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Theorem hlsupr2 32385
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 2402 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 hlsupr2.j . . . 4  |-  .\/  =  ( join `  K )
3 hlsupr2.a . . . 4  |-  A  =  ( Atoms `  K )
41, 2, 3hlsupr 32384 . . 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 432 . 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 1000 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  HL )
7 hlcvl 32358 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CvLat )
86, 7syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  CvLat
)
9 simpl2 1001 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  P  e.  A )
10 simpl3 1002 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  Q  e.  A )
11 simpr 459 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  r  e.  A )
123, 1, 2cvlsupr3 32343 . . . . 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 1232 . . . 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 2914 . . 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 3743 . . . . 5  |-  ( P  e.  A  ->  A  =/=  (/) )
16153ad2ant2 1019 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  A  =/=  (/) )
17 r19.37zv 3868 . . . 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 17 . . 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 253 . 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 232 1  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  E. r  e.  A  ( P  .\/  r )  =  ( Q  .\/  r ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   (/)c0 3737   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   lecple 14808   joincjn 15789   Atomscatm 32262   CvLatclc 32264   HLchlt 32349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-preset 15773  df-poset 15791  df-plt 15804  df-lub 15820  df-glb 15821  df-join 15822  df-meet 15823  df-p0 15885  df-lat 15892  df-covers 32265  df-ats 32266  df-atl 32297  df-cvlat 32321  df-hlat 32350
This theorem is referenced by:  4atexlemex6  33072
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