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Theorem jidd 25295
Description: Join is idempotent.  ( P  \/  P )  =  P. (Contributed by FL, 12-Dec-2009.)
Hypothesis
Ref Expression
jop1  |-  X  =  dom  dom  J
Assertion
Ref Expression
jidd  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( P  e.  X  ->  ( P J P )  =  P ) )

Proof of Theorem jidd
StepHypRef Expression
1 jop1 . . . . . . 7  |-  X  =  dom  dom  J
21labss1 25292 . . . . . 6  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( ( P  e.  X  /\  P  e.  X )  ->  ( P M ( P J P ) )  =  P ) )
32imp 418 . . . . 5  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P M ( P J 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 )
61cljo 25289 . . . . . . . 8  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( ( P  e.  X  /\  P  e.  X )  ->  ( P J P )  e.  X ) )
76imp 418 . . . . . . 7  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P J 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 J P )  e.  X ) )
91labss2 25294 . . . . . 6  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( ( P  e.  X  /\  ( P J P )  e.  X )  -> 
( P J ( P M ( P J P ) ) )  =  P ) )
104, 8, 9sylc 56 . . . . 5  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P J ( P M ( P J P ) ) )  =  P )
11 oveq2 5882 . . . . . 6  |-  ( ( P M ( P J P ) )  =  P  ->  ( P J ( P M ( P J P ) ) )  =  ( P J P ) )
12 eqtr 2313 . . . . . . . 8  |-  ( ( ( P J P )  =  ( P J ( P M ( P J P ) ) )  /\  ( P J ( P M ( P J P ) ) )  =  P )  -> 
( P J P )  =  P )
1312ex 423 . . . . . . 7  |-  ( ( P J P )  =  ( P J ( P M ( P J P ) ) )  ->  (
( P J ( P M ( P J P ) ) )  =  P  -> 
( P J P )  =  P ) )
1413eqcoms 2299 . . . . . 6  |-  ( ( P J ( P M ( P J P ) ) )  =  ( P J P )  ->  (
( P J ( P M ( P J P ) ) )  =  P  -> 
( P J P )  =  P ) )
1511, 14syl 15 . . . . 5  |-  ( ( P M ( P J P ) )  =  P  ->  (
( P J ( P M ( P J P ) ) )  =  P  -> 
( P J P )  =  P ) )
163, 10, 15sylc 56 . . . 4  |-  ( ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  /\  ( P  e.  X  /\  P  e.  X )
)  ->  ( P J P )  =  P )
1716expcom 424 . . 3  |-  ( ( P  e.  X  /\  P  e.  X )  ->  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  -> 
( P J P )  =  P ) )
1817anidms 626 . 2  |-  ( P  e.  X  ->  (
( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e.  LatAlg )  ->  ( P J P )  =  P ) )
1918com12 27 1  |-  ( ( J  e.  A  /\  M  e.  B  /\  <. J ,  M >.  e. 
LatAlg )  ->  ( P  e.  X  ->  ( P J P )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   dom cdm 4705  (class class class)co 5874   LatAlgclatalg 25284
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-sbc 3005  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-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-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-latalg 25285
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