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Theorem idlnegcl 30337
Description: An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idlnegcl.1  |-  G  =  ( 1st `  R
)
idlnegcl.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
idlnegcl  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  e.  I )

Proof of Theorem idlnegcl
StepHypRef Expression
1 idlnegcl.1 . . . 4  |-  G  =  ( 1st `  R
)
2 eqid 2467 . . . 4  |-  ran  G  =  ran  G
31, 2idlss 30331 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  G )
4 ssel2 3504 . . . . 5  |-  ( ( I  C_  ran  G  /\  A  e.  I )  ->  A  e.  ran  G
)
5 eqid 2467 . . . . . 6  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 idlnegcl.2 . . . . . 6  |-  N  =  ( inv `  G
)
7 eqid 2467 . . . . . 6  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 5, 2, 6, 7rngonegmn1l 30270 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G )  -> 
( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
94, 8sylan2 474 . . . 4  |-  ( ( R  e.  RingOps  /\  (
I  C_  ran  G  /\  A  e.  I )
)  ->  ( N `  A )  =  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A ) )
109anassrs 648 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  C_  ran  G )  /\  A  e.  I
)  ->  ( N `  A )  =  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A ) )
113, 10syldanl 30120 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
121rneqi 5235 . . . . . 6  |-  ran  G  =  ran  ( 1st `  R
)
1312, 5, 7rngo1cl 25245 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  G )
141, 2, 6rngonegcl 30266 . . . . 5  |-  ( ( R  e.  RingOps  /\  (GId `  ( 2nd `  R
) )  e.  ran  G )  ->  ( N `  (GId `  ( 2nd `  R ) ) )  e.  ran  G )
1513, 14mpdan 668 . . . 4  |-  ( R  e.  RingOps  ->  ( N `  (GId `  ( 2nd `  R
) ) )  e. 
ran  G )
1615ad2antrr 725 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  (GId `  ( 2nd `  R
) ) )  e. 
ran  G )
171, 5, 2idllmulcl 30335 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  ( N `  (GId `  ( 2nd `  R
) ) )  e. 
ran  G ) )  ->  ( ( N `
 (GId `  ( 2nd `  R ) ) ) ( 2nd `  R
) A )  e.  I )
1817anassrs 648 . . 3  |-  ( ( ( ( R  e.  RingOps 
/\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  /\  ( N `  (GId `  ( 2nd `  R ) ) )  e.  ran  G
)  ->  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R
) A )  e.  I )
1916, 18mpdan 668 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A )  e.  I
)
2011, 19eqeltrd 2555 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  e.  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   ran crn 5006   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794  GIdcgi 25003   invcgn 25004   RingOpscrngo 25191   Idlcidl 30322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-1st 6795  df-2nd 6796  df-grpo 25007  df-gid 25008  df-ginv 25009  df-ablo 25098  df-ass 25129  df-exid 25131  df-mgmOLD 25135  df-sgrOLD 25147  df-mndo 25154  df-rngo 25192  df-idl 30325
This theorem is referenced by:  idlsubcl  30338
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