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Theorem idllmulcl 30619
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 2392 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
51, 2, 3, 4isidl 30613 . . . . 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 482 . . . 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 1008 . . 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 455 . . . . . 6  |-  ( ( ( z H x )  e.  I  /\  ( x H z )  e.  I )  ->  ( z H x )  e.  I
)
98ralimi 2785 . . . . 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 464 . . . 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 2785 . . 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 6222 . . . 4  |-  ( x  =  A  ->  (
z H x )  =  ( z H A ) )
1413eleq1d 2461 . . 3  |-  ( x  =  A  ->  (
( z H x )  e.  I  <->  ( z H A )  e.  I
) )
15 oveq1 6221 . . . 4  |-  ( z  =  B  ->  (
z H A )  =  ( B H A ) )
1615eleq1d 2461 . . 3  |-  ( z  =  B  ->  (
( z H A )  e.  I  <->  ( B H A )  e.  I
) )
1714, 16rspc2v 3157 . 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 467 1  |-  ( ( ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  /\  ( A  e.  I  /\  B  e.  X
) )  ->  ( B H A )  e.  I )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836   A.wral 2742    C_ wss 3402   ran crn 4927   ` cfv 5509  (class class class)co 6214   1stc1st 6715   2ndc2nd 6716  GIdcgi 25327   RingOpscrngo 25515   Idlcidl 30606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-iota 5473  df-fun 5511  df-fv 5517  df-ov 6217  df-idl 30609
This theorem is referenced by:  idlnegcl  30621  divrngidl  30627  intidl  30628  unichnidl  30630  prnc  30666  ispridlc  30669
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