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Theorem idlrmulcl 25969
Description: An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
idllmulcl.1  |-  G  =  ( 1st `  R
)
idllmulcl.2  |-  H  =  ( 2nd `  R
)
idllmulcl.3  |-  X  =  ran  G
Assertion
Ref Expression
idlrmulcl  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  X
) )  ->  ( A H B )  e.  I )

Proof of Theorem idlrmulcl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idllmulcl.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 idllmulcl.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
3 idllmulcl.3 . . . . . 6  |-  X  =  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_  X  /\  (GId `  G
)  e.  I  /\  A. x  e.  I  ( A. y  e.  I 
( x G y )  e.  I  /\  A. z  e.  X  ( ( z H x )  e.  I  /\  ( x H z )  e.  I ) ) ) ) )
65biimpa 470 . . . 4  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  (
I  C_  X  /\  (GId `  G )  e.  I  /\  A. x  e.  I  ( A. y  e.  I  (
x G y )  e.  I  /\  A. z  e.  X  (
( z H x )  e.  I  /\  ( x H 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.  X  (
( z H x )  e.  I  /\  ( x H z )  e.  I ) ) )
8 simpr 447 . . . . . 6  |-  ( ( ( z H x )  e.  I  /\  ( x H z )  e.  I )  ->  ( x H z )  e.  I
)
98ralimi 2694 . . . . 5  |-  ( A. z  e.  X  (
( z H x )  e.  I  /\  ( x H z )  e.  I )  ->  A. z  e.  X  ( x H z )  e.  I )
109adantl 452 . . . 4  |-  ( ( A. y  e.  I 
( x G y )  e.  I  /\  A. z  e.  X  ( ( z H x )  e.  I  /\  ( x H z )  e.  I ) )  ->  A. z  e.  X  ( x H z )  e.  I )
1110ralimi 2694 . . 3  |-  ( A. x  e.  I  ( A. y  e.  I 
( x G y )  e.  I  /\  A. z  e.  X  ( ( z H x )  e.  I  /\  ( x H z )  e.  I ) )  ->  A. x  e.  I  A. z  e.  X  ( x H z )  e.  I )
127, 11syl 15 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  A. x  e.  I  A. z  e.  X  ( x H z )  e.  I )
13 oveq1 5952 . . . 4  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1413eleq1d 2424 . . 3  |-  ( x  =  A  ->  (
( x H z )  e.  I  <->  ( A H z )  e.  I ) )
15 oveq2 5953 . . . 4  |-  ( z  =  B  ->  ( A H z )  =  ( A H B ) )
1615eleq1d 2424 . . 3  |-  ( z  =  B  ->  (
( A H z )  e.  I  <->  ( A H B )  e.  I
) )
1714, 16rspc2v 2966 . 2  |-  ( ( A  e.  I  /\  B  e.  X )  ->  ( A. x  e.  I  A. z  e.  X  ( x H z )  e.  I  ->  ( A H B )  e.  I ) )
1812, 17mpan9 455 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  X
) )  ->  ( A H 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 4772   ` cfv 5337  (class class class)co 5945   1stc1st 6207   2ndc2nd 6208  GIdcgi 20966   RingOpscrngo 21154   Idlcidl 25955
This theorem is referenced by:  1idl  25974  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 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
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 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-idl 25958
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