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Theorem 3atlem2 32757
Description: Lemma for 3at 32763. (Contributed by NM, 22-Jun-2012.)
Hypotheses
Ref Expression
3at.l  |-  .<_  =  ( le `  K )
3at.j  |-  .\/  =  ( join `  K )
3at.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
3atlem2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )

Proof of Theorem 3atlem2
StepHypRef Expression
1 simp3 1007 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )
2 simp11 1035 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  K  e.  HL )
3 hllat 32637 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  K  e.  Lat )
5 simp121 1137 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  P  e.  A )
6 simp122 1138 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  e.  A )
7 eqid 2429 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
8 3at.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
9 3at.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 32640 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1264 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
12 simp123 1139 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  R  e.  A )
137, 9atbase 32563 . . . . . . . 8  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1412, 13syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  R  e.  ( Base `  K )
)
15 simp131 1140 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  S  e.  A )
16 simp132 1141 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  T  e.  A )
177, 8, 9hlatjcl 32640 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
182, 15, 16, 17syl3anc 1264 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( S  .\/  T )  e.  (
Base `  K )
)
19 simp133 1142 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  U  e.  A )
207, 9atbase 32563 . . . . . . . . 9  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2119, 20syl 17 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  U  e.  ( Base `  K )
)
227, 8latjcl 16248 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( S  .\/  T )  .\/  U )  e.  ( Base `  K ) )
234, 18, 21, 22syl3anc 1264 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( S  .\/  T )  .\/  U )  e.  ( Base `  K ) )
24 3at.l . . . . . . . 8  |-  .<_  =  ( le `  K )
257, 24, 8latjle12 16259 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  R  e.  ( Base `  K
)  /\  ( ( S  .\/  T )  .\/  U )  e.  ( Base `  K ) ) )  ->  ( ( ( P  .\/  Q ) 
.<_  ( ( S  .\/  T )  .\/  U )  /\  R  .<_  ( ( S  .\/  T ) 
.\/  U ) )  <-> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U ) ) )
264, 11, 14, 23, 25syl13anc 1266 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( (
( P  .\/  Q
)  .<_  ( ( S 
.\/  T )  .\/  U )  /\  R  .<_  ( ( S  .\/  T
)  .\/  U )
)  <->  ( ( P 
.\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) ) )
271, 26mpbird 235 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .<_  ( ( S  .\/  T )  .\/  U )  /\  R  .<_  ( ( S  .\/  T ) 
.\/  U ) ) )
2827simprd 464 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  R  .<_  ( ( S  .\/  T
)  .\/  U )
)
298, 9hlatjass 32643 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A
) )  ->  (
( S  .\/  T
)  .\/  U )  =  ( S  .\/  ( T  .\/  U ) ) )
302, 15, 16, 19, 29syl13anc 1266 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( S  .\/  T )  .\/  U )  =  ( S 
.\/  ( T  .\/  U ) ) )
31 simp22r 1125 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  P  .<_  ( T  .\/  U ) )
32 simp22l 1124 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  P  =/=  U )
3324, 8, 9hlatexchb2 32667 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  T  e.  A  /\  U  e.  A
)  /\  P  =/=  U )  ->  ( P  .<_  ( T  .\/  U
)  <->  ( P  .\/  U )  =  ( T 
.\/  U ) ) )
342, 5, 16, 19, 32, 33syl131anc 1277 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  .<_  ( T  .\/  U
)  <->  ( P  .\/  U )  =  ( T 
.\/  U ) ) )
3531, 34mpbid 213 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  .\/  U )  =  ( T  .\/  U ) )
3635oveq2d 6321 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( S  .\/  ( P  .\/  U
) )  =  ( S  .\/  ( T 
.\/  U ) ) )
3730, 36eqtr4d 2473 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( S  .\/  T )  .\/  U )  =  ( S 
.\/  ( P  .\/  U ) ) )
388, 9hlatjass 32643 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  U )  =  ( P  .\/  ( Q  .\/  U ) ) )
392, 5, 6, 19, 38syl13anc 1266 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  U )  =  ( P 
.\/  ( Q  .\/  U ) ) )
408, 9hlatj12 32644 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  U  e.  A
) )  ->  ( P  .\/  ( Q  .\/  U ) )  =  ( Q  .\/  ( P 
.\/  U ) ) )
412, 5, 6, 19, 40syl13anc 1266 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  .\/  ( Q  .\/  U
) )  =  ( Q  .\/  ( P 
.\/  U ) ) )
