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Theorem idllmulcl 26632
Description: An ideal is closed under multiplication on the left. (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
idllmulcl  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  X
) )  ->  ( B H A )  e.  I )

Proof of Theorem idllmulcl
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 2438 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4isidl 26626 . . . . 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 472 . . . 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 972 . . 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 simpl 445 . . . . . 6  |-  ( ( ( z H x )  e.  I  /\  ( x H z )  e.  I )  ->  ( z H x )  e.  I
)
98ralimi 2783 . . . . 5  |-  ( A. z  e.  X  (
( z H x )  e.  I  /\  ( x H z )  e.  I )  ->  A. z  e.  X  ( z H x )  e.  I )
109adantl 454 . . . 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  ( z H x )  e.  I )
1110ralimi 2783 . . 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  ( z H x )  e.  I )
127, 11syl 16 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  A. x  e.  I  A. z  e.  X  ( z H x )  e.  I )
13 oveq2 6091 . . . 4  |-  ( x  =  A  ->  (
z H x )  =  ( z H A ) )
1413eleq1d 2504 . . 3  |-  ( x  =  A  ->  (
( z H x )  e.  I  <->  ( z H A )  e.  I
) )
15 oveq1 6090 . . . 4  |-  ( z  =  B  ->  (
z H A )  =  ( B H A ) )
1615eleq1d 2504 . . 3  |-  ( z  =  B  ->  (
( z H A )  e.  I  <->  ( B H A )  e.  I
) )
1714, 16rspc2v 3060 . 2  |-  ( ( A  e.  I  /\  B  e.  X )  ->  ( A. x  e.  I  A. z  e.  X  ( z H x )  e.  I  ->  ( B H A )  e.  I ) )
1812, 17mpan9 457 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  X
) )  ->  ( B H A )  e.  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ran crn 4881   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350  GIdcgi 21777   RingOpscrngo 21965   Idlcidl 26619
This theorem is referenced by:  idlnegcl  26634  divrngidl  26640  intidl  26641  unichnidl  26643  prnc  26679  ispridlc  26682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-idl 26622
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