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Theorem idlsubcl 26751
Description: An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
idlsubcl.1  |-  G  =  ( 1st `  R
)
idlsubcl.2  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
idlsubcl  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  e.  I )

Proof of Theorem idlsubcl
StepHypRef Expression
1 idlsubcl.1 . . . . 5  |-  G  =  ( 1st `  R
)
2 eqid 2296 . . . . 5  |-  ran  G  =  ran  G
31, 2idlcl 26745 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  A  e.  ran  G
)
41, 2idlcl 26745 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  B  e.  ran  G
)
53, 4anim12da 26435 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  ran  G  /\  B  e.  ran  G ) )
6 eqid 2296 . . . . . 6  |-  ( inv `  G )  =  ( inv `  G )
7 idlsubcl.2 . . . . . 6  |-  D  =  (  /g  `  G
)
81, 2, 6, 7rngosub 26682 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G  /\  B  e.  ran  G )  -> 
( A D B )  =  ( A G ( ( inv `  G ) `  B
) ) )
983expb 1152 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  ran  G  /\  B  e.  ran  G ) )  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
109adantlr 695 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  ran  G  /\  B  e.  ran  G ) )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
115, 10syldan 456 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
12 simprl 732 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  A  e.  I )
131, 6idlnegcl 26750 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  ( ( inv `  G
) `  B )  e.  I )
1413adantrl 696 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  (
( inv `  G
) `  B )  e.  I )
1512, 14jca 518 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  I  /\  ( ( inv `  G
) `  B )  e.  I ) )
161idladdcl 26747 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  ( ( inv `  G
) `  B )  e.  I ) )  -> 
( A G ( ( inv `  G
) `  B )
)  e.  I )
1715, 16syldan 456 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A G ( ( inv `  G ) `  B
) )  e.  I
)
1811, 17eqeltrd 2370 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   invcgn 20871    /g cgs 20872   RingOpscrngo 21058   Idlcidl 26735
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-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  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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