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Theorem cdleme21c 29205
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 995 . 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 1071 . . . . . . 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 28238 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  CvLat )
42, 3syl 17 . . . . . 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 1073 . . . . . 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 993 . . . . . 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 988 . . . . . 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 992 . . . . . . 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 28257 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  =/=  P )
1312necomd 2495 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  S )
142, 6, 5, 8, 1, 13syl131anc 1200 . . . . . 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 989 . . . . . 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 28227 . . . . . 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 1205 . . . . 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 28246 . . . . . 6  |-  ( ( K  e.  HL  /\  z  e.  A  /\  S  e.  A )  ->  ( z  .\/  S
)  =  ( S 
.\/  z ) )
192, 7, 6, 18syl3anc 1187 . . . . 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 2285 . . . 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 3932 . . 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 1072 . . . . . 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 1074 . . . . . 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 994 . . . . . 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 29089 . . . . . 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 1203 . . . . 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 28242 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
312, 30syl 17 . . . . . . . 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 2253 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
3332, 10, 11hlatjcl 28245 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
342, 5, 8, 33syl3anc 1187 . . . . . . . 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 28876 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3622, 35syl 17 . . . . . . . 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 14027 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
3831, 34, 36, 37syl3anc 1187 . . . . . . 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 3953 . . . . . 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 3938 . . . . . 6  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
4139, 23, 40syl2anc 645 . . . . 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 28215 . . . . 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 1200 . . . 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 28253 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
452, 5, 8, 44syl3anc 1187 . . . . . 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 29098 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) )  ->  U  .<_  ( P  .\/  Q ) )
472, 22, 5, 8, 46syl22anc 1188 . . . . . 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 28168 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
495, 48syl 17 . . . . . . 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 28168 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5129, 50syl 17 . . . . . . 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 14012 . . . . . . 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 1189 . . . . . 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 892 . . . . 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 28168 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
566, 55syl 17 . . . . . 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 28245 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  ->  ( P  .\/  U
)  e.  ( Base `  K ) )
582, 5, 29, 57syl3anc 1187 . . . . . 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 14006 . . . . . 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 1189 . . . . 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 658 . . . 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 228 . 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995   Atomscatm 28142   CvLatclc 28144   HLchlt 28229   LHypclh 28862
This theorem is referenced by:  cdleme21at  29206  cdleme21ct  29207  cdleme21d  29208
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-lhyp 28866
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