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Theorem hlsupr 28842
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 2284 . . . 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 28837 . . 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 447 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) ) )
76imp 420 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 5    -> wi 6    /\ wa 360    /\ w3a 936    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   Atomscatm 28720   HLchlt 28807
This theorem is referenced by:  hlsupr2  28843  atbtwnexOLDN  28903  atbtwnex  28904  cdlemb  29250  lhpexle2lem  29465  lhpexle3lem  29467  cdlemf1  30017  cdlemg35  30169
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5822  df-cvlat 28779  df-hlat 28808
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