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Theorem hlsupr 30197
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 2296 . . . 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 30192 . . 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 445 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ) ) ) )
76imp 418 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  hlsupr2  30198  atbtwnexOLDN  30258  atbtwnex  30259  cdlemb  30605  lhpexle2lem  30820  lhpexle3lem  30822  cdlemf1  31372  cdlemg35  31524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-cvlat 30134  df-hlat 30163
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