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Theorem idlnegcl 26647
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 2283 . . . 4  |-  ran  G  =  ran  G
31, 2idlss 26641 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  G )
4 ssel2 3175 . . . . 5  |-  ( ( I  C_  ran  G  /\  A  e.  I )  ->  A  e.  ran  G
)
5 eqid 2283 . . . . . 6  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 idlnegcl.2 . . . . . 6  |-  N  =  ( inv `  G
)
7 eqid 2283 . . . . . 6  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 5, 2, 6, 7rngonegmn1l 26580 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G )  -> 
( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
94, 8sylan2 460 . . . 4  |-  ( ( R  e.  RingOps  /\  (
I  C_  ran  G  /\  A  e.  I )
)  ->  ( N `  A )  =  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A ) )
109anassrs 629 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  C_  ran  G )  /\  A  e.  I
)  ->  ( N `  A )  =  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A ) )
113, 10syldanl 26334 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
121rneqi 4905 . . . . . 6  |-  ran  G  =  ran  ( 1st `  R
)
1312, 5, 7rngo1cl 21096 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  G )
141, 2, 6rngonegcl 26576 . . . . 5  |-  ( ( R  e.  RingOps  /\  (GId `  ( 2nd `  R
) )  e.  ran  G )  ->  ( N `  (GId `  ( 2nd `  R ) ) )  e.  ran  G )
1513, 14mpdan 649 . . . 4  |-  ( R  e.  RingOps  ->  ( N `  (GId `  ( 2nd `  R
) ) )  e. 
ran  G )
1615ad2antrr 706 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  (GId `  ( 2nd `  R
) ) )  e. 
ran  G )
171, 5, 2idllmulcl 26645 . . . 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 629 . . 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 649 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A )  e.  I
)
2011, 19eqeltrd 2357 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   invcgn 20855   RingOpscrngo 21042   Idlcidl 26632
This theorem is referenced by:  idlsubcl  26648
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-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  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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