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Theorem 2llnjaN 30377
Description: The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 30378 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 2296 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 2llnja.l . 2  |-  .<_  =  ( le `  K )
3 simpl1l 1006 . . 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 30175 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 15 . 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 1033 . . . 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 1034 . . . 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 30178 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 1182 . . 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 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 ) ) )  ->  S  e.  A )
13 simpl32 1037 . . . 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 30178 . . . 4  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
153, 12, 13, 14syl3anc 1182 . . 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 14172 . . 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 1182 . 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 1007 . . 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 30345 . . 3  |-  ( W  e.  P  ->  W  e.  ( Base `  K
) )
2118, 20syl 15 . 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 961 . . 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 962 . . 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 14184 . . . 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 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
)  .<_  W  /\  ( S  .\/  T )  .<_  W )  <->  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) )  .<_  W )
)
2622, 23, 25mpbi2and 887 . 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 30101 . . . . . . . . . 10  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
2813, 27syl 15 . . . . . . . . 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 14172 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  T )  e.  ( Base `  K ) )
305, 11, 28, 29syl3anc 1182 . . . . . . . 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 30101 . . . . . . . . . . 11  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
3212, 31syl 15 . . . . . . . . . 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 14183 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) )  ->  T  .<_  ( S  .\/  T
) )
345, 32, 28, 33syl3anc 1182 . . . . . . . . 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 14188 . . . . . . . . . 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 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
)  ->  ( ( Q  .\/  R )  .\/  T )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) ) )
3734, 36mpd 14 . . . . . . . 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 14180 . . . . . . 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 975 . . . . . 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 1066 . . . . . . 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 1087 . . . . . . . 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 1088 . . . . . . . 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 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 ) )  ->  T  e.  A
)
44 simp123 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  =/=  R
)
45 simp23 990 . . . . . . . . 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 960 . . . . . . . . . . . . . 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 447 . . . . . . . . . . . . . 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 14184 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . 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 975 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 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 887 . . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . 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 30288 . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . 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 975 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 201 . . . . . . . . . . . 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 2301 . . . . . . . . . . 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 423 . . . . . . . . . 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 2495 . . . . . . . . 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 14 . . . . . . . 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 30348 . . . . . . . 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 1200 . . . . . . 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 1067 . . . . . . 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 30373 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( ( Q  .\/  R )  .\/  T )  e.  P  /\  W  e.  P )  ->  (
( ( Q  .\/  R )  .\/  T ) 
.<_  W  <->  ( ( Q 
.\/  R )  .\/  T )  =  W ) )
6740, 64, 65, 66syl3anc 1182 . . . . . 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 201 . . . . 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 975 . . . . 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 4061 . . . 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 1153 . . 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 14172 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
735, 11, 32, 72syl3anc 1182 . . . . . . . 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 14182 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) )  ->  S  .<_  ( S  .\/  T
) )
755, 32, 28, 74syl3anc 1182 . . . . . . . . 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 14188 . . . . . . . . . 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 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
)  ->  ( ( Q  .\/  R )  .\/  S )  .<_  ( ( Q  .\/  R )  .\/  ( S  .\/  T ) ) ) )
7875, 77mpd 14 . . . . . . . 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 14180 . . . . . . 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 975 . . . . . 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 1066 . . . . . . 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 1087 . . . . . . . 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 1088 . . . . . . . 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 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 ) )  ->  S  e.  A )
85 simp123 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  =/=  R )
86 simp3 957 . . . . . . . 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 30348 . . . . . . . 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 1200 . . . . . . 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 1067 . . . . . . 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 30373 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( ( Q  .\/  R )  .\/  S )  e.  P  /\  W  e.  P )  ->  (
( ( Q  .\/  R )  .\/  S ) 
.<_  W  <->  ( ( Q 
.\/  R )  .\/  S )  =  W ) )
9181, 88, 89, 90syl3anc 1182 . . . . . 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 201 . . . . 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 975 . . . . 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 4061 . . . 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 1153 . . 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 150 . 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 14179 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162   LLinesclln 30302   LPlanesclpl 30303
This theorem is referenced by:  2llnjN  30378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310
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