Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2llnjaN Unicode version

Theorem 2llnjaN 29731
Description: The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 29732 in terms of atoms). (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2llnja.l  |-  .<_  =  ( le `  K )
2llnja.j  |-  .\/  =  ( join `  K )
2llnja.a  |-  A  =  ( Atoms `  K )
2llnja.n  |-  N  =  ( LLines `  K )
2llnja.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
2llnjaN  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  =  W )

Proof of Theorem 2llnjaN
StepHypRef Expression
1 eqid 2380 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 2llnja.l . 2  |-  .<_  =  ( le `  K )
3 simpl1l 1008 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  K  e.  HL )
4 hllat 29529 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  K  e.  Lat )
6 simpl21 1035 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  Q  e.  A )
7 simpl22 1036 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  R  e.  A )
8 2llnja.j . . . . 5  |-  .\/  =  ( join `  K )
9 2llnja.a . . . . 5  |-  A  =  ( Atoms `  K )
101, 8, 9hlatjcl 29532 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( Q  .\/  R )  e.  (
Base `  K )
)
12 simpl31 1038 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  S  e.  A )
13 simpl32 1039 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  T  e.  A )
141, 8, 9hlatjcl 29532 . . . 4  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
153, 12, 13, 14syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( S  .\/  T )  e.  (
Base `  K )
)
161, 8latjcl 14399 . . 3  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  e.  ( Base `  K ) )
175, 11, 15, 16syl3anc 1184 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  e.  ( Base `  K ) )
18 simpl1r 1009 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  W  e.  P )
19 2llnja.p . . . 4  |-  P  =  ( LPlanes `  K )
201, 19lplnbase 29699 . . 3  |-  ( W  e.  P  ->  W  e.  ( Base `  K
) )
2118, 20syl 16 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  W  e.  ( Base `  K )
)
22 simpr1 963 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( Q  .\/  R )  .<_  W )
23 simpr2 964 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( S  .\/  T )  .<_  W )
241, 2, 8latjle12 14411 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  R )  e.  ( Base `  K )  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( Q  .\/  R )  .<_  W  /\  ( S  .\/  T ) 
.<_  W )  <->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  .<_  W )
)
255, 11, 15, 21, 24syl13anc 1186 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W )  <->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  .<_  W )
)
2622, 23, 25mpbi2and 888 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  .<_  W )
271, 9atbase 29455 . . . . . . . . . 10  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
2813, 27syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  T  e.  ( Base `  K )
)
291, 8latjcl 14399 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  T )  e.  ( Base `  K ) )
305, 11, 28, 29syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  T )  e.  ( Base `  K ) )
311, 9atbase 29455 . . . . . . . . . . 11  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
3212, 31syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  S  e.  ( Base `  K )
)
331, 2, 8latlej2 14410 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) )  ->  T  .<_  ( S  .\/  T
) )
345, 32, 28, 33syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  T  .<_  ( S  .\/  T ) )
351, 2, 8latjlej2 14415 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( T  e.  ( Base `  K )  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  ( Q  .\/  R )  e.  (
Base `  K )
) )  ->  ( T  .<_  ( S  .\/  T )  ->  ( ( Q  .\/  R )  .\/  T )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) ) )
365, 28, 15, 11, 35syl13anc 1186 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( T  .<_  ( S  .\/  T
)  ->  ( ( Q  .\/  R )  .\/  T )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) ) )
3734, 36mpd 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  T )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) )
