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Theorem hlsupr2 29635
Description: A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
hlsupr2.j  |-  .\/  =  ( join `  K )
hlsupr2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlsupr2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r ) )
Distinct variable groups:    A, r    K, r    P, r    Q, r
Allowed substitution hint:    .\/ ( r)

Proof of Theorem hlsupr2
StepHypRef Expression
1 eqid 2366 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 hlsupr2.j . . . 4  |-  .\/  =  ( join `  K )
3 hlsupr2.a . . . 4  |-  A  =  ( Atoms `  K )
41, 2, 3hlsupr 29634 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) )
54ex 423 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r ( le `  K ) ( P 
.\/  Q ) ) ) )
6 simpl1 959 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  HL )
7 hlcvl 29608 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CvLat )
86, 7syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  CvLat
)
9 simpl2 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  P  e.  A )
10 simpl3 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  Q  e.  A )
11 simpr 447 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  r  e.  A )
123, 1, 2cvlsupr3 29593 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  Q  e.  A  /\  r  e.  A )
)  ->  ( ( P  .\/  r )  =  ( Q  .\/  r
)  <->  ( P  =/= 
Q  ->  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
138, 9, 10, 11, 12syl13anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  ( ( P  .\/  r )  =  ( Q  .\/  r
)  <->  ( P  =/= 
Q  ->  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
1413rexbidva 2645 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r )  <->  E. r  e.  A  ( P  =/=  Q  ->  ( r  =/=  P  /\  r  =/=  Q  /\  r ( le `  K ) ( P 
.\/  Q ) ) ) ) )
15 ne0i 3549 . . . . 5  |-  ( P  e.  A  ->  A  =/=  (/) )
16153ad2ant2 978 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  A  =/=  (/) )
17 r19.37zv 3639 . . . 4  |-  ( A  =/=  (/)  ->  ( E. r  e.  A  ( P  =/=  Q  ->  (
r  =/=  P  /\  r  =/=  Q  /\  r
( le `  K
) ( P  .\/  Q ) ) )  <->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
1816, 17syl 15 . . 3  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( E. r  e.  A  ( P  =/= 
Q  ->  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) )  <->  ( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/= 
Q  /\  r ( le `  K ) ( P  .\/  Q ) ) ) ) )
1914, 18bitrd 244 . 2  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r )  <-> 
( P  =/=  Q  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  Q  /\  r ( le `  K ) ( P 
.\/  Q ) ) ) ) )
205, 19mpbird 223 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   E.wrex 2629   (/)c0 3543   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   lecple 13423   joincjn 14288   Atomscatm 29512   CvLatclc 29514   HLchlt 29599
This theorem is referenced by:  4atexlemex6  30322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-join 14320  df-lat 14362  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600
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