Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme22e Unicode version

Theorem cdleme22e 30533
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 4th line on p. 115.  F,  N,  O represent f(z), fz(s), fz(t) respectively. When t  \/ v = p  \/ q, fz(s)  <_ fz(t)  \/ v. (Contributed by NM, 6-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l  |-  .<_  =  ( le `  K )
cdleme22.j  |-  .\/  =  ( join `  K )
cdleme22.m  |-  ./\  =  ( meet `  K )
cdleme22.a  |-  A  =  ( Atoms `  K )
cdleme22.h  |-  H  =  ( LHyp `  K
)
cdleme22e.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme22e.f  |-  F  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
cdleme22e.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  z )  ./\  W
) ) )
cdleme22e.o  |-  O  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( T  .\/  z )  ./\  W
) ) )
Assertion
Ref Expression
cdleme22e  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  N  .<_  ( O  .\/  V
) )

Proof of Theorem cdleme22e
StepHypRef Expression
1 cdleme22e.n . . 3  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  z )  ./\  W
) ) )
2 simp1l 979 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  K  e.  HL )
3 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  K  e.  Lat )
5 simp21l 1072 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  P  e.  A )
6 simp22l 1074 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  Q  e.  A )
7 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme22.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdleme22.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 29556 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
12 simp1r 980 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  W  e.  H )
13 simp33l 1082 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  z  e.  A )
14 cdleme22.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 cdleme22.m . . . . . . 7  |-  ./\  =  ( meet `  K )
16 cdleme22.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
17 cdleme22e.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme22e.f . . . . . . 7  |-  F  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 30415 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  z  e.  A ) )  ->  F  e.  ( Base `  K ) )
202, 12, 5, 6, 13, 19syl23anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  F  e.  ( Base `  K
) )
21 simp23l 1076 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  S  e.  A )
227, 8, 9hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  z  e.  A )  ->  ( S  .\/  z
)  e.  ( Base `  K ) )
232, 21, 13, 22syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( S  .\/  z )  e.  ( Base `  K
) )
247, 16lhpbase 30187 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2512, 24syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  W  e.  ( Base `  K
) )
267, 15latmcl 14157 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  .\/  z )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( S  .\/  z )  ./\  W )  e.  ( Base `  K ) )
274, 23, 25, 26syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( S  .\/  z
)  ./\  W )  e.  ( Base `  K
) )
287, 8latjcl 14156 . . . . 5  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  (
( S  .\/  z
)  ./\  W )  e.  ( Base `  K
) )  ->  ( F  .\/  ( ( S 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)
294, 20, 27, 28syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( F  .\/  ( ( S 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)
307, 14, 15latmle1 14182 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( F  .\/  ( ( S  .\/  z )  ./\  W
) )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  z ) 
./\  W ) ) )  .<_  ( P  .\/  Q ) )
314, 11, 29, 30syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( S  .\/  z )  ./\  W
) ) )  .<_  ( P  .\/  Q ) )
321, 31syl5eqbr 4056 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  N  .<_  ( P  .\/  Q
) )
33 simp1 955 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
34 simp21 988 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
35 simp23r 1077 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  T  e.  A )
36 simp31 991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( V  e.  A  /\  V  .<_  W ) )
