Metamath Proof Explorer

Theorem paddasslem16

Description: Lemma for paddass . Use elpaddn0 to eliminate x and r from paddasslem15 . (Contributed by NM, 11-Jan-2012)

Ref Expression
Hypotheses paddasslem.l 
|- .<_ = ( le ` K )
paddasslem.j 
|- .\/ = ( join ` K )
paddasslem.a 
|- A = ( Atoms ` K )
paddasslem.p 
|- .+ = ( +P ` K )
Assertion paddasslem16 
|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )

Proof

Step Hyp Ref Expression
1 paddasslem.l 
 |-  .<_ = ( le ` K )
2 paddasslem.j 
 |-  .\/ = ( join ` K )
3 paddasslem.a 
 |-  A = ( Atoms ` K )
4 paddasslem.p 
 |-  .+ = ( +P ` K )
5 hllat 
 |-  ( K e. HL -> K e. Lat )
6 5 3ad2ant1 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. Lat )
7 simp21 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> X C_ A )
8 simp1 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> K e. HL )
9 simp22 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Y C_ A )
10 simp23 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> Z C_ A )
11 3 4 paddssat 
 |-  ( ( K e. HL /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) C_ A )
12 8 9 10 11 syl3anc 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( Y .+ Z ) C_ A )
13 simp3l 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) )
14 1 2 3 4 elpaddn0 
 |-  ( ( ( K e. Lat /\ X C_ A /\ ( Y .+ Z ) C_ A ) /\ ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) <-> ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) ) )
15 6 7 12 13 14 syl31anc 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) <-> ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) ) )
16 simpr 
 |-  ( ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) -> ( Y =/= (/) /\ Z =/= (/) ) )
17 1 2 3 4 paddasslem15 
 |-  ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) /\ ( p e. A /\ ( x e. X /\ r e. ( Y .+ Z ) ) /\ p .<_ ( x .\/ r ) ) ) -> p e. ( ( X .+ Y ) .+ Z ) )
18 16 17 syl3anl3 
 |-  ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ ( p e. A /\ ( x e. X /\ r e. ( Y .+ Z ) ) /\ p .<_ ( x .\/ r ) ) ) -> p e. ( ( X .+ Y ) .+ Z ) )
19 18 3exp2 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. A -> ( ( x e. X /\ r e. ( Y .+ Z ) ) -> ( p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) ) )
20 19 imp 
 |-  ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ p e. A ) -> ( ( x e. X /\ r e. ( Y .+ Z ) ) -> ( p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) ) )
21 20 rexlimdvv 
 |-  ( ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) /\ p e. A ) -> ( E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) -> p e. ( ( X .+ Y ) .+ Z ) ) )
22 21 expimpd 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( ( p e. A /\ E. x e. X E. r e. ( Y .+ Z ) p .<_ ( x .\/ r ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) )
23 15 22 sylbid 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( p e. ( X .+ ( Y .+ Z ) ) -> p e. ( ( X .+ Y ) .+ Z ) ) )
24 23 ssrdv 
 |-  ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )