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Theorem cdleme28a 34353
Description: Lemma for cdleme25b 34337. 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 1099 . . . 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 33347 . . . 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 1100 . . . 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 1019 . . . 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 1020 . . . 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 1022 . . . 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 1021 . . . 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 34349 . . . 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 1235 . . 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 1023 . . . . 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 34349 . . . . 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 1235 . . . 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 1018 . . . . . . 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 1168 . . . . . 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 1026 . . . . . 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 1024 . . . . . . 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 1113 . . . . . . 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 1114 . . . . . . 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 1168 . . . . . 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 34333 . . . . . 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 1219 . . . . 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 33273 . . . . 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 15341 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  V  e.  B )  ->  ( Y  .\/  V
)  e.  B )
435, 29, 41, 42syl3anc 1219 . . 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 1115 . . . . 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 33981 . . . . . 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 15342 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
485, 44, 46, 47syl3anc 1219 . . . 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 15341 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( Y  .\/  ( X  ./\  W ) )  e.  B )
505, 29, 48, 49syl3anc 1219 . . 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 34334 . . . . . 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 1219 . . . . 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 532 . . . 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 34332 . . . . . 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 1219 . . . . 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 532 . . . 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 34350 . . . 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 1250 . . 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 1107 . . . . . . 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 1109 . . . . . . 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 33350 . . . . . . 7  |-  ( ( K  e.  HL  /\  s  e.  A  /\  t  e.  A )  ->  ( s  .\/  t
)  e.  B )
623, 59, 60, 61syl3anc 1219 . . . . . 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 15367 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( s  .\/  t
)  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( ( s  .\/  t )  ./\  ( X  ./\  W ) ) 
.<_  ( X  ./\  W
) )
645, 62, 48, 63syl3anc 1219 . . . . 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 4434 . . . 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 15356 . . . . 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 1221 . . . 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 15348 . 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 15351 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  ./\  W )  e.  B )  -> 
( X  ./\  W
)  .<_  ( Y  .\/  ( X  ./\  W ) ) )
715, 29, 48, 70syl3anc 1219 . 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 15352 . . 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 1221 . 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 912 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   ifcif 3900   class class class wbr 4401   ` cfv 5527   iota_crio 6161  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Latclat 15335   Atomscatm 33247   HLchlt 33334   LHypclh 33967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-riotaBAD 32943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-iin 4283  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-undef 6903  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-llines 33481  df-lplanes 33482  df-lvols 33483  df-lines 33484  df-psubsp 33486  df-pmap 33487  df-padd 33779  df-lhyp 33971
This theorem is referenced by:  cdleme28b  34354
  Copyright terms: Public domain W3C validator