37 simp32l 1080 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  P  =/=  Q )
38 simp32r 1081 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( T  .\/  V )  =  ( P  .\/  Q
) )
3914, 8, 15, 9, 16, 17cdleme22a 30529 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  T  e.  A )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  P  =/= 
Q  /\  ( T  .\/  V )  =  ( P  .\/  Q ) ) )  ->  V  =  U )
4033, 34, 6, 35, 36, 37, 38, 39syl133anc 1205 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  V  =  U )
4140oveq2d 5874 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( O  .\/  V )  =  ( O  .\/  U
) )
42 cdleme22e.o . . . . . 6  |-  O  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( T  .\/  z )  ./\  W
) ) )
4342oveq1i 5868 . . . . 5  |-  ( O 
.\/  U )  =  ( ( ( P 
.\/  Q )  ./\  ( F  .\/  ( ( T  .\/  z ) 
./\  W ) ) )  .\/  U )
44 simp21r 1073 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  -.  P  .<_  W )
4514, 8, 15, 9, 16, 17cdleme0a 30400 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
462, 12, 5, 44, 6, 37, 45syl222anc 1198 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  U  e.  A )
477, 8, 9hlatjcl 29556 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  T  e.  A  /\  z  e.  A )  ->  ( T  .\/  z
)  e.  ( Base `  K ) )
482, 35, 13, 47syl3anc 1182 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( T  .\/  z )  e.  ( Base `  K
) )
497, 15latmcl 14157 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( T  .\/  z )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( T  .\/  z )  ./\  W )  e.  ( Base `  K ) )
504, 48, 25, 49syl3anc 1182 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( T  .\/  z
)  ./\  W )  e.  ( Base `  K
) )
517, 8latjcl 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  (
( T  .\/  z
)  ./\  W )  e.  ( Base `  K
) )  ->  ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)
524, 20, 50, 51syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)
5314, 8, 15, 9, 16, 17cdlemeulpq 30409 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
542, 12, 5, 6, 53syl22anc 1183 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  U  .<_  ( P  .\/  Q
) )
557, 14, 8, 15, 9atmod2i1 30050 . . . . . 6  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)  /\  U  .<_  ( P  .\/  Q ) )  ->  ( (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( T  .\/  z )  ./\  W
) ) )  .\/  U )  =  ( ( P  .\/  Q ) 
./\  ( ( F 
.\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) ) )
562, 46, 11, 52, 54, 55syl131anc 1195 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( ( P  .\/  Q )  ./\  ( F  .\/  ( ( T  .\/  z )  ./\  W
) ) )  .\/  U )  =  ( ( P  .\/  Q ) 
./\  ( ( F 
.\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) ) )
5743, 56syl5req 2328 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  ( ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) )  =  ( O  .\/  U ) )
5841, 57eqtr4d 2318 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( O  .\/  V )  =  ( ( P  .\/  Q )  ./\  ( ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) ) )
5940oveq2d 5874 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( T  .\/  V )  =  ( T  .\/  U
) )
6038, 59eqtr3d 2317 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  Q )  =  ( T  .\/  U
) )
617, 8, 9hlatjcl 29556 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
622, 35, 46, 61syl3anc 1182 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( T  .\/  U )  e.  ( Base `  K
) )
637, 9atbase 29479 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  ( Base `  K
) )
6413, 63syl 15 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  z  e.  ( Base `  K
) )
657, 14, 8latlej1 14166 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( T  .\/  U )  e.  ( Base `  K
)  /\  z  e.  ( Base `  K )
)  ->  ( T  .\/  U )  .<_  ( ( T  .\/  U ) 
.\/  z ) )
664, 62, 64, 65syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( T  .\/  U )  .<_  ( ( T  .\/  U )  .\/  z ) )
678, 9hlatj32 29561 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  U  e.  A  /\  z  e.  A
) )  ->  (
( T  .\/  U
)  .\/  z )  =  ( ( T 
.\/  z )  .\/  U ) )
682, 35, 46, 13, 67syl13anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( T  .\/  U
