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Theorem cdleme28a 30856
Description: Lemma for cdleme25b 30840. TODO: FIX COMMENT (Contributed by NM, 4-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b  |-  B  =  ( Base `  K
)
cdleme26.l  |-  .<_  =  ( le `  K )
cdleme26.j  |-  .\/  =  ( join `  K )
cdleme26.m  |-  ./\  =  ( meet `  K )
cdleme26.a  |-  A  =  ( Atoms `  K )
cdleme26.h  |-  H  =  ( LHyp `  K
)
cdleme27.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme27.f  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
cdleme27.z  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
cdleme27.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
cdleme27.d  |-  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
cdleme27.c  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
cdleme27.g  |-  G  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdleme27.o  |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
cdleme27.e  |-  E  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
cdleme27.y  |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G
)
cdleme28a.v  |-  V  =  ( ( s  .\/  t )  ./\  ( X  ./\  W ) )
Assertion
Ref Expression
cdleme28a  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( C  .\/  ( X  ./\  W ) )  .<_  ( Y 
.\/  ( X  ./\  W ) ) )
Distinct variable groups:    t, s, u, z, A    B, s,
t, u, z    u, F    u, G    H, s,
t, z    .\/ , s, t, u, z    K, s, t, z    .<_ , s, t, u, z    ./\ , s,
t, u, z    t, N, u    O, s, u    P, s, t, u, z    Q, s, t, u, z    U, s, t, u, z   
z, V    W, s,
t, u, z    X, s
Allowed substitution hints:    C( z, u, t, s)    D( z, u, t, s)    E( z, u, t, s)    F( z, t, s)    G( z, t, s)    H( u)    K( u)    N( z, s)    O( z, t)    V( u, t, s)    X( z, u, t)    Y( z, u, t, s)    Z( z, u, t, s)

Proof of Theorem cdleme28a
StepHypRef Expression
1 cdleme26.b . . 3  |-  B  =  ( Base `  K
)
2 cdleme26.l . . 3  |-  .<_  =  ( le `  K )
3 simp11l 1068 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  K  e.  HL )
4 hllat 29850 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  K  e.  Lat )
6 simp11r 1069 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  W  e.  H )
7 simp12 988 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp13 989 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
9 simp22 991 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
10 simp21 990 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  P  =/=  Q )
11 cdleme26.j . . . . 5  |-  .\/  =  ( join `  K )
12 cdleme26.m . . . . 5  |-  ./\  =  ( meet `  K )
13 cdleme26.a . . . . 5  |-  A  =  ( Atoms `  K )
14 cdleme26.h . . . . 5  |-  H  =  ( LHyp `  K
)
15 cdleme27.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
16 cdleme27.f . . . . 5  |-  F  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
17 cdleme27.z . . . . 5  |-  Z  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W
) ) )
18 cdleme27.n . . . . 5  |-  N  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( s  .\/  z )  ./\  W
) ) )
19 cdleme27.d . . . . 5  |-  D  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  N ) )
20 cdleme27.c . . . . 5  |-  C  =  if ( s  .<_  ( P  .\/  Q ) ,  D ,  F
)
211, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20cdleme27cl 30852 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( s  e.  A  /\  -.  s  .<_  W )  /\  P  =/=  Q
) )  ->  C  e.  B )
223, 6, 7, 8, 9, 10, 21syl222anc 1200 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  C  e.  B )
23 simp23 992 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
t  e.  A  /\  -.  t  .<_  W ) )
24 cdleme27.g . . . . . 6  |-  G  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
25 cdleme27.o . . . . . 6  |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
26 cdleme27.e . . . . . 6  |-  E  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
27 cdleme27.y . . . . . 6  |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G
)
281, 2, 11, 12, 13, 14, 15, 24, 17, 25, 26, 27cdleme27cl 30852 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( t  e.  A  /\  -.  t  .<_  W )  /\  P  =/=  Q
) )  ->  Y  e.  B )
293, 6, 7, 8, 23, 10, 28syl222anc 1200 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  Y  e.  B )
30 simp11 987 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3130, 9, 233jca 1134 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  (
t  e.  A  /\  -.  t  .<_  W ) ) )
32 simp33 995 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
33 simp31 993 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  s  =/=  t )
34 simp32l 1082 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
s  .\/  ( X  ./\ 
W ) )  =  X )
35 simp32r 1083 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
t  .\/  ( X  ./\ 
W ) )  =  X )
3633, 34, 353jca 1134 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
s  =/=  t  /\  ( s  .\/  ( X  ./\  W ) )  =  X  /\  (
t  .\/  ( X  ./\ 
W ) )  =  X ) )
37 cdleme28a.v . . . . . . 7  |-  V  =  ( ( s  .\/  t )  ./\  ( X  ./\  W ) )
381, 2, 11, 12, 13, 14, 37cdleme23b 30836 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( s  =/=  t  /\  ( s 
.\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X ) )  ->  V  e.  A )
