 17eqeltrd 2357 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 358    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   invcgn 20855    /g cgs 20856   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-rep 4131  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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-idl 26635
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