)  .\/  z )  =  ( ( T 
.\/  z )  .\/  U ) )
697, 9atbase 29479 . . . . . . . . . 10  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
7046, 69syl 15 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  U  e.  ( Base `  K
) )
717, 8latj32 14203 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( z  e.  (
Base `  K )  /\  U  e.  ( Base `  K )  /\  ( ( T  .\/  z )  ./\  W
)  e.  ( Base `  K ) ) )  ->  ( ( z 
.\/  U )  .\/  ( ( T  .\/  z )  ./\  W
) )  =  ( ( z  .\/  (
( T  .\/  z
)  ./\  W )
)  .\/  U )
)
724, 64, 70, 50, 71syl13anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( z  .\/  U
)  .\/  ( ( T  .\/  z )  ./\  W ) )  =  ( ( z  .\/  (
( T  .\/  z
)  ./\  W )
)  .\/  U )
)
737, 8latj32 14203 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( F  e.  ( Base `  K )  /\  ( ( T  .\/  z )  ./\  W
)  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) ) )  -> 
( ( F  .\/  ( ( T  .\/  z )  ./\  W
) )  .\/  U
)  =  ( ( F  .\/  U ) 
.\/  ( ( T 
.\/  z )  ./\  W ) ) )
744, 20, 50, 70, 73syl13anc 1184 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( F  .\/  (
( T  .\/  z
)  ./\  W )
)  .\/  U )  =  ( ( F 
.\/  U )  .\/  ( ( T  .\/  z )  ./\  W
) ) )
757, 8, 9hlatjcl 29556 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  HL  /\  P  e.  A  /\  z  e.  A )  ->  ( P  .\/  z
)  e.  ( Base `  K ) )
762, 5, 13, 75syl3anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  z )  e.  ( Base `  K
) )
7714, 8, 9hlatlej1 29564 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  HL  /\  P  e.  A  /\  z  e.  A )  ->  P  .<_  ( P  .\/  z ) )
782, 5, 13, 77syl3anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  P  .<_  ( P  .\/  z
) )
797, 14, 8, 15, 9atmod3i1 30053 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  z
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  z
) )  ->  ( P  .\/  ( ( P 
.\/  z )  ./\  W ) )  =  ( ( P  .\/  z
)  ./\  ( P  .\/  W ) ) )
802, 5, 76, 25, 78, 79syl131anc 1195 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  ( ( P 
.\/  z )  ./\  W ) )  =  ( ( P  .\/  z
)  ./\  ( P  .\/  W ) ) )
81 eqid 2283 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1.
`  K )  =  ( 1. `  K
)
8214, 8, 81, 9, 16lhpjat2 30210 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
832, 12, 34, 82syl21anc 1181 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  W )  =  ( 1. `  K
) )
8483oveq2d 5874 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  z
)  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  z )  ./\  ( 1. `  K ) ) )
85 hlol 29551 . . . . . . . . . . . . . . . . . . 19  |-  ( K  e.  HL  ->  K  e.  OL )
862, 85syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  K  e.  OL )
877, 15, 81olm11 29417 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  OL  /\  ( P  .\/  z )  e.  ( Base `  K
) )  ->  (
( P  .\/  z
)  ./\  ( 1. `  K ) )  =  ( P  .\/  z
) )
8886, 76, 87syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  z
)  ./\  ( 1. `  K ) )  =  ( P  .\/  z
) )
8980, 84, 883eqtrd 2319 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  ( ( P 
.\/  z )  ./\  W ) )  =  ( P  .\/  z ) )
9089oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  (
( P  .\/  z
)  ./\  W )
)  .\/  Q )  =  ( ( P 
.\/  z )  .\/  Q ) )
9117oveq2i 5869 . . . . . . . . . . . . . . . . . . 19  |-  ( Q 
.\/  U )  =  ( Q  .\/  (
( P  .\/  Q
)  ./\  W )
)
9214, 8, 9hlatlej2 29565 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( P  .\/  Q ) )
932, 5, 6, 92syl3anc 1182 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  Q  .<_  ( P  .\/  Q
) )
947, 14, 8, 15, 9atmod3i1 30053 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  Q  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  ( Q  .\/  W ) ) )
952, 6, 11, 25, 93, 94syl131anc 1195 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( Q  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  ( Q  .\/  W ) ) )
9691, 95syl5eq 2327 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( ( P  .\/  Q )  ./\  ( Q  .\/  W ) ) )
97 simp22 989 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
9814, 8, 81, 9, 16lhpjat2 30210 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  .\/  W
)  =  ( 1.
