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Theorem hlsupr2 30258
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 2438 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 hlsupr2.j . . . 4  |-  .\/  =  ( join `  K )
3 hlsupr2.a . . . 4  |-  A  =  ( Atoms `  K )
41, 2, 3hlsupr 30257 . . 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 425 . 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 961 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  HL )
7 hlcvl 30231 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CvLat )
86, 7syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  CvLat
)
9 simpl2 962 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  P  e.  A )
10 simpl3 963 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  Q  e.  A )
11 simpr 449 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  r  e.  A )
123, 1, 2cvlsupr3 30216 . . . . 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 1187 . . . 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 2724 . . 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 3636 . . . . 5  |-  ( P  e.  A  ->  A  =/=  (/) )
16153ad2ant2 980 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  A  =/=  (/) )
17 r19.37zv 3726 . . . 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 16 . . 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 246 . 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 225 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   (/)c0 3630   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   Atomscatm 30135   CvLatclc 30137   HLchlt 30222
This theorem is referenced by:  4atexlemex6  30945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-join 14438  df-lat 14480  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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