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Theorem idlnegcl 28990
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 2454 . . . 4  |-  ran  G  =  ran  G
31, 2idlss 28984 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  I  C_ 
ran  G )
4 ssel2 3462 . . . . 5  |-  ( ( I  C_  ran  G  /\  A  e.  I )  ->  A  e.  ran  G
)
5 eqid 2454 . . . . . 6  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 idlnegcl.2 . . . . . 6  |-  N  =  ( inv `  G
)
7 eqid 2454 . . . . . 6  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
81, 5, 2, 6, 7rngonegmn1l 28923 . . . . 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 28773 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  ( N `  A
)  =  ( ( N `  (GId `  ( 2nd `  R ) ) ) ( 2nd `  R ) A ) )
121rneqi 5177 . . . . . 6  |-  ran  G  =  ran  ( 1st `  R
)
1312, 5, 7rngo1cl 24088 . . . . 5  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  G )
141, 2, 6rngonegcl 28919 . . . . 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 28988 . . . 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 2542 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 1370    e. wcel 1758    C_ wss 3439   ran crn 4952   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689  GIdcgi 23846   invcgn 23847   RingOpscrngo 24034   Idlcidl 28975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-1st 6690  df-2nd 6691  df-grpo 23850  df-gid 23851  df-ginv 23852  df-ablo 23941  df-ass 23972  df-exid 23974  df-mgm 23978  df-sgr 23990  df-mndo 23997  df-rngo 24035  df-idl 28978
This theorem is referenced by:  idlsubcl  28991
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