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Theorem 4atexlemex6 30885
Description: Lemma for 4atexlem7 30886. (Contributed by NM, 25-Nov-2012.)
Hypotheses
Ref Expression
4thatleme.l  |-  .<_  =  ( le `  K )
4thatleme.j  |-  .\/  =  ( join `  K )
4thatleme.m  |-  ./\  =  ( meet `  K )
4thatleme.a  |-  A  =  ( Atoms `  K )
4thatleme.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
4atexlemex6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
Distinct variable groups:    z, A    z, 
.\/    z,  .<_    z,  ./\    z, P    z, Q    z, R    z, S    z, W
Allowed substitution hints:    H( z)    K( z)

Proof of Theorem 4atexlemex6
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 simp11l 1066 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
2 simp11 985 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
3 simp12 986 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
4 simp13l 1070 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  A )
5 simp32 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  Q )
6 4thatleme.l . . . . 5  |-  .<_  =  ( le `  K )
7 4thatleme.j . . . . 5  |-  .\/  =  ( join `  K )
8 4thatleme.m . . . . 5  |-  ./\  =  ( meet `  K )
9 4thatleme.a . . . . 5  |-  A  =  ( Atoms `  K )
10 4thatleme.h . . . . 5  |-  H  =  ( LHyp `  K
)
116, 7, 8, 9, 10lhpat 30854 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P 
.\/  Q )  ./\  W )  e.  A )
122, 3, 4, 5, 11syl112anc 1186 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  -> 
( ( P  .\/  Q )  ./\  W )  e.  A )
13 simp2r 982 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  S  e.  A )
14 simp12l 1068 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  P  e.  A )
15 simp33 993 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  -.  S  .<_  ( P 
.\/  Q ) )
166, 7, 9atnlej1 30190 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  =/=  P )
171, 13, 14, 4, 15, 16syl131anc 1195 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  S  =/=  P )
1817necomd 2542 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  S )
196, 7, 8, 9, 10lhpat 30854 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( S  e.  A  /\  P  =/=  S ) )  ->  ( ( P 
.\/  S )  ./\  W )  e.  A )
202, 3, 13, 18, 19syl112anc 1186 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  -> 
( ( P  .\/  S )  ./\  W )  e.  A )
217, 9hlsupr2 30198 . . 3  |-  ( ( K  e.  HL  /\  ( ( P  .\/  Q )  ./\  W )  e.  A  /\  (
( P  .\/  S
)  ./\  W )  e.  A )  ->  E. t  e.  A  ( (
( P  .\/  Q
)  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S )  ./\  W )  .\/  t ) )
221, 12, 20, 21syl3anc 1182 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  E. t  e.  A  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )
23 simp111 1084 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
24 simp112 1085 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
25 simp113 1086 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
26 simp12r 1069 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  ->  S  e.  A )
27 simp2ll 1022 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
28273ad2ant1 976 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  ->  R  e.  A )
29 simp2lr 1023 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  -.  R  .<_  W )
30293ad2ant1 976 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  ->  -.  R  .<_  W )
31 simp131 1090 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  -> 
( P  .\/  R
)  =  ( Q 
.\/  R ) )
3228, 30, 313jca 1132 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  -> 
( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) ) )
33 3simpc 954 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  -> 
( t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) ) )
34 simp132 1091 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  ->  P  =/=  Q )
35 simp133 1092 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  ->  -.  S  .<_  ( P 
.\/  Q ) )
36 biid 227 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  (
t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) ) )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) ) )  <->  ( (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  (
t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) ) )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) ) ) )
37 eqid 2296 . . . . . 6  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
38 eqid 2296 . . . . . 6  |-  ( ( P  .\/  S ) 
./\  W )  =  ( ( P  .\/  S )  ./\  W )
39 eqid 2296 . . . . . 6  |-  ( ( Q  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( Q  .\/  t
)  ./\  ( P  .\/  S ) )
40 eqid 2296 . . . . . 6  |-  ( ( R  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( R  .\/  t
)  ./\  ( P  .\/  S ) )
4136, 6, 7, 8, 9, 10, 37, 38, 39, 404atexlemex4 30884 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  (
t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) ) )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  ( ( Q  .\/  t )  ./\  ( P  .\/  S ) )  =  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
4236, 6, 7, 8, 9, 10, 37, 38, 394atexlemex2 30882 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  (
t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) ) )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) ) )  /\  ( ( Q  .\/  t )  ./\  ( P  .\/  S ) )  =/=  S )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
4341, 42pm2.61dane 2537 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W  /\  ( P  .\/  R )  =  ( Q  .\/  R ) )  /\  (
t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) ) )  /\  ( P  =/= 
Q  /\  -.  S  .<_  ( P  .\/  Q
) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
4423, 24, 25, 26, 32, 33, 34, 35, 43syl332anc 1213 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  /\  t  e.  A  /\  ( ( ( P 
.\/  Q )  ./\  W )  .\/  t )  =  ( ( ( P  .\/  S ) 
./\  W )  .\/  t ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
4544rexlimdv3a 2682 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  -> 
( E. t  e.  A  ( ( ( P  .\/  Q ) 
./\  W )  .\/  t )  =  ( ( ( P  .\/  S )  ./\  W )  .\/  t )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z
) ) ) )
4622, 45mpd 14 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  S  e.  A )  /\  (
( P  .\/  R
)  =  ( Q 
.\/  R )  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  4atexlem7  30886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lhyp 30799
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