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Theorem hlsupr 30022
Description: A Hilbert lattice has the superposition property. Theorem 13.2 in [Crawley] p. 107. (Contributed by NM, 30-Jan-2012.)
Hypotheses
Ref Expression
hlsupr.l  |-  .<_  =  ( le `  K )
hlsupr.j  |-  .\/  =  ( join `  K )
hlsupr.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlsupr  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
Distinct variable groups:    A, r    K, r    P, r    Q, r
Allowed substitution hints:    .\/ ( r)    .<_ ( r)

Proof of Theorem hlsupr
StepHypRef Expression
1 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 hlsupr.l . . . 4  |-  .<_  =  ( le `  K )
3 hlsupr.j . . . 4  |-  .\/  =  ( join `  K )
4 hlsupr.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4hlsuprexch 30017 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( ( P  =/= 
Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )  /\  A. r  e.  ( Base `  K ) ( ( -.  P  .<_  r  /\  P  .<_  ( r  .\/  Q ) )  ->  Q  .<_  ( r  .\/  P
) ) ) )
65simpld 446 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) ) )
76imp 419 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r  .<_  ( P  .\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   Atomscatm 29900   HLchlt 29987
This theorem is referenced by:  hlsupr2  30023  atbtwnexOLDN  30083  atbtwnex  30084  cdlemb  30430  lhpexle2lem  30645  lhpexle3lem  30647  cdlemf1  31197  cdlemg35  31349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5409  df-fv 5453  df-ov 6075  df-cvlat 29959  df-hlat 29988
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