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Theorem idladdcl 25967
Description: An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
idladdcl.1  |-  G  =  ( 1st `  R
)
Assertion
Ref Expression
idladdcl  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A G B )  e.  I )

Proof of Theorem idladdcl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idladdcl.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 eqid 2358 . . . . . 6  |-  ( 2nd `  R )  =  ( 2nd `  R )
3 eqid 2358 . . . . . 6  |-  ran  G  =  ran  G
4 eqid 2358 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4isidl 25962 . . . . 5  |-  ( R  e.  RingOps  ->  ( I  e.  ( Idl `  R
)  <->  ( I  C_  ran  G  /\  (GId `  G )  e.  I  /\  A. x  e.  I 
( A. y  e.  I  ( x G y )  e.  I  /\  A. z  e.  ran  G ( ( z ( 2nd `  R ) x )  e.  I  /\  ( x ( 2nd `  R ) z )  e.  I ) ) ) ) )
65biimpa 470 . . . 4  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  (
I  C_  ran  G  /\  (GId `  G )  e.  I  /\  A. x  e.  I  ( A. y  e.  I  (
x G y )  e.  I  /\  A. z  e.  ran  G ( ( z ( 2nd `  R ) x )  e.  I  /\  (
x ( 2nd `  R
) z )  e.  I ) ) ) )
76simp3d 969 . . 3  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  A. x  e.  I  ( A. y  e.  I  (
x G y )  e.  I  /\  A. z  e.  ran  G ( ( z ( 2nd `  R ) x )  e.  I  /\  (
x ( 2nd `  R
) z )  e.  I ) ) )
8 simpl 443 . . . 4  |-  ( ( A. y  e.  I 
( x G y )  e.  I  /\  A. z  e.  ran  G
( ( z ( 2nd `  R ) x )  e.  I  /\  ( x ( 2nd `  R ) z )  e.  I ) )  ->  A. y  e.  I 
( x G y )  e.  I )
98ralimi 2694 . . 3  |-  ( A. x  e.  I  ( A. y  e.  I 
( x G y )  e.  I  /\  A. z  e.  ran  G
( ( z ( 2nd `  R ) x )  e.  I  /\  ( x ( 2nd `  R ) z )  e.  I ) )  ->  A. x  e.  I  A. y  e.  I 
( x G y )  e.  I )
107, 9syl 15 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  A. x  e.  I  A. y  e.  I  ( x G y )  e.  I )
11 oveq1 5949 . . . 4  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
1211eleq1d 2424 . . 3  |-  ( x  =  A  ->  (
( x G y )  e.  I  <->  ( A G y )  e.  I ) )
13 oveq2 5950 . . . 4  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
1413eleq1d 2424 . . 3  |-  ( y  =  B  ->  (
( A G y )  e.  I  <->  ( A G B )  e.  I
) )
1512, 14rspc2v 2966 . 2  |-  ( ( A  e.  I  /\  B  e.  I )  ->  ( A. x  e.  I  A. y  e.  I  ( x G y )  e.  I  ->  ( A G B )  e.  I ) )
1610, 15mpan9 455 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  I
) )  ->  ( A G B )  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   ran crn 4769   ` cfv 5334  (class class class)co 5942   1stc1st 6204   2ndc2nd 6205  GIdcgi 20960   RingOpscrngo 21148   Idlcidl 25955
This theorem is referenced by:  idlsubcl  25971  intidl  25977  unichnidl  25979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-idl 25958
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