428, 9hlatj32 32645 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( P 
.\/  R )  .\/  Q ) )
432, 5, 6, 12, 42syl13anc 1266 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( P  .\/  R ) 
.\/  Q ) )
441, 43, 303brtr3d 4455 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  R )  .\/  Q )  .<_  ( S  .\/  ( T  .\/  U
) ) )
457, 8, 9hlatjcl 32640 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
462, 5, 12, 45syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  .\/  R )  e.  (
Base `  K )
)
477, 9atbase 32563 . . . . . . . . . . . 12  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
486, 47syl 17 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  e.  ( Base `  K )
)
497, 9atbase 32563 . . . . . . . . . . . . 13  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
5015, 49syl 17 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  S  e.  ( Base `  K )
)
517, 8, 9hlatjcl 32640 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
522, 16, 19, 51syl3anc 1264 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( T  .\/  U )  e.  (
Base `  K )
)
537, 8latjcl 16248 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
) )  ->  ( S  .\/  ( T  .\/  U ) )  e.  (
Base `  K )
)
544, 50, 52, 53syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( S  .\/  ( T  .\/  U
) )  e.  (
Base `  K )
)
557, 24, 8latjle12 16259 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  R )  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
)  /\  ( S  .\/  ( T  .\/  U
) )  e.  (
Base `  K )
) )  ->  (
( ( P  .\/  R )  .<_  ( S  .\/  ( T  .\/  U
) )  /\  Q  .<_  ( S  .\/  ( T  .\/  U ) ) )  <->  ( ( P 
.\/  R )  .\/  Q )  .<_  ( S  .\/  ( T  .\/  U
) ) ) )
564, 46, 48, 54, 55syl13anc 1266 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( (
( P  .\/  R
)  .<_  ( S  .\/  ( T  .\/  U ) )  /\  Q  .<_  ( S  .\/  ( T 
.\/  U ) ) )  <->  ( ( P 
.\/  R )  .\/  Q )  .<_  ( S  .\/  ( T  .\/  U
) ) ) )
5744, 56mpbird 235 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  R )  .<_  ( S  .\/  ( T 
.\/  U ) )  /\  Q  .<_  ( S 
.\/  ( T  .\/  U ) ) ) )
5857simprd 464 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  .<_  ( S  .\/  ( T 
.\/  U ) ) )
5958, 36breqtrrd 4452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  Q  .<_  ( S  .\/  ( P 
.\/  U ) ) )
607, 8, 9hlatjcl 32640 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
612, 5, 19, 60syl3anc 1264 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( P  .\/  U )  e.  (
Base `  K )
)
62 simp23 1040 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  -.  Q  .<_  ( P  .\/  U
) )
637, 24, 8, 9hlexchb2 32658 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  S  e.  A  /\  ( P  .\/  U
)  e.  ( Base `  K ) )  /\  -.  Q  .<_  ( P 
.\/  U ) )  ->  ( Q  .<_  ( S  .\/  ( P 
.\/  U ) )  <-> 
( Q  .\/  ( P  .\/  U ) )  =  ( S  .\/  ( P  .\/  U ) ) ) )
642, 6, 15, 61, 62, 63syl131anc 1277 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( Q  .<_  ( S  .\/  ( P  .\/  U ) )  <-> 
( Q  .\/  ( P  .\/  U ) )  =  ( S  .\/  ( P  .\/  U ) ) ) )
6559, 64mpbid 213 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( Q  .\/  ( P  .\/  U
) )  =  ( S  .\/  ( P 
.\/  U ) ) )
6639, 41, 653eqtrd 2474 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  U )  =  ( S 
.\/  ( P  .\/  U ) ) )
6737, 66eqtr4d 2473 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( S  .\/  T )  .\/  U )  =  ( ( P  .\/  Q ) 
.\/  U ) )
6828, 67breqtrd 4450 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  R  .<_  ( ( P  .\/  Q
)  .\/  U )
)
69 simp21 1038 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  -.  R  .<_  ( P  .\/  Q
) )
707, 24, 8, 9hlexchb1 32657 . . . 4  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  U  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K ) )  /\  -.  R  .<_  ( P 
.\/  Q ) )  ->  ( R  .<_  ( ( P  .\/  Q
)  .\/  U )  <->  ( ( P  .\/  Q
)  .\/  R )  =  ( ( P 
.\/  Q )  .\/  U ) ) )
712, 12, 19, 11, 69, 70syl131anc 1277 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( R  .<_  ( ( P  .\/  Q )  .\/  U )  <-> 
( ( P  .\/  Q )  .\/  R )  =  ( ( P 
.\/  Q )  .\/  U ) ) )
7268, 71mpbid 213 . 2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( P  .\/  Q ) 
.\/  U ) )
7372, 67eqtr4d 2473 1  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  ( P  =/= 
U  /\  P  .<_  ( T  .\/  U ) )  /\  -.  Q  .<_  ( P  .\/  U
) )  /\  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  ( ( P  .\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   Latclat 16242   Atomscatm 32537   HLchlt 32624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-covers 32540  df-ats 32541  df-atl 32572  df-cvlat 32596  df-hlat 32625
This theorem is referenced by:  3atlem3  32758
  Copyright terms: Public domain W3C validator