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Theorem hlsupr 32367
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 2400 . . . 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 32362 . . 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 457 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) ) )
76imp 427 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   A.wral 2751   E.wrex 2752   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   Atomscatm 32245   HLchlt 32332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-iota 5487  df-fv 5531  df-ov 6235  df-cvlat 32304  df-hlat 32333
This theorem is referenced by:  hlsupr2  32368  atbtwnexOLDN  32428  atbtwnex  32429  cdlemb  32775  lhpexle2lem  32990  lhpexle3lem  32992  cdlemf1  33544  cdlemg35  33696
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