381, 2, 5, 30, 17, 21, 37, 26lattrd 14407 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  T )  .<_  W )
39383adant3 977 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( Q 
.\/  R )  .\/  T )  .<_  W )
40 simp11l 1068 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  K  e.  HL )
41 simp121 1089 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  Q  e.  A
)
42 simp122 1090 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  R  e.  A
)
43 simp132 1093 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  T  e.  A
)
44 simp123 1091 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  Q  =/=  R
)
45 simp23 992 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )
46 simpl3 962 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  /\  T  .<_  ( Q 
.\/  R ) )  ->  S  .<_  ( Q 
.\/  R ) )
47 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  /\  T  .<_  ( Q 
.\/  R ) )  ->  T  .<_  ( Q 
.\/  R ) )
481, 2, 8latjle12 14411 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( Q  .\/  R )  /\  T  .<_  ( Q 
.\/  R ) )  <-> 
( S  .\/  T
)  .<_  ( Q  .\/  R ) ) )
495, 32, 28, 11, 48syl13anc 1186 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( S  .<_  ( Q  .\/  R )  /\  T  .<_  ( Q  .\/  R ) )  <->  ( S  .\/  T )  .<_  ( Q  .\/  R ) ) )
50493adant3 977 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( S 
.<_  ( Q  .\/  R
)  /\  T  .<_  ( Q  .\/  R ) )  <->  ( S  .\/  T )  .<_  ( Q  .\/  R ) ) )
5150adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  /\  T  .<_  ( Q 
.\/  R ) )  ->  ( ( S 
.<_  ( Q  .\/  R
)  /\  T  .<_  ( Q  .\/  R ) )  <->  ( S  .\/  T )  .<_  ( Q  .\/  R ) ) )
5246, 47, 51mpbi2and 888 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  /\  T  .<_  ( Q 
.\/  R ) )  ->  ( S  .\/  T )  .<_  ( Q  .\/  R ) )
53 simpl3 962 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( S  e.  A  /\  T  e.  A  /\  S  =/= 
T ) )
542, 8, 9ps-1 29642 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
)  /\  ( Q  e.  A  /\  R  e.  A ) )  -> 
( ( S  .\/  T )  .<_  ( Q  .\/  R )  <->  ( S  .\/  T )  =  ( Q  .\/  R ) ) )
553, 53, 6, 7, 54syl112anc 1188 . . . . . . . . . . . . . . 15  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( S  .\/  T )  .<_  ( Q  .\/  R )  <-> 
( S  .\/  T
)  =  ( Q 
.\/  R ) ) )
56553adant3 977 . . . . . . . . . . . . . 14  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( S 
.\/  T )  .<_  ( Q  .\/  R )  <-> 
( S  .\/  T
)  =  ( Q 
.\/  R ) ) )
5756adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  /\  T  .<_  ( Q 
.\/  R ) )  ->  ( ( S 
.\/  T )  .<_  ( Q  .\/  R )  <-> 
( S  .\/  T
)  =  ( Q 
.\/  R ) ) )
5852, 57mpbid 202 . . . . . . . . . . . 12  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  /\  T  .<_  ( Q 
.\/  R ) )  ->  ( S  .\/  T )  =  ( Q 
.\/  R ) )
5958eqcomd 2385 . . . . . . . . . . 11  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  /\  T  .<_  ( Q 
.\/  R ) )  ->  ( Q  .\/  R )  =  ( S 
.\/  T ) )
6059ex 424 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( T  .<_  ( Q  .\/  R )  ->  ( Q  .\/  R )  =  ( S 
.\/  T ) ) )
6160necon3ad 2579 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( Q 
.\/  R )  =/=  ( S  .\/  T
)  ->  -.  T  .<_  ( Q  .\/  R
) ) )
6245, 61mpd 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  -.  T  .<_  ( Q  .\/  R ) )
632, 8, 9, 19lplni2 29702 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  T  e.  A
)  /\  ( Q  =/=  R  /\  -.  T  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  T )  e.  P )
6440, 41, 42, 43, 44, 62, 63syl132anc 1202 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( Q 
.\/  R )  .\/  T )  e.  P )
65 simp11r 1069 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  W  e.  P
)
662, 19lplncmp 29727 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( ( Q  .\/  R )  .\/  T )  e.  P  /\  W  e.  P )  ->  (
( ( Q  .\/  R )  .\/  T ) 
.<_  W  <->  ( ( Q 
.\/  R )  .\/  T )  =  W ) )
6740, 64, 65, 66syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( ( Q  .\/  R ) 
.\/  T )  .<_  W 
<->  ( ( Q  .\/  R )  .\/  T )  =  W ) )
6839, 67mpbid 202 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( Q 
