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Theorem 2llnjaN 30300
Description: The join of two different lattice lines in a lattice plane equals the plane (version of 2llnjN 30301 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 2435 . 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 30098 . . 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 30101 . . . 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 30101 . . . 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 14471 . . 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 30268 . . 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 14483 . . . 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 30024 . . . . . . . . . 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 14471 . . . . . . . . 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 30024 . . . . . . . . . . 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 14482 . . . . . . . . . 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 14487 . . . . . . . . . 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 14479 . . . . . . 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 14483 . . . . . . . . . . . . . . . . 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 30211 . . . . . . . . . . . . . . . 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 2440 . . . . . . . . . . 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 2634 . . . . . . . . 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 30271 . . . . . . . 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 30296 . . . . . . 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 4226 . . . 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 14471 . . . . . . . . 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 14481 . . . . . . . . . 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 14487 . . . . . . . . . 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 14479 . . . . . . 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 30271 . . . . . . . 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 30296 . . . . . . 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 4226 . . . 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 14478 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29998   HLchlt 30085   LLinesclln 30225   LPlanesclpl 30226
This theorem is referenced by:  2llnjN  30301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233
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