`  K ) )
992, 12, 97, 98syl21anc 1181 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( Q  .\/  W )  =  ( 1. `  K
) )
10099oveq2d 5874 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  ( Q  .\/  W ) )  =  ( ( P  .\/  Q )  ./\  ( 1. `  K ) ) )
1017, 15, 81olm11 29417 . . . . . . . . . . . . . . . . . . 19  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
10286, 11, 101syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
10396, 100, 1023eqtrd 2319 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q
) )
104103oveq1d 5873 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( Q  .\/  U
)  .\/  ( ( P  .\/  z )  ./\  W ) )  =  ( ( P  .\/  Q
)  .\/  ( ( P  .\/  z )  ./\  W ) ) )
1057, 9atbase 29479 . . . . . . . . . . . . . . . . . 18  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1065, 105syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  P  e.  ( Base `  K
) )
1077, 15latmcl 14157 . . . . . . . . . . . . . . . . . 18  |-  ( ( K  e.  Lat  /\  ( P  .\/  z )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  z )  ./\  W )  e.  ( Base `  K ) )
1084, 76, 25, 107syl3anc 1182 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  z
)  ./\  W )  e.  ( Base `  K
) )
1097, 9atbase 29479 . . . . . . . . . . . . . . . . . 18  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1106, 109syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  Q  e.  ( Base `  K
) )
1117, 8latj32 14203 . . . . . . . . . . . . . . . . 17  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( ( P  .\/  z )  ./\  W
)  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) ) )  -> 
( ( P  .\/  ( ( P  .\/  z )  ./\  W
) )  .\/  Q
)  =  ( ( P  .\/  Q ) 
.\/  ( ( P 
.\/  z )  ./\  W ) ) )
1124, 106, 108, 110, 111syl13anc 1184 . . . . . . . . . . . . . . . 16  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  (
( P  .\/  z
)  ./\  W )
)  .\/  Q )  =  ( ( P 
.\/  Q )  .\/  ( ( P  .\/  z )  ./\  W
) ) )
113104, 112eqtr4d 2318 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( Q  .\/  U
)  .\/  ( ( P  .\/  z )  ./\  W ) )  =  ( ( P  .\/  (
( P  .\/  z
)  ./\  W )
)  .\/  Q )
)
1148, 9hlatj32 29561 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  z  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  z )  =  ( ( P 
.\/  z )  .\/  Q ) )
1152, 5, 6, 13, 114syl13anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  .\/  z )  =  ( ( P 
.\/  z )  .\/  Q ) )
11690, 113, 1153eqtr4rd 2326 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  .\/  z )  =  ( ( Q 
.\/  U )  .\/  ( ( P  .\/  z )  ./\  W
) ) )
1177, 8latj32 14203 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  (
( P  .\/  z
)  ./\  W )  e.  ( Base `  K
) ) )  -> 
( ( Q  .\/  U )  .\/  ( ( P  .\/  z ) 
./\  W ) )  =  ( ( Q 
.\/  ( ( P 
.\/  z )  ./\  W ) )  .\/  U
) )
1184, 110, 70, 108, 117syl13anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( Q  .\/  U
)  .\/  ( ( P  .\/  z )  ./\  W ) )  =  ( ( Q  .\/  (
( P  .\/  z
)  ./\  W )
)  .\/  U )
)
119116, 118eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  .\/  z )  =  ( ( Q 
.\/  ( ( P 
.\/  z )  ./\  W ) )  .\/  U
) )
120119oveq2d 5874 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( z  .\/  U
)  ./\  ( ( P  .\/  Q )  .\/  z ) )  =  ( ( z  .\/  U )  ./\  ( ( Q  .\/  ( ( P 
.\/  z )  ./\  W ) )  .\/  U
) ) )
1217, 8latjcl 14156 . . . . . . . . . . . . . 14  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  z  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  z )  e.  (
Base `  K )
)
1224, 11, 64, 121syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  .\/  z )  e.  ( Base `  K
) )
1237, 14, 8latlej2 14167 . . . . . . . . . . . . . 14  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  z  e.  ( Base `  K )
)  ->  z  .<_  ( ( P  .\/  Q
)  .\/  z )
)
1244, 11, 64, 123syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  z  .<_  ( ( P  .\/  Q )  .\/  z ) )
1257, 14, 8, 15, 9atmod1i1 30046 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( z  e.  A  /\  U  e.  ( Base `  K )  /\  ( ( P  .\/  Q )  .\/  z )  e.  ( Base `  K
) )  /\  z  .<_  ( ( P  .\/  Q )  .\/  z ) )  ->  ( z  .\/  ( U  ./\  (
( P  .\/  Q
)  .\/  z )
) )  =  ( ( z  .\/  U
)  ./\  ( ( P  .\/  Q )  .\/  z ) ) )
1262, 13, 70, 122, 124, 125syl131anc 1195 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
z  .\/  ( U  ./\  ( ( P  .\/  Q )  .\/  z ) ) )  =  ( ( z  .\/  U
