Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idlrmulcl Unicode version

Theorem idlrmulcl 26646
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 2283 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4isidl 26639 . . . . 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 2618 . . . . 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 2618 . . 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 5865 . . . 4  |-  ( x  =  A  ->  (
x H z )  =  ( A H z ) )
1413eleq1d 2349 . . 3  |-  ( x  =  A  ->  (
( x H z )  e.  I  <->  ( A H z )  e.  I ) )
15 oveq2 5866 . . . 4  |-  ( z  =  B  ->  ( A H z )  =  ( A H B ) )
1615eleq1d 2349 . . 3  |-  ( z  =  B  ->  (
( A H z )  e.  I  <->  ( A H B )  e.  I
) )
1714, 16rspc2v 2890 . 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   Idlcidl 26632
This theorem is referenced by:  1idl  26651  intidl  26654  unichnidl  26656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-idl 26635
  Copyright terms: Public domain W3C validator