3931, 32, 36, 38syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  V  e.  A )
401, 13atbase 29776 . . . . 5  |-  ( V  e.  A  ->  V  e.  B )
4139, 40syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  V  e.  B )
421, 11latjcl 14438 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  V  e.  B )  ->  ( Y  .\/  V
)  e.  B )
435, 29, 41, 42syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( Y  .\/  V )  e.  B )
44 simp33l 1084 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  X  e.  B )
451, 14lhpbase 30484 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
466, 45syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  W  e.  B )
471, 12latmcl 14439 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
485, 44, 46, 47syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( X  ./\  W )  e.  B )
491, 11latjcl 14438 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( Y  .\/  ( X  ./\  W ) )  e.  B )
505, 29, 48, 49syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( Y  .\/  ( X  ./\  W ) )  e.  B
)
511, 2, 11, 12, 13, 14, 37cdleme23c 30837 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( s  =/=  t  /\  ( s 
.\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X ) )  ->  s  .<_  ( t  .\/  V ) )
5231, 32, 36, 51syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  s  .<_  ( t  .\/  V
) )
5333, 52jca 519 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
s  =/=  t  /\  s  .<_  ( t  .\/  V ) ) )
541, 2, 11, 12, 13, 14, 37cdleme23a 30835 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( s  =/=  t  /\  ( s 
.\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X ) )  ->  V  .<_  W )
5531, 32, 36, 54syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  V  .<_  W )
5639, 55jca 519 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( V  e.  A  /\  V  .<_  W ) )
571, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27cdleme27a 30853 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W ) )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( ( s  =/=  t  /\  s  .<_  ( t  .\/  V
) )  /\  ( V  e.  A  /\  V  .<_  W ) ) )  ->  C  .<_  ( Y  .\/  V ) )
5830, 10, 9, 7, 8, 23, 53, 56, 57syl332anc 1215 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  C  .<_  ( Y  .\/  V
) )
59 simp22l 1076 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  s  e.  A )
60 simp23l 1078 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  t  e.  A )
611, 11, 13hlatjcl 29853 . . . . . . 7  |-  ( ( K  e.  HL  /\  s  e.  A  /\  t  e.  A )  ->  ( s  .\/  t
)  e.  B )
623, 59, 60, 61syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
s  .\/  t )  e.  B )
631, 2, 12latmle2 14465 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( s  .\/  t
)  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( ( s  .\/  t )  ./\  ( X  ./\  W ) ) 
.<_  ( X  ./\  W
) )
645, 62, 48, 63syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
( s  .\/  t
)  ./\  ( X  ./\ 
W ) )  .<_  ( X  ./\  W ) )
6537, 64syl5eqbr 4209 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  V  .<_  ( X  ./\  W
) )
661, 2, 11latjlej2 14454 . . . . 5  |-  ( ( K  e.  Lat  /\  ( V  e.  B  /\  ( X  ./\  W
)  e.  B  /\  Y  e.  B )
)  ->  ( V  .<_  ( X  ./\  W
)  ->  ( Y  .\/  V )  .<_  ( Y 
.\/  ( X  ./\  W ) ) ) )
675, 41, 48, 29, 66syl13anc 1186 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( V  .<_  ( X  ./\  W )  ->  ( Y  .\/  V )  .<_  ( Y 
.\/  ( X  ./\  W ) ) ) )
6865, 67mpd 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( Y  .\/  V )  .<_  ( Y  .\/  ( X 
./\  W ) ) )
691, 2, 5, 22, 43, 50, 58, 68lattrd 14446 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  C  .<_  ( Y  .\/  ( X  ./\  W ) ) )
701, 2, 11latlej2 14449 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( X  ./\  W
)  .<_  ( Y  .\/  ( X  ./\  W ) ) )
715, 29, 48, 70syl3anc 1184 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( X  ./\  W )  .<_  ( Y  .\/  ( X 
./\  W ) ) )
721, 2, 11latjle12 14450 . . 3  |-  ( ( K  e.  Lat  /\  ( C  e.  B  /\  ( X  ./\  W
)  e.  B  /\  ( Y  .\/  ( X 
./\  W ) )  e.  B ) )  ->  ( ( C 
.<_  ( Y  .\/  ( X  ./\  W ) )  /\  ( X  ./\  W )  .<_  ( Y  .\/  ( X  ./\  W
) ) )  <->  ( C  .\/  ( X  ./\  W
) )  .<_  ( Y 
.\/  ( X  ./\  W ) ) ) )
735, 22, 48, 50, 72syl13anc 1186 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  (
( C  .<_  ( Y 
.\/  ( X  ./\  W ) )  /\  ( X  ./\  W )  .<_  ( Y  .\/  ( X 
./\  W ) ) )  <->  ( C  .\/  ( X  ./\  W ) )  .<_  ( Y  .\/  ( X  ./\  W
) ) ) )
7469, 71, 73mpbi2and 888 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( s  e.  A  /\  -.  s  .<_  W )  /\  ( t  e.  A  /\  -.  t  .<_  W ) )  /\  ( s  =/=  t  /\  ( ( s  .\/  ( X  ./\  W ) )  =  X  /\  ( t  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( C  .\/  ( X  ./\  W ) )  .<_  ( Y 
.\/  ( X  ./\  W ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   ifcif 3703   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   iota_crio 6505   Basecbs 13428   lecple 13495   joincjn 14360   meetcmee 14361   Latclat 14433   Atomscatm 29750   HLchlt 29837   LHypclh 30470
This theorem is referenced by:  cdleme28b  30857
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474
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