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Theorem labs2 25293
Description: Absorption law.  ( x  \/  ( x  /\  y ) )  =  x. (Contributed by FL, 12-Dec-2009.)
Hypothesis
Ref Expression
jop1  |-  X  =  dom  dom  J
Assertion
Ref Expression
labs2  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  A. x  e.  X  A. y  e.  X  ( x J ( x M y ) )  =  x )
Distinct variable groups:    x, J, y    x, M, y    x, X, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem labs2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 jop1 . . . 4  |-  X  =  dom  dom  J
21islatalg 25286 . . 3  |-  ( ( J  e.  A  /\  M  e.  B )  ->  ( <. J ,  M >.  e.  LatAlg 
<->  ( J : ( X  X.  X ) --> X  /\  M :
( X  X.  X
) --> X  /\  A. x  e.  X  A. y  e.  X  (
( x J y )  =  ( y J x )  /\  ( x M y )  =  ( y M x )  /\  ( ( x M ( x J y ) )  =  x  /\  ( x J ( x M y ) )  =  x  /\  A. z  e.  X  ( ( x M ( y M z ) )  =  ( ( x M y ) M z )  /\  ( x J ( y J z ) )  =  ( ( x J y ) J z ) ) ) ) ) ) )
3 simp32 992 . . . . . 6  |-  ( ( ( x J y )  =  ( y J x )  /\  ( x M y )  =  ( y M x )  /\  ( ( x M ( x J y ) )  =  x  /\  ( x J ( x M y ) )  =  x  /\  A. z  e.  X  ( ( x M ( y M z ) )  =  ( ( x M y ) M z )  /\  ( x J ( y J z ) )  =  ( ( x J y ) J z ) ) ) )  ->  ( x J ( x M y ) )  =  x )
43ralimi 2631 . . . . 5  |-  ( A. y  e.  X  (
( x J y )  =  ( y J x )  /\  ( x M y )  =  ( y M x )  /\  ( ( x M ( x J y ) )  =  x  /\  ( x J ( x M y ) )  =  x  /\  A. z  e.  X  ( ( x M ( y M z ) )  =  ( ( x M y ) M z )  /\  ( x J ( y J z ) )  =  ( ( x J y ) J z ) ) ) )  ->  A. y  e.  X  ( x J ( x M y ) )  =  x )
54ralimi 2631 . . . 4  |-  ( A. x  e.  X  A. y  e.  X  (
( x J y )  =  ( y J x )  /\  ( x M y )  =  ( y M x )  /\  ( ( x M ( x J y ) )  =  x  /\  ( x J ( x M y ) )  =  x  /\  A. z  e.  X  ( ( x M ( y M z ) )  =  ( ( x M y ) M z )  /\  ( x J ( y J z ) )  =  ( ( x J y ) J z ) ) ) )  ->  A. x  e.  X  A. y  e.  X  ( x J ( x M y ) )  =  x )
653ad2ant3 978 . . 3  |-  ( ( J : ( X  X.  X ) --> X  /\  M : ( X  X.  X ) --> X  /\  A. x  e.  X  A. y  e.  X  ( (
x J y )  =  ( y J x )  /\  (
x M y )  =  ( y M x )  /\  (
( x M ( x J y ) )  =  x  /\  ( x J ( x M y ) )  =  x  /\  A. z  e.  X  ( ( x M ( y M z ) )  =  ( ( x M y ) M z )  /\  ( x J ( y J z ) )  =  ( ( x J y ) J z ) ) ) ) )  ->  A. x  e.  X  A. y  e.  X  ( x J ( x M y ) )  =  x )
72, 6syl6bi 219 . 2  |-  ( ( J  e.  A  /\  M  e.  B )  ->  ( <. J ,  M >.  e.  LatAlg  ->  A. x  e.  X  A. y  e.  X  ( x J ( x M y ) )  =  x ) )
873impia 1148 1  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  A. x  e.  X  A. y  e.  X  ( x J ( x M y ) )  =  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656    X. cxp 4703   dom cdm 4705   -->wf 5267  (class class class)co 5874   LatAlgclatalg 25284
This theorem is referenced by:  labss2  25294
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-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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  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-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-latalg 25285
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