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Theorem cdlemg18d 34215
Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg18d  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  A )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r
Allowed substitution hints:    R( r)    T( r)    H( r)    K( r)    ./\ ( r)

Proof of Theorem cdlemg18d
StepHypRef Expression
1 simp1 1006 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp21r 1124 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  G  e.  T )
3 simp22 1040 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  P  =/=  Q )
4 simp23 1041 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( G `  P
)  =/=  P )
5 simp31 1042 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( R `  G
)  .<_  ( P  .\/  Q ) )
6 simp33 1044 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
7 cdlemg12.l . . . . . . 7  |-  .<_  =  ( le `  K )
8 cdlemg12.j . . . . . . 7  |-  .\/  =  ( join `  K )
9 cdlemg12.m . . . . . . 7  |-  ./\  =  ( meet `  K )
10 cdlemg12.a . . . . . . 7  |-  A  =  ( Atoms `  K )
11 cdlemg12.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
12 cdlemg12.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemg12b.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
147, 8, 9, 10, 11, 12, 13cdlemg17b 34196 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( G  e.  T  /\  P  =/=  Q
)  /\  ( ( G `  P )  =/=  P  /\  ( R `
 G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( G `  P )  =  Q )
151, 2, 3, 4, 5, 6, 14syl123anc 1282 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( G `  P
)  =  Q )
1615fveq2d 5884 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  ( G `  P )
)  =  ( F `
 Q ) )
1716oveq2d 6320 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( P  .\/  ( F `  ( G `  P ) ) )  =  ( P  .\/  ( F `  Q ) ) )
18 simp21l 1123 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  F  e.  T )
197, 8, 9, 10, 11, 12, 13cdlemg17bq 34207 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( ( G `  P )  =/=  P  /\  ( R `
 G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( G `  Q )  =  P )
201, 18, 2, 3, 4, 5, 6, 19syl133anc 1288 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( G `  Q
)  =  P )
2120fveq2d 5884 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  ( G `  Q )
)  =  ( F `
 P ) )
2221oveq2d 6320 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( Q  .\/  ( F `  ( G `  Q ) ) )  =  ( Q  .\/  ( F `  P ) ) )
2317, 22oveq12d 6322 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  =  ( ( P  .\/  ( F `  Q ) )  ./\  ( Q  .\/  ( F `  P
) ) ) )
24 simp11 1036 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
25 simp12 1037 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
26 simp13 1038 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
27 simp32 1043 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =/=  ( P  .\/  Q ) )
287, 8, 9, 10, 11, 12cdlemg11aq 34172 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  (
( F `  ( G `  P )
)  .\/  ( F `  ( G `  Q
) ) )  =/=  ( P  .\/  Q
) ) )  -> 
( F `  ( G `  Q )
)  =/=  Q )
2924, 25, 26, 18, 2, 27, 28syl123anc 1282 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  ( G `  Q )
)  =/=  Q )
3021, 29eqnetrrd 2719 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  P
)  =/=  Q )
317, 8, 9, 10, 11, 12, 13cdlemg17irq 34209 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( F  e.  T  /\  G  e.  T  /\  P  =/=  Q
)  /\  ( ( G `  P )  =/=  P  /\  ( R `
 G )  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( F `  ( G `  Q ) )  =  ( F `
 P ) )
321, 18, 2, 3, 4, 5, 6, 31syl133anc 1288 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( F `  ( G `  Q )
)  =  ( F `
 P ) )
3316, 32oveq12d 6322 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( F `  ( G `  P ) )  .\/  ( F `
 ( G `  Q ) ) )  =  ( ( F `
 Q )  .\/  ( F `  P ) ) )
3433, 27eqnetrrd 2719 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) )
35 eqid 2423 . . . 4  |-  ( ( P  .\/  Q ) 
./\  W )  =  ( ( P  .\/  Q )  ./\  W )
367, 8, 9, 10, 11, 12, 13, 35cdlemg18c 34214 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  F  e.  T
)  /\  ( P  =/=  Q  /\  ( F `
 P )  =/= 
Q  /\  ( ( F `  Q )  .\/  ( F `  P
) )  =/=  ( P  .\/  Q ) ) )  ->  ( ( P  .\/  ( F `  Q ) )  ./\  ( Q  .\/  ( F `
 P ) ) )  e.  A )
3724, 25, 26, 18, 3, 30, 34, 36syl133anc 1288 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  Q ) )  ./\  ( Q  .\/  ( F `  P
) ) )  e.  A )
3823, 37eqeltrd 2511 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q  /\  ( G `
 P )  =/= 
P )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( ( F `  ( G `
 P ) ) 
.\/  ( F `  ( G `  Q ) ) )  =/=  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  -> 
( ( P  .\/  ( F `  ( G `
 P ) ) )  ./\  ( Q  .\/  ( F `  ( G `  Q )
) ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   E.wrex 2777   class class class wbr 4422   ` cfv 5600  (class class class)co 6304   lecple 15194   joincjn 16186   meetcmee 16187   Atomscatm 32796   HLchlt 32883   LHypclh 33516   LTrncltrn 33633   trLctrl 33691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4535  ax-sep 4545  ax-nul 4554  ax-pow 4601  ax-pr 4659  ax-un 6596  ax-riotaBAD 32492
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3302  df-csb 3398  df-dif 3441  df-un 3443  df-in 3445  df-ss 3452  df-nul 3764  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4219  df-iun 4300  df-iin 4301  df-br 4423  df-opab 4482  df-mpt 4483  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6266  df-ov 6307  df-oprab 6308  df-mpt2 6309  df-1st 6806  df-2nd 6807  df-undef 7030  df-map 7484  df-preset 16170  df-poset 16188  df-plt 16201  df-lub 16217  df-glb 16218  df-join 16219  df-meet 16220  df-p0 16282  df-p1 16283  df-lat 16289  df-clat 16351  df-oposet 32709  df-ol 32711  df-oml 32712  df-covers 32799  df-ats 32800  df-atl 32831  df-cvlat 32855  df-hlat 32884  df-llines 33030  df-lplanes 33031  df-lvols 33032  df-lines 33033  df-psubsp 33035  df-pmap 33036  df-padd 33328  df-lhyp 33520  df-laut 33521  df-ldil 33636  df-ltrn 33637  df-trl 33692
This theorem is referenced by:  cdlemg18  34216  cdlemg19a  34217
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