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Theorem mop 25185
Description: Meet is a binary internal operation. (Contributed by FL, 12-Dec-2009.)
Hypothesis
Ref Expression
jop1  |-  X  =  dom  dom  J
Assertion
Ref Expression
mop  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  M :
( X  X.  X
) --> X )

Proof of Theorem mop
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 jop1 . . . 4  |-  X  =  dom  dom  J
21islatalg 25183 . . 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 simp2 956 . . 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 ) ) ) ) )  ->  M : ( X  X.  X ) --> X )
42, 3syl6bi 219 . 2  |-  ( ( J  e.  A  /\  M  e.  B )  ->  ( <. J ,  M >.  e.  LatAlg  ->  M : ( X  X.  X ) --> X ) )
543impia 1148 1  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  M :
( X  X.  X
) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643    X. cxp 4687   dom cdm 4689   -->wf 5251  (class class class)co 5858   LatAlgclatalg 25181
This theorem is referenced by:  clme  25187
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-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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  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-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-latalg 25182
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