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Theorem idllmulcl 26748
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 2296 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4isidl 26742 . . . . 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 simpl 443 . . . . . 6  |-  ( ( ( z H x )  e.  I  /\  ( x H z )  e.  I )  ->  ( z H x )  e.  I
)
98ralimi 2631 . . . . 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 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  ( z H x )  e.  I )
1110ralimi 2631 . . 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 15 . 2  |-  ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R
) )  ->  A. x  e.  I  A. z  e.  X  ( z H x )  e.  I )
13 oveq2 5882 . . . 4  |-  ( x  =  A  ->  (
z H x )  =  ( z H A ) )
1413eleq1d 2362 . . 3  |-  ( x  =  A  ->  (
( z H x )  e.  I  <->  ( z H A )  e.  I
) )
15 oveq1 5881 . . . 4  |-  ( z  =  B  ->  (
z H A )  =  ( B H A ) )
1615eleq1d 2362 . . 3  |-  ( z  =  B  ->  (
( z H A )  e.  I  <->  ( B H A )  e.  I
) )
1714, 16rspc2v 2903 . 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 455 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Idlcidl 26735
This theorem is referenced by:  idlnegcl  26750  divrngidl  26756  intidl  26757  unichnidl  26759  prnc  26795  ispridlc  26798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-idl 26738
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