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Theorem midd 25193
Description: Meet is idempotent.  ( P  /\  P )  =  P. (Contributed by FL, 12-Dec-2009.)
Hypothesis
Ref Expression
jop1  |-  X  =  dom  dom  J
Assertion
Ref Expression
midd  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( P  e.  X  ->  ( P M P )  =  P ) )

Proof of Theorem midd
StepHypRef Expression
1 jop1 . . . . . . 7  |-  X  =  dom  dom  J
21labss2 25191 . . . . . 6  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( ( P  e.  X  /\  P  e.  X )  ->  ( P J ( P M P ) )  =  P ) )
32imp 418 . . . . 5  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P J ( P M P ) )  =  P )
4 simpl 443 . . . . . 6  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg ) )
5 simprl 732 . . . . . . 7  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  P  e.  X )
61clme 25187 . . . . . . . 8  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( ( P  e.  X  /\  P  e.  X )  ->  ( P M P )  e.  X ) )
76imp 418 . . . . . . 7  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P M P )  e.  X
)
85, 7jca 518 . . . . . 6  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P  e.  X  /\  ( P M P )  e.  X ) )
91labss1 25189 . . . . . 6  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( ( P  e.  X  /\  ( P M P )  e.  X )  -> 
( P M ( P J ( P M P ) ) )  =  P ) )
104, 8, 9sylc 56 . . . . 5  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P M ( P J ( P M P ) ) )  =  P )
11 oveq2 5866 . . . . . 6  |-  ( ( P J ( P M P ) )  =  P  ->  ( P M ( P J ( P M P ) ) )  =  ( P M P ) )
12 eqtr 2300 . . . . . . . 8  |-  ( ( ( P M P )  =  ( P M ( P J ( P M P ) ) )  /\  ( P M ( P J ( P M P ) ) )  =  P )  -> 
( P M P )  =  P )
1312ex 423 . . . . . . 7  |-  ( ( P M P )  =  ( P M ( P J ( P M P ) ) )  ->  (
( P M ( P J ( P M P ) ) )  =  P  -> 
( P M P )  =  P ) )
1413eqcoms 2286 . . . . . 6  |-  ( ( P M ( P J ( P M P ) ) )  =  ( P M P )  ->  (
( P M ( P J ( P M P ) ) )  =  P  -> 
( P M P )  =  P ) )
1511, 14syl 15 . . . . 5  |-  ( ( P J ( P M P ) )  =  P  ->  (
( P M ( P J ( P M P ) ) )  =  P  -> 
( P M P )  =  P ) )
163, 10, 15sylc 56 . . . 4  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P M P )  =  P )
1716expcom 424 . . 3  |-  ( ( P  e.  X  /\  P  e.  X )  ->  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  -> 
( P M P )  =  P ) )
1817anidms 626 . 2  |-  ( P  e.  X  ->  (
( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  ( P M P )  =  P ) )
1918com12 27 1  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( P  e.  X  ->  ( P M P )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   dom cdm 4689  (class class class)co 5858   LatAlgclatalg 25181
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-sbc 2992  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-id 4309  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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