Metamath Proof Explorer

Theorem hlsupr2

Description: A Hilbert lattice has the superposition property. (Contributed by NM, 25-Nov-2012)

Ref Expression
Hypotheses hlsupr2.j 
|- .\/ = ( join ` K )
hlsupr2.a 
|- A = ( Atoms ` K )
Assertion hlsupr2 
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> E. r e. A ( P .\/ r ) = ( Q .\/ r ) )

Proof

Step Hyp Ref Expression
1 hlsupr2.j 
 |-  .\/ = ( join ` K )
2 hlsupr2.a 
 |-  A = ( Atoms ` K )
3 eqid 
 |-  ( le ` K ) = ( le ` K )
4 3 1 2 hlsupr 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> E. r e. A ( r =/= P /\ r =/= Q /\ r ( le ` K ) ( P .\/ Q ) ) )
5 4 ex 
 |-  ( ( 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 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ r e. A ) -> K e. HL )
7 hlcvl 
 |-  ( K e. HL -> K e. CvLat )
8 6 7 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ r e. A ) -> K e. CvLat )
9 simpl2 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ r e. A ) -> P e. A )
10 simpl3 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ r e. A ) -> Q e. A )
11 simpr 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ r e. A ) -> r e. A )
12 2 3 1 cvlsupr3 
 |-  ( ( 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 ) ) ) ) )
13 8 9 10 11 12 syl13anc 
 |-  ( ( ( 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 ) ) ) ) )
14 13 rexbidva 
 |-  ( ( 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 
 |-  ( P e. A -> A =/= (/) )
16 15 3ad2ant2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> A =/= (/) )
17 r19.37zv 
 |-  ( 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 ) ) ) ) )
18 16 17 syl 
 |-  ( ( 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 ) ) ) ) )
19 14 18 bitrd 
 |-  ( ( 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 ) ) ) ) )
20 5 19 mpbird 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> E. r e. A ( P .\/ r ) = ( Q .\/ r ) )