)  ./\  ( ( P  .\/  Q )  .\/  z ) ) )
12718oveq1i 5868 . . . . . . . . . . . . 13  |-  ( F 
.\/  U )  =  ( ( ( z 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z ) 
./\  W ) ) )  .\/  U )
1287, 8, 9hlatjcl 29556 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  z  e.  A  /\  U  e.  A )  ->  ( z  .\/  U
)  e.  ( Base `  K ) )
1292, 13, 46, 128syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
z  .\/  U )  e.  ( Base `  K
) )
1307, 8latjcl 14156 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  z
)  ./\  W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( P 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)
1314, 110, 108, 130syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( Q  .\/  ( ( P 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)
13214, 8, 9hlatlej2 29565 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  z  e.  A  /\  U  e.  A )  ->  U  .<_  ( z  .\/  U ) )
1332, 13, 46, 132syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  U  .<_  ( z  .\/  U
) )
1347, 14, 8, 15, 9atmod2i1 30050 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  ( z  .\/  U
)  e.  ( Base `  K )  /\  ( Q  .\/  ( ( P 
.\/  z )  ./\  W ) )  e.  (
Base `  K )
)  /\  U  .<_  ( z  .\/  U ) )  ->  ( (
( z  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  U )  =  ( ( z  .\/  U ) 
./\  ( ( Q 
.\/  ( ( P 
.\/  z )  ./\  W ) )  .\/  U
) ) )
1352, 46, 129, 131, 133, 134syl131anc 1195 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )  .\/  U )  =  ( ( z  .\/  U ) 
./\  ( ( Q 
.\/  ( ( P 
.\/  z )  ./\  W ) )  .\/  U
) ) )
136127, 135syl5eq 2327 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( F  .\/  U )  =  ( ( z  .\/  U )  ./\  ( ( Q  .\/  ( ( P 
.\/  z )  ./\  W ) )  .\/  U
) ) )
137120, 126, 1363eqtr4rd 2326 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( F  .\/  U )  =  ( z  .\/  ( U  ./\  ( ( P 
.\/  Q )  .\/  z ) ) ) )
1387, 14, 8latlej1 14166 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  z  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  z ) )
1394, 11, 64, 138syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q )  .\/  z ) )
1407, 14, 4, 70, 11, 122, 54, 139lattrd 14164 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  U  .<_  ( ( P  .\/  Q )  .\/  z ) )
1417, 14, 15latleeqm1 14185 . . . . . . . . . . . . . 14  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  z )  e.  ( Base `  K
) )  ->  ( U  .<_  ( ( P 
.\/  Q )  .\/  z )  <->  ( U  ./\  ( ( P  .\/  Q )  .\/  z ) )  =  U ) )
1424, 70, 122, 141syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( U  .<_  ( ( P 
.\/  Q )  .\/  z )  <->  ( U  ./\  ( ( P  .\/  Q )  .\/  z ) )  =  U ) )
143140, 142mpbid 201 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( U  ./\  ( ( P 
.\/  Q )  .\/  z ) )  =  U )
144143oveq2d 5874 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
z  .\/  ( U  ./\  ( ( P  .\/  Q )  .\/  z ) ) )  =  ( z  .\/  U ) )
145137, 144eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( F  .\/  U )  =  ( z  .\/  U
) )
146145oveq1d 5873 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( F  .\/  U
)  .\/  ( ( T  .\/  z )  ./\  W ) )  =  ( ( z  .\/  U
)  .\/  ( ( T  .\/  z )  ./\  W ) ) )
14774, 146eqtrd 2315 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( F  .\/  (
( T  .\/  z
)  ./\  W )
)  .\/  U )  =  ( ( z 
.\/  U )  .\/  ( ( T  .\/  z )  ./\  W
) ) )
14814, 8, 9hlatlej2 29565 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  T  e.  A  /\  z  e.  A )  ->  z  .<_  ( T  .\/  z ) )
1492, 35, 13, 148syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  z  .<_  ( T  .\/  z
) )
1507, 14, 8, 15, 9atmod3i1 30053 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( z  e.  A  /\  ( T  .\/  z
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  z  .<_  ( T  .\/  z
) )  ->  (
z  .\/  ( ( T  .\/  z )  ./\  W ) )  =  ( ( T  .\/  z
)  ./\  ( z  .\/  W ) ) )
1512, 13, 48, 25, 149, 150syl131anc 1195 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
z  .\/  ( ( T  .\/  z )  ./\  W ) )  =  ( ( T  .\/  z
)  ./\  ( z  .\/  W ) ) )
152 simp33 993 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
z  e.  A  /\  -.  z  .<_  W ) )
15314, 8, 81, 9, 16lhpjat2 30210 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( z  e.  A  /\  -.  z  .<_  W ) )  -> 
( z  .\/  W
)  =  ( 1.
