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Theorem 4atexlemex6 30556
Description: Lemma for 4atexlem7 30557. (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 1068 . . 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 987 . . . 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 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 ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
4 simp13l 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 ) ) )  ->  Q  e.  A )
5 simp32 994 . . . 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 30525 . . . 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 1188 . . 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 984 . . . 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 1070 . . . . . 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 995 . . . . . 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 29861 . . . . . 6  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  P  e.  A  /\  Q  e.  A
)  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  =/=  P )
171, 13, 14, 4, 15, 16syl131anc 1197 . . . . 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 2650 . . . 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 30525 . . . 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 1188 . . 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 29869 . . 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 1184 . 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 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 ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
24 simp112 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 ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
25 simp113 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 ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
26 simp12r 1071 . . . 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 1024 . . . . . 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 978 . . . . 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 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  .<_  W )
30293ad2ant1 978 . . . . 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 1092 . . . . 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 1134 . . . 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 956 . . . 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 1093 . . . 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 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 ) )  ->  -.  S  .<_  ( P 
.\/  Q ) )
36 biid 228 . . . . . 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 2404 . . . . . 6  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
38 eqid 2404 . . . . . 6  |-  ( ( P  .\/  S ) 
./\  W )  =  ( ( P  .\/  S )  ./\  W )
39 eqid 2404 . . . . . 6  |-  ( ( Q  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( Q  .\/  t
)  ./\  ( P  .\/  S ) )
40 eqid 2404 . . . . . 6  |-  ( ( R  .\/  t ) 
./\  ( P  .\/  S ) )  =  ( ( R  .\/  t
)  ./\  ( P  .\/  S ) )
4136, 6, 7, 8, 9, 10, 37, 38, 39, 404atexlemex4 30555 . . . . 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 30553 . . . . 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 2645 . . . 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 1215 . . 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 2792 . 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  4atexlem7  30557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lhyp 30470
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