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Theorem 4atexlemex6 30263
Description: Lemma for 4atexlem7 30264. (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 30232 . . . 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 29568 . . . . . 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 2529 . . . 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 30232 . . . 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 29576 . . 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 2283 . . . . . 6  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
38 eqid 2283 . . . . . 6  |-  ( ( P  .\/  S ) 
./\  W )  =  ( ( P  .\/  S )  ./\  W )
39 eqid 2283 . . . . . 6  |-  ( ( Q  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( Q  .\/  t
)  ./\  ( P  .\/  S ) )
40 eqid 2283 . . . . . 6  |-  ( ( R  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( R  .\/  t
)  ./\  ( P  .\/  S ) )
4136, 6, 7, 8, 9, 10, 37, 38, 39, 404atexlemex4 30262 . . . . 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 30260 . . . . 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 2524 . . . 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 2669 . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  4atexlem7  30264
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-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-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-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-lhyp 30177
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