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Theorem cdleme28b 30560
Description: Lemma for cdleme25b 30543. TODO: FIX COMMENT (Contributed by NM, 6-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
)
Assertion
Ref Expression
cdleme28b  |-  ( ( ( ( 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    W, s, t, u, z    X, s, z, t
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)    X( u)    Y( z, u, t, s)    Z( z, u, t, s)

Proof of Theorem cdleme28b
StepHypRef Expression
1 cdleme26.b . 2  |-  B  =  ( Base `  K
)
2 cdleme26.l . 2  |-  .<_  =  ( le `  K )
3 simp11l 1066 . . 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.  HL )
4 hllat 29553 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 15 . 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 ) ) )  ->  K  e.  Lat )
6 simp11r 1067 . . . 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 986 . . . 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 987 . . . 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 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 ) ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
10 simp21 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  =/=  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 30555 . . . 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 1198 . . 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 simp33l 1082 . . . 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  e.  B )
241, 14lhpbase 30187 . . . . 5  |-  ( W  e.  H  ->  W  e.  B )
256, 24syl 15 . . . 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.  B )
261, 12latmcl 14157 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
275, 23, 25, 26syl3anc 1182 . . 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 ) ) )  ->  ( X  ./\  W )  e.  B )
281, 11latjcl 14156 . . 3  |-  ( ( K  e.  Lat  /\  C  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( C  .\/  ( X  ./\  W ) )  e.  B )
295, 22, 27, 28syl3anc 1182 . 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  .\/  ( X  ./\  W ) )  e.  B
)
30 simp23 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 ) ) )  ->  (
t  e.  A  /\  -.  t  .<_  W ) )
31 cdleme27.g . . . . 5  |-  G  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
32 cdleme27.o . . . . 5  |-  O  =  ( ( P  .\/  Q )  ./\  ( Z  .\/  ( ( t  .\/  z )  ./\  W
) ) )
33 cdleme27.e . . . . 5  |-  E  =  ( iota_ u  e.  B A. z  e.  A  ( ( -.  z  .<_  W  /\  -.  z  .<_  ( P  .\/  Q
) )  ->  u  =  O ) )
34 cdleme27.y . . . . 5  |-  Y  =  if ( t  .<_  ( P  .\/  Q ) ,  E ,  G
)
351, 2, 11, 12, 13, 14, 15, 31, 17, 32, 33, 34cdleme27cl 30555 . . . 4  |-  ( ( ( 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 )
363, 6, 7, 8, 30, 10, 35syl222anc 1198 . . 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  e.  B )
371, 11latjcl 14156 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( Y  .\/  ( X  ./\  W ) )  e.  B )
385, 36, 27, 37syl3anc 1182 . 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 ) ) )  ->  ( Y  .\/  ( X  ./\  W ) )  e.  B
)
39 eqid 2283 . . 3  |-  ( ( s  .\/  t ) 
./\  ( X  ./\  W ) )  =  ( ( s  .\/  t
)  ./\  ( X  ./\ 
W ) )
401, 2, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 31, 32, 33, 34, 39cdleme28a 30559 . 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  .\/  ( X  ./\  W ) )  .<_  ( Y 
.\/  ( X  ./\  W ) ) )
41 simp11 985 . . 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.  HL  /\  W  e.  H ) )
42 simp31 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  =/=  t )
4342necomd 2529 . . 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 ) ) )  ->  t  =/=  s )
44 simp32 992 . . . 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  .\/  ( X  ./\  W ) )  =  X  /\  (
t  .\/  ( X  ./\ 
W ) )  =  X ) )
4544ancomd 438 . . 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 ) ) )  ->  (
( t  .\/  ( X  ./\  W ) )  =  X  /\  (
s  .\/  ( X  ./\ 
W ) )  =  X ) )
46 simp33 993 . . 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 ) ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
47 eqid 2283 . . . 4  |-  ( ( t  .\/  s ) 
./\  ( X  ./\  W ) )  =  ( ( t  .\/  s
)  ./\  ( X  ./\ 
W ) )
481, 2, 11, 12, 13, 14, 15, 31, 17, 32, 33, 34, 16, 18, 19, 20, 47cdleme28a 30559 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( t  e.  A  /\  -.  t  .<_  W )  /\  ( s  e.  A  /\  -.  s  .<_  W ) )  /\  ( t  =/=  s  /\  ( ( t  .\/  ( X  ./\  W ) )  =  X  /\  ( s  .\/  ( X  ./\  W ) )  =  X )  /\  ( X  e.  B  /\  -.  X  .<_  W ) ) )  ->  ( Y  .\/  ( X  ./\  W ) )  .<_  ( C 
.\/  ( X  ./\  W ) ) )
4941, 7, 8, 10, 30, 9, 43, 45, 46, 48syl333anc 1214 . 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 ) ) )  ->  ( Y  .\/  ( X  ./\  W ) )  .<_  ( C 
.\/  ( X  ./\  W ) ) )
501, 2, 5, 29, 38, 40, 49latasymd 14163 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    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme28c  30561
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-3or 935  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-rmo 2551  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-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177
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