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Theorem 4atexlemex6 30872
Description: Lemma for 4atexlem7 30873. (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 1069 . . 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 988 . . . 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 989 . . . 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 1073 . . . 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 995 . . . 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 30841 . . . 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 1189 . . 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 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 ) ) )  ->  S  e.  A )
14 simp12l 1071 . . . . . 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 996 . . . . . 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 30177 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  =/=  P )
171, 13, 14, 4, 15, 16syl131anc 1198 . . . . 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 2688 . . . 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 30841 . . . 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 1189 . . 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 30185 . . 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 1185 . 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 1087 . . . 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 1088 . . . 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 1089 . . . 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 1072 . . . 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 1025 . . . . . 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 979 . . . . 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 1026 . . . . . 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 979 . . . . 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 1093 . . . . 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 1135 . . . 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 957 . . . 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 1094 . . . 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 1095 . . . 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 229 . . . . . 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 2437 . . . . . 6  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
38 eqid 2437 . . . . . 6  |-  ( ( P  .\/  S ) 
./\  W )  =  ( ( P  .\/  S )  ./\  W )
39 eqid 2437 . . . . . 6  |-  ( ( Q  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( Q  .\/  t
)  ./\  ( P  .\/  S ) )
40 eqid 2437 . . . . . 6  |-  ( ( R  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( R  .\/  t
)  ./\  ( P  .\/  S ) )
4136, 6, 7, 8, 9, 10, 37, 38, 39, 404atexlemex4 30871 . . . . 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 30869 . . . . 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 2683 . . . 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 1216 . . 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 2833 . 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 15 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   lecple 13537   joincjn 14402   meetcmee 14403   Atomscatm 30062   HLchlt 30149   LHypclh 30782
This theorem is referenced by:  4atexlem7  30873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-p1 14470  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297  df-lhyp 30786
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