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Theorem 4atlem12b 30093
Description: Lemma for 4at 30095. Substitute  T for  P (cont.). (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
4at.l  |-  .<_  =  ( le `  K )
4at.j  |-  .\/  =  ( join `  K )
4at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
4atlem12b  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )

Proof of Theorem 4atlem12b
StepHypRef Expression
1 simp11 987 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
2 simp121 1089 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  R  e.  A
)
3 simp122 1090 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  S  e.  A
)
42, 3jca 519 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( R  e.  A  /\  S  e.  A ) )
5 simp13 989 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )
61, 4, 53jca 1134 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) ) )
7 simp2l 983 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
)  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )
86, 7jca 519 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A
)  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) ) ) )
9 simp3lr 1029 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  Q  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )
10 simp3rl 1030 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  R  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )
11 simp3rr 1031 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  S  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )
12 simp111 1086 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  K  e.  HL )
13 hllat 29846 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1412, 13syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  K  e.  Lat )
15 eqid 2404 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
16 4at.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1715, 16atbase 29772 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
182, 17syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  R  e.  (
Base `  K )
)
1915, 16atbase 29772 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
203, 19syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  S  e.  (
Base `  K )
)
21 simp123 1091 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  T  e.  A
)
22 simp131 1092 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  U  e.  A
)
23 4at.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
2415, 23, 16hlatjcl 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
2512, 21, 22, 24syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( T  .\/  U )  e.  ( Base `  K ) )
26 simp132 1093 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  V  e.  A
)
27 simp133 1094 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  W  e.  A
)
2815, 23, 16hlatjcl 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  V  e.  A  /\  W  e.  A )  ->  ( V  .\/  W
)  e.  ( Base `  K ) )
2912, 26, 27, 28syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( V  .\/  W )  e.  ( Base `  K ) )
3015, 23latjcl 14434 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( T  .\/  U )  e.  ( Base `  K
)  /\  ( V  .\/  W )  e.  (
Base `  K )
)  ->  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  e.  ( Base `  K ) )
3114, 25, 29, 30syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  e.  ( Base `  K ) )
32 4at.l . . . . . . . 8  |-  .<_  =  ( le `  K )
3315, 32, 23latjle12 14446 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  (
( T  .\/  U
)  .\/  ( V  .\/  W ) )  e.  ( Base `  K
) ) )  -> 
( ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) )  <->  ( R  .\/  S )  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) ) )
3414, 18, 20, 31, 33syl13anc 1186 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( R 
.<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  /\  S  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )  <->  ( R  .\/  S )  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) ) )
3510, 11, 34mpbi2and 888 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( R  .\/  S )  .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) ) )
36 simp113 1088 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  Q  e.  A
)
3715, 16atbase 29772 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3836, 37syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  Q  e.  (
Base `  K )
)
3915, 23, 16hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
4012, 2, 3, 39syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( R  .\/  S )  e.  ( Base `  K ) )
4115, 32, 23latjle12 14446 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  ( ( T  .\/  U )  .\/  ( V  .\/  W ) )  e.  ( Base `  K ) ) )  ->  ( ( Q 
.<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  /\  ( R  .\/  S )  .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) ) )  <->  ( Q  .\/  ( R  .\/  S
) )  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) ) )
4214, 38, 40, 31, 41syl13anc 1186 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( Q 
.<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  /\  ( R  .\/  S )  .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) ) )  <->  ( Q  .\/  ( R  .\/  S
) )  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) ) )
439, 35, 42mpbi2and 888 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( Q  .\/  ( R  .\/  S ) )  .<_  ( ( T  .\/  U )  .\/  ( V  .\/  W ) ) )
44 simp3ll 1028 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  P  .<_  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )
45 simp112 1087 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  P  e.  A
)
46 simp2r 984 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)
4732, 23, 164atlem12a 30092 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
)  /\  -.  P  .<_  ( ( U  .\/  V )  .\/  W ) )  ->  ( P  .<_  ( ( T  .\/  U )  .\/  ( V 
.\/  W ) )  <-> 
( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  =  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) )
4812, 45, 21, 5, 46, 47syl311anc 1198 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( P  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  <->  ( ( P  .\/  U )  .\/  ( V  .\/  W ) )  =  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) ) )
4944, 48mpbid 202 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( P 
.\/  U )  .\/  ( V  .\/  W ) )  =  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )
5043, 49breqtrrd 4198 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( Q  .\/  ( R  .\/  S ) )  .<_  ( ( P  .\/  U )  .\/  ( V  .\/  W ) ) )
5132, 23, 164atlem11 30091 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q )  .\/  R ) ) )  ->  (
( Q  .\/  ( R  .\/  S ) ) 
.<_  ( ( P  .\/  U )  .\/  ( V 
.\/  W ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  U ) 
.\/  ( V  .\/  W ) ) ) )
528, 50, 51sylc 58 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  U ) 
.\/  ( V  .\/  W ) ) )
5352, 49eqtrd 2436 1  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( U  e.  A  /\  V  e.  A  /\  W  e.  A
) )  /\  (
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q )  /\  -.  S  .<_  ( ( P  .\/  Q ) 
.\/  R ) )  /\  -.  P  .<_  ( ( U  .\/  V
)  .\/  W )
)  /\  ( ( P  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) )  /\  Q  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) ) )  /\  ( R  .<_  ( ( T  .\/  U
)  .\/  ( V  .\/  W ) )  /\  S  .<_  ( ( T 
.\/  U )  .\/  ( V  .\/  W ) ) ) ) )  ->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  =  ( ( T  .\/  U ) 
.\/  ( V  .\/  W ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   Latclat 14429   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  4atlem12  30094
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982
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