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Theorem hlsupr2 29576
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 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 hlsupr2.j . . . 4  |-  .\/  =  ( join `  K )
3 hlsupr2.a . . . 4  |-  A  =  ( Atoms `  K )
41, 2, 3hlsupr 29575 . . 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 958 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  K  e.  HL )
7 hlcvl 29549 . . . . . 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 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  r  e.  A
)  ->  P  e.  A )
10 simpl3 960 . . . . 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 29534 . . . . 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 1184 . . . 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 2560 . . 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 3461 . . . . 5  |-  ( P  e.  A  ->  A  =/=  (/) )
16153ad2ant2 977 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  A  =/=  (/) )
17 r19.37zv 3550 . . . 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 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   CvLatclc 29455   HLchlt 29540
This theorem is referenced by:  4atexlemex6  30263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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