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Theorem idlnegcl 26750
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 2296 . . . 4  |-  ran  G  =  ran  G
31, 2idlss 26744 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  G )
4 ssel2 3188 . . . . 5  |-  ( ( I  C_  ran  G  /\  A  e.  I )  ->  A  e.  ran  G
)
5 eqid 2296 . . . . . 6  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 idlnegcl.2 . . . . . 6  |-  N  =  ( inv `  G
)
7 eqid 2296 . . . . . 6  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 5, 2, 6, 7rngonegmn1l 26683 . . . . 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 26437 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
121rneqi 4921 . . . . . 6  |-  ran  G  =  ran  ( 1st `  R
)
1312, 5, 7rngo1cl 21112 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  G )
141, 2, 6rngonegcl 26679 . . . . 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 26748 . . . 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 2370 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 1632    e. wcel 1696    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   invcgn 20871   RingOpscrngo 21058   Idlcidl 26735
This theorem is referenced by:  idlsubcl  26751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-idl 26738
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