Metamath Proof Explorer

Theorem paddasslem6

Description: Lemma for paddass . (Contributed by NM, 8-Jan-2012)

Ref Expression
Hypotheses paddasslem.l 
|- .<_ = ( le ` K )
paddasslem.j 
|- .\/ = ( join ` K )
paddasslem.a 
|- A = ( Atoms ` K )
Assertion paddasslem6 
|- ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> p .<_ ( s .\/ z ) )

Proof

Step Hyp Ref Expression
1 paddasslem.l 
 |-  .<_ = ( le ` K )
2 paddasslem.j 
 |-  .\/ = ( join ` K )
3 paddasslem.a 
 |-  A = ( Atoms ` K )
4 simpl1 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> K e. HL )
5 simpl2r 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> s e. A )
6 simpl2l 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> p e. A )
7 simpl3 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> z e. A )
8 5 6 7 3jca 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> ( s e. A /\ p e. A /\ z e. A ) )
9 simprl 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> s =/= z )
10 4 8 9 3jca 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> ( K e. HL /\ ( s e. A /\ p e. A /\ z e. A ) /\ s =/= z ) )
11 simprr 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> s .<_ ( p .\/ z ) )
12 1 2 3 hlatexch2 
 |-  ( ( K e. HL /\ ( s e. A /\ p e. A /\ z e. A ) /\ s =/= z ) -> ( s .<_ ( p .\/ z ) -> p .<_ ( s .\/ z ) ) )
13 10 11 12 sylc 
 |-  ( ( ( K e. HL /\ ( p e. A /\ s e. A ) /\ z e. A ) /\ ( s =/= z /\ s .<_ ( p .\/ z ) ) ) -> p .<_ ( s .\/ z ) )