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Theorem cdleme28b 30619
Description: Lemma for cdleme25b 30602. 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 1067 . . 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 29612 . . 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 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 ) ) )  ->  W  e.  H )
7 simp12 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 ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
8 simp13 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 ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
9 simp22 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 ) ) )  ->  (
s  e.  A  /\  -.  s  .<_  W ) )
10 simp21 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 ) ) )  ->  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 30614 . . . 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 1199 . . 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 1083 . . . 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 30246 . . . . 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 14367 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
275, 23, 25, 26syl3anc 1183 . . 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 14366 . . 3  |-  ( ( K  e.  Lat  /\  C  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( C  .\/  ( X  ./\  W ) )  e.  B )
295, 22, 27, 28syl3anc 1183 . 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 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 ) ) )  ->  (
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 30614 . . . 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 1199 . . 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 14366 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( Y  .\/  ( X  ./\  W ) )  e.  B )
385, 36, 27, 37syl3anc 1183 . 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 2366 . . 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 30618 . 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 986 . . 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 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  =/=  t )
4342necomd 2612 . . 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 993 . . . 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 994 . . 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 2366 . . . 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 30618 . . 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 1215 . 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 14373 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   ifcif 3654   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   iota_crio 6439   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   Latclat 14361   Atomscatm 29512   HLchlt 29599   LHypclh 30232
This theorem is referenced by:  cdleme28c  30620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-llines 29746  df-lplanes 29747  df-lvols 29748  df-lines 29749  df-psubsp 29751  df-pmap 29752  df-padd 30044  df-lhyp 30236
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