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Theorem hlsupr 34059
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 2462 . . . 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 34054 . . 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 459 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) ) )
76imp 429 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   Atomscatm 33937   HLchlt 34024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-iota 5544  df-fv 5589  df-ov 6280  df-cvlat 33996  df-hlat 34025
This theorem is referenced by:  hlsupr2  34060  atbtwnexOLDN  34120  atbtwnex  34121  cdlemb  34467  lhpexle2lem  34682  lhpexle3lem  34684  cdlemf1  35234  cdlemg35  35386
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