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Metamath Proof Explorer

Theorem paddasslem13

Description: Lemma for paddass . The case when r .<_ ( x .\/ y ) . (Unlike the proof in Maeda and Maeda, we don't need x =/= y .) (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 paddasslem13 
|- ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> p e. ( ( 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 simpl1l 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> K e. HL )
6 simpl21 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> X C_ A )
7 simpl22 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> Y C_ A )
8 3 4 paddssat 
 |-  ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) C_ A )
9 5 6 7 8 syl3anc 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> ( X .+ Y ) C_ A )
10 simpl23 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> Z C_ A )
11 3 4 sspadd1 
 |-  ( ( K e. HL /\ ( X .+ Y ) C_ A /\ Z C_ A ) -> ( X .+ Y ) C_ ( ( X .+ Y ) .+ Z ) )
12 5 9 10 11 syl3anc 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> ( X .+ Y ) C_ ( ( X .+ Y ) .+ Z ) )
13 5 hllatd 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> K e. Lat )
14 simprll 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> x e. X )
15 simprlr 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> y e. Y )
16 simpl3l 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> p e. A )
17 eqid 
 |-  ( Base ` K ) = ( Base ` K )
18 17 3 atbase 
 |-  ( p e. A -> p e. ( Base ` K ) )
19 16 18 syl 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> p e. ( Base ` K ) )
20 6 14 sseldd 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> x e. A )
21 17 3 atbase 
 |-  ( x e. A -> x e. ( Base ` K ) )
22 20 21 syl 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> x e. ( Base ` K ) )
23 simpl3r 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> r e. A )
24 17 3 atbase 
 |-  ( r e. A -> r e. ( Base ` K ) )
25 23 24 syl 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> r e. ( Base ` K ) )
26 17 2 latjcl 
 |-  ( ( K e. Lat /\ x e. ( Base ` K ) /\ r e. ( Base ` K ) ) -> ( x .\/ r ) e. ( Base ` K ) )
27 13 22 25 26 syl3anc 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> ( x .\/ r ) e. ( Base ` K ) )
28 7 15 sseldd 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> y e. A )
29 17 3 atbase 
 |-  ( y e. A -> y e. ( Base ` K ) )
30 28 29 syl 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> y e. ( Base ` K ) )
31 17 2 latjcl 
 |-  ( ( K e. Lat /\ x e. ( Base ` K ) /\ y e. ( Base ` K ) ) -> ( x .\/ y ) e. ( Base ` K ) )
32 13 22 30 31 syl3anc 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> ( x .\/ y ) e. ( Base ` K ) )
33 simprrr 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> p .<_ ( x .\/ r ) )
34 17 1 2 latlej1 
 |-  ( ( K e. Lat /\ x e. ( Base ` K ) /\ y e. ( Base ` K ) ) -> x .<_ ( x .\/ y ) )
35 13 22 30 34 syl3anc 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> x .<_ ( x .\/ y ) )
36 simprrl 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> r .<_ ( x .\/ y ) )
37 17 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( x e. ( Base ` K ) /\ r e. ( Base ` K ) /\ ( x .\/ y ) e. ( Base ` K ) ) ) -> ( ( x .<_ ( x .\/ y ) /\ r .<_ ( x .\/ y ) ) <-> ( x .\/ r ) .<_ ( x .\/ y ) ) )
38 13 22 25 32 37 syl13anc 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> ( ( x .<_ ( x .\/ y ) /\ r .<_ ( x .\/ y ) ) <-> ( x .\/ r ) .<_ ( x .\/ y ) ) )
39 35 36 38 mpbi2and 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> ( x .\/ r ) .<_ ( x .\/ y ) )
40 17 1 13 19 27 32 33 39 lattrd 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> p .<_ ( x .\/ y ) )
41 1 2 3 4 elpaddri 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( x e. X /\ y e. Y ) /\ ( p e. A /\ p .<_ ( x .\/ y ) ) ) -> p e. ( X .+ Y ) )
42 13 6 7 14 15 16 40 41 syl322anc 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> p e. ( X .+ Y ) )
43 12 42 sseldd 
 |-  ( ( ( ( K e. HL /\ p =/= z ) /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ ( p e. A /\ r e. A ) ) /\ ( ( x e. X /\ y e. Y ) /\ ( r .<_ ( x .\/ y ) /\ p .<_ ( x .\/ r ) ) ) ) -> p e. ( ( X .+ Y ) .+ Z ) )