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Theorem idlsubcl 32170
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 2422 . . . . 5  |-  ran  G  =  ran  G
31, 2idlcl 32164 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  A  e.  I )  ->  A  e.  ran  G
)
41, 2idlcl 32164 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  B  e.  ran  G
)
53, 4anim12da 31952 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  ran  G  /\  B  e.  ran  G ) )
6 eqid 2422 . . . . . 6  |-  ( inv `  G )  =  ( inv `  G )
7 idlsubcl.2 . . . . . 6  |-  D  =  (  /g  `  G
)
81, 2, 6, 7rngosub 32101 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  ran  G  /\  B  e.  ran  G )  -> 
( A D B )  =  ( A G ( ( inv `  G ) `  B
) ) )
983expb 1206 . . . 4  |-  ( ( R  e.  RingOps  /\  ( A  e.  ran  G  /\  B  e.  ran  G ) )  ->  ( A D B )  =  ( A G ( ( inv `  G ) `
 B ) ) )
109adantlr 719 . . 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 472 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
12 simprl 762 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  A  e.  I )
131, 6idlnegcl 32169 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  B  e.  I )  ->  ( ( inv `  G
) `  B )  e.  I )
1413adantrl 720 . . . 4  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  (
( inv `  G
) `  B )  e.  I )
1512, 14jca 534 . . 3  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A  e.  I  /\  ( ( inv `  G
) `  B )  e.  I ) )
161idladdcl 32166 . . 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 472 . 2  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A G ( ( inv `  G ) `  B
) )  e.  I
)
1811, 17eqeltrd 2510 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A D B )  e.  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   ran crn 4851   ` cfv 5598  (class class class)co 6302   1stc1st 6802   invcgn 25902    /g cgs 25903   RingOpscrngo 26089   Idlcidl 32154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-1st 6804  df-2nd 6805  df-grpo 25905  df-gid 25906  df-ginv 25907  df-gdiv 25908  df-ablo 25996  df-ass 26027  df-exid 26029  df-mgmOLD 26033  df-sgrOLD 26045  df-mndo 26052  df-rngo 26090  df-idl 32157
This theorem is referenced by: (None)
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