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Theorem cdleme21c 30809
Description: Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
Hypotheses
Ref Expression
cdleme21.l  |-  .<_  =  ( le `  K )
cdleme21.j  |-  .\/  =  ( join `  K )
cdleme21.m  |-  ./\  =  ( meet `  K )
cdleme21.a  |-  A  =  ( Atoms `  K )
cdleme21.h  |-  H  =  ( LHyp `  K
)
cdleme21.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
cdleme21c  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  -.  U  .<_  ( S  .\/  z ) )

Proof of Theorem cdleme21c
StepHypRef Expression
1 simp23 992 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  -.  S  .<_  ( P  .\/  Q ) )
2 simp11l 1068 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  K  e.  HL )
3 hlcvl 29842 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CvLat )
42, 3syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  K  e.  CvLat )
5 simp12l 1070 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  P  e.  A
)
6 simp21 990 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  S  e.  A
)
7 simp3l 985 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  z  e.  A
)
8 simp13 989 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  Q  e.  A
)
9 cdleme21.l . . . . . . . . 9  |-  .<_  =  ( le `  K )
10 cdleme21.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
11 cdleme21.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
129, 10, 11atnlej1 29861 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  =/=  P )
1312necomd 2650 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  S )
142, 6, 5, 8, 1, 13syl131anc 1197 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  P  =/=  S
)
15 simp3r 986 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( P  .\/  z )  =  ( S  .\/  z ) )
1611, 10cvlsupr7 29831 . . . . . 6  |-  ( ( K  e.  CvLat  /\  ( P  e.  A  /\  S  e.  A  /\  z  e.  A )  /\  ( P  =/=  S  /\  ( P  .\/  z
)  =  ( S 
.\/  z ) ) )  ->  ( P  .\/  S )  =  ( z  .\/  S ) )
174, 5, 6, 7, 14, 15, 16syl132anc 1202 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( P  .\/  S )  =  ( z 
.\/  S ) )
1810, 11hlatjcom 29850 . . . . . 6  |-  ( ( K  e.  HL  /\  z  e.  A  /\  S  e.  A )  ->  ( z  .\/  S
)  =  ( S 
.\/  z ) )
192, 7, 6, 18syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( z  .\/  S )  =  ( S 
.\/  z ) )
2017, 19eqtrd 2436 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( P  .\/  S )  =  ( S 
.\/  z ) )
2120breq2d 4184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( U  .<_  ( P  .\/  S )  <-> 
U  .<_  ( S  .\/  z ) ) )
22 simp11r 1069 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  W  e.  H
)
23 simp12r 1071 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  -.  P  .<_  W )
24 simp22 991 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  P  =/=  Q
)
25 cdleme21.m . . . . . . 7  |-  ./\  =  ( meet `  K )
26 cdleme21.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
27 cdleme21.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
289, 10, 25, 11, 26, 27cdleme0a 30693 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
292, 22, 5, 23, 8, 24, 28syl222anc 1200 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  U  e.  A
)
30 hllat 29846 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
312, 30syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  K  e.  Lat )
32 eqid 2404 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
3332, 10, 11hlatjcl 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
342, 5, 8, 33syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K ) )
3532, 26lhpbase 30480 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3622, 35syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  W  e.  (
Base `  K )
)
3732, 9, 25latmle2 14461 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3831, 34, 36, 37syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( ( P 
.\/  Q )  ./\  W )  .<_  W )
3927, 38syl5eqbr 4205 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  U  .<_  W )
40 nbrne2 4190 . . . . . 6  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
4139, 23, 40syl2anc 643 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  U  =/=  P
)
429, 10, 11cvlatexch1 29819 . . . . 5  |-  ( ( K  e.  CvLat  /\  ( U  e.  A  /\  S  e.  A  /\  P  e.  A )  /\  U  =/=  P
)  ->  ( U  .<_  ( P  .\/  S
)  ->  S  .<_  ( P  .\/  U ) ) )
434, 29, 6, 5, 41, 42syl131anc 1197 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( U  .<_  ( P  .\/  S )  ->  S  .<_  ( P 
.\/  U ) ) )
449, 10, 11hlatlej1 29857 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
452, 5, 8, 44syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  P  .<_  ( P 
.\/  Q ) )
469, 10, 25, 11, 26, 27cdlemeulpq 30702 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
472, 22, 5, 8, 46syl22anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  U  .<_  ( P 
.\/  Q ) )
4832, 11atbase 29772 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
495, 48syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  P  e.  (
Base `  K )
)
5032, 11atbase 29772 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5129, 50syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  U  e.  (
Base `  K )
)
5232, 9, 10latjle12 14446 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  Q )  /\  U  .<_  ( P 
.\/  Q ) )  <-> 
( P  .\/  U
)  .<_  ( P  .\/  Q ) ) )
5331, 49, 51, 34, 52syl13anc 1186 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( ( P 
.<_  ( P  .\/  Q
)  /\  U  .<_  ( P  .\/  Q ) )  <->  ( P  .\/  U )  .<_  ( P  .\/  Q ) ) )
5445, 47, 53mpbi2and 888 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( P  .\/  U )  .<_  ( P  .\/  Q ) )
5532, 11atbase 29772 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
566, 55syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  S  e.  (
Base `  K )
)
5732, 10, 11hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
582, 5, 29, 57syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( P  .\/  U )  e.  ( Base `  K ) )
5932, 9lattr 14440 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( P  .\/  U )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( S  .<_  ( P 
.\/  U )  /\  ( P  .\/  U ) 
.<_  ( P  .\/  Q
) )  ->  S  .<_  ( P  .\/  Q
) ) )
6031, 56, 58, 34, 59syl13anc 1186 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( ( S 
.<_  ( P  .\/  U
)  /\  ( P  .\/  U )  .<_  ( P 
.\/  Q ) )  ->  S  .<_  ( P 
.\/  Q ) ) )
6154, 60mpan2d 656 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( S  .<_  ( P  .\/  U )  ->  S  .<_  ( P 
.\/  Q ) ) )
6243, 61syld 42 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( U  .<_  ( P  .\/  S )  ->  S  .<_  ( P 
.\/  Q ) ) )
6321, 62sylbird 227 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  ( U  .<_  ( S  .\/  z )  ->  S  .<_  ( P 
.\/  Q ) ) )
641, 63mtod 170 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( S  e.  A  /\  P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) )  /\  (
z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  -.  U  .<_  ( S  .\/  z ) )
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   meetcmee 14357   Latclat 14429   Atomscatm 29746   CvLatclc 29748   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme21at  30810  cdleme21ct  30811  cdleme21d  30812
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-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-p1 14424  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-lhyp 30470
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