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Theorem idlnegcl 26632
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 2436 . . . 4  |-  ran  G  =  ran  G
31, 2idlss 26626 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  G )
4 ssel2 3343 . . . . 5  |-  ( ( I  C_  ran  G  /\  A  e.  I )  ->  A  e.  ran  G
)
5 eqid 2436 . . . . . 6  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 idlnegcl.2 . . . . . 6  |-  N  =  ( inv `  G
)
7 eqid 2436 . . . . . 6  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 5, 2, 6, 7rngonegmn1l 26565 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G )  -> 
( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
94, 8sylan2 461 . . . 4  |-  ( ( R  e.  RingOps  /\  (
I  C_  ran  G  /\  A  e.  I )
)  ->  ( N `  A )  =  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A ) )
109anassrs 630 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  C_  ran  G )  /\  A  e.  I
)  ->  ( N `  A )  =  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A ) )
113, 10syldanl 26413 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
121rneqi 5096 . . . . . 6  |-  ran  G  =  ran  ( 1st `  R
)
1312, 5, 7rngo1cl 22017 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  G )
141, 2, 6rngonegcl 26561 . . . . 5  |-  ( ( R  e.  RingOps  /\  (GId `  ( 2nd `  R
) )  e.  ran  G )  ->  ( N `  (GId `  ( 2nd `  R ) ) )  e.  ran  G )
1513, 14mpdan 650 . . . 4  |-  ( R  e.  RingOps  ->  ( N `  (GId `  ( 2nd `  R
) ) )  e. 
ran  G )
1615ad2antrr 707 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  (GId `  ( 2nd `  R
) ) )  e. 
ran  G )
171, 5, 2idllmulcl 26630 . . . 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 630 . . 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 650 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( ( N `  (GId `  ( 2nd `  R
) ) ) ( 2nd `  R ) A )  e.  I
)
2011, 19eqeltrd 2510 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 359    = wceq 1652    e. wcel 1725    C_ wss 3320   ran crn 4879   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348  GIdcgi 21775   invcgn 21776   RingOpscrngo 21963   Idlcidl 26617
This theorem is referenced by:  idlsubcl  26633
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 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781  df-ablo 21870  df-ass 21901  df-exid 21903  df-mgm 21907  df-sgr 21919  df-mndo 21926  df-rngo 21964  df-idl 26620
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