.\/  R )  .\/  T )  =  W )
69373adant3 977 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  ( ( Q 
.\/  R )  .\/  T )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) )
7068, 69eqbrtrrd 4168 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  S  .<_  ( Q 
.\/  R ) )  ->  W  .<_  ( ( Q  .\/  R ) 
.\/  ( S  .\/  T ) ) )
71703expia 1155 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( S  .<_  ( Q  .\/  R
)  ->  W  .<_  ( ( Q  .\/  R
)  .\/  ( S  .\/  T ) ) ) )
721, 8latjcl 14399 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
735, 11, 32, 72syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
741, 2, 8latlej1 14409 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) )  ->  S  .<_  ( S  .\/  T
) )
755, 32, 28, 74syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  S  .<_  ( S  .\/  T ) )
761, 2, 8latjlej2 14415 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  ( Q  .\/  R )  e.  (
Base `  K )
) )  ->  ( S  .<_  ( S  .\/  T )  ->  ( ( Q  .\/  R )  .\/  S )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) ) )
775, 32, 15, 11, 76syl13anc 1186 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( S  .<_  ( S  .\/  T
)  ->  ( ( Q  .\/  R )  .\/  S )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) ) )
7875, 77mpd 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  S )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) )
791, 2, 5, 73, 17, 21, 78, 26lattrd 14407 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  S )  .<_  W )
80793adant3 977 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  ( ( Q  .\/  R )  .\/  S )  .<_  W )
81 simp11l 1068 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  K  e.  HL )
82 simp121 1089 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  Q  e.  A )
83 simp122 1090 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  R  e.  A )
84 simp131 1092 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  S  e.  A )
85 simp123 1091 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  Q  =/=  R )
86 simp3 959 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  -.  S  .<_  ( Q  .\/  R
) )
872, 8, 9, 19lplni2 29702 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  P )
8881, 82, 83, 84, 85, 86, 87syl132anc 1202 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  ( ( Q  .\/  R )  .\/  S )  e.  P )
89 simp11r 1069 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  W  e.  P )
902, 19lplncmp 29727 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( ( Q  .\/  R )  .\/  S )  e.  P  /\  W  e.  P )  ->  (
( ( Q  .\/  R )  .\/  S ) 
.<_  W  <->  ( ( Q 
.\/  R )  .\/  S )  =  W ) )
9181, 88, 89, 90syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  ( (
( Q  .\/  R
)  .\/  S )  .<_  W  <->  ( ( Q 
.\/  R )  .\/  S )  =  W ) )
9280, 91mpbid 202 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  ( ( Q  .\/  R )  .\/  S )  =  W )
93783adant3 977 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  ( ( Q  .\/  R )  .\/  S )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) )
9492, 93eqbrtrrd 4168 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) )  /\  -.  S  .<_  ( Q  .\/  R ) )  ->  W  .<_  ( ( Q  .\/  R
)  .\/  ( S  .\/  T ) ) )
95943expia 1155 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( -.  S  .<_  ( Q  .\/  R )  ->  W  .<_  ( ( Q  .\/  R
)  .\/  ( S  .\/  T ) ) ) )
9671, 95pm2.61d 152 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  W  .<_  ( ( Q  .\/  R
)  .\/  ( S  .\/  T ) ) )
971, 2, 5, 17, 21, 26, 96latasymd 14406 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  P )  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  =/=  R )  /\  ( S  e.  A  /\  T  e.  A  /\  S  =/=  T
) )  /\  (
( Q  .\/  R
)  .<_  W  /\  ( S  .\/  T )  .<_  W  /\  ( Q  .\/  R )  =/=  ( S 
.\/  T ) ) )  ->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  =  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   Latclat 14394   Atomscatm 29429   HLchlt 29516   LLinesclln 29656   LPlanesclpl 29657
This theorem is referenced by:  2llnjN  29732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664
  Copyright terms: Public domain W3C validator