`  K ) )
1542, 12, 152, 153syl21anc 1181 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
z  .\/  W )  =  ( 1. `  K ) )
155154oveq2d 5874 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( T  .\/  z
)  ./\  ( z  .\/  W ) )  =  ( ( T  .\/  z )  ./\  ( 1. `  K ) ) )
156151, 155eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
z  .\/  ( ( T  .\/  z )  ./\  W ) )  =  ( ( T  .\/  z
)  ./\  ( 1. `  K ) ) )
1577, 15, 81olm11 29417 . . . . . . . . . . 11  |-  ( ( K  e.  OL  /\  ( T  .\/  z )  e.  ( Base `  K
) )  ->  (
( T  .\/  z
)  ./\  ( 1. `  K ) )  =  ( T  .\/  z
) )
15886, 48, 157syl2anc 642 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( T  .\/  z
)  ./\  ( 1. `  K ) )  =  ( T  .\/  z
) )
159156, 158eqtr2d 2316 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( T  .\/  z )  =  ( z  .\/  (
( T  .\/  z
)  ./\  W )
) )
160159oveq1d 5873 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( T  .\/  z
)  .\/  U )  =  ( ( z 
.\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) )
16172, 147, 1603eqtr4rd 2326 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( T  .\/  z
)  .\/  U )  =  ( ( F 
.\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) )
16268, 161eqtrd 2315 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( T  .\/  U
)  .\/  z )  =  ( ( F 
.\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) )
16366, 162breqtrd 4047 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( T  .\/  U )  .<_  ( ( F  .\/  ( ( T  .\/  z )  ./\  W
) )  .\/  U
) )
16460, 163eqbrtrd 4043 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  Q )  .<_  ( ( F  .\/  ( ( T  .\/  z )  ./\  W
) )  .\/  U
) )
1657, 8latjcl 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( F  .\/  ( ( T  .\/  z ) 
./\  W ) )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
)  e.  ( Base `  K ) )
1664, 52, 70, 165syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( F  .\/  (
( T  .\/  z
)  ./\  W )
)  .\/  U )  e.  ( Base `  K
) )
1677, 14, 15latleeqm1 14185 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
)  e.  ( Base `  K ) )  -> 
( ( P  .\/  Q )  .<_  ( ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
)  <->  ( ( P 
.\/  Q )  ./\  ( ( F  .\/  ( ( T  .\/  z )  ./\  W
) )  .\/  U
) )  =  ( P  .\/  Q ) ) )
1684, 11, 166, 167syl3anc 1182 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  .<_  ( ( F 
.\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
)  <->  ( ( P 
.\/  Q )  ./\  ( ( F  .\/  ( ( T  .\/  z )  ./\  W
) )  .\/  U
) )  =  ( P  .\/  Q ) ) )
169164, 168mpbid 201 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  (
( P  .\/  Q
)  ./\  ( ( F  .\/  ( ( T 
.\/  z )  ./\  W ) )  .\/  U
) )  =  ( P  .\/  Q ) )
17058, 169eqtr2d 2316 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  ( P  .\/  Q )  =  ( O  .\/  V
) )
17132, 170breqtrd 4047 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  T  e.  A ) )  /\  ( ( V  e.  A  /\  V  .<_  W )  /\  ( P  =/=  Q  /\  ( T  .\/  V )  =  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  -.  z  .<_  W ) ) )  ->  N  .<_  ( O  .\/  V
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   1.cp1 14144   Latclat 14151   OLcol 29364   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme26e  30548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177
  Copyright terms: Public domain W3C validator