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Theorem hlsupr2 29869
Description: A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
hlsupr2.j  |-  .\/  =  ( join `  K )
hlsupr2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlsupr2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r ) )
Distinct variable groups:    A, r    K, r    P, r    Q, r
Allowed substitution hint:    .\/ ( r)

Proof of Theorem hlsupr2
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 hlsupr2.j . . . 4  |-  .\/  =  ( join `  K )
3 hlsupr2.a . . . 4  |-  A  =  ( Atoms `  K )
41, 2, 3hlsupr 29868 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) )
54ex 424 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r ( le `  K ) ( P 
.\/  Q ) ) ) )
6 simpl1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  HL )
7 hlcvl 29842 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CvLat )
86, 7syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  CvLat
)
9 simpl2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  P  e.  A )
10 simpl3 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  Q  e.  A )
11 simpr 448 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  r  e.  A )
123, 1, 2cvlsupr3 29827 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  r  e.  A )
)  ->  ( ( P  .\/  r )  =  ( Q  .\/  r
)  <->  ( P  =/= 
Q  ->  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
138, 9, 10, 11, 12syl13anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  ( ( P  .\/  r )  =  ( Q  .\/  r
)  <->  ( P  =/= 
Q  ->  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
1413rexbidva 2683 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r )  <->  E. r  e.  A  ( P  =/=  Q  ->  ( r  =/=  P  /\  r  =/=  Q  /\  r ( le `  K ) ( P 
.\/  Q ) ) ) ) )
15 ne0i 3594 . . . . 5  |-  ( P  e.  A  ->  A  =/=  (/) )
16153ad2ant2 979 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  A  =/=  (/) )
17 r19.37zv 3684 . . . 4  |-  ( A  =/=  (/)  ->  ( E. r  e.  A  ( P  =/=  Q  ->  (
r  =/=  P  /\  r  =/=  Q  /\  r
( le `  K
) ( P  .\/  Q ) ) )  <->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
1816, 17syl 16 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( E. r  e.  A  ( P  =/= 
Q  ->  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) )  <->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
1914, 18bitrd 245 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r )  <-> 
( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r ( le `  K ) ( P 
.\/  Q ) ) ) ) )
205, 19mpbird 224 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   (/)c0 3588   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   Atomscatm 29746   CvLatclc 29748   HLchlt 29833
This theorem is referenced by:  4atexlemex6  30556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-join 14388  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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