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Theorem cdlemg33 31205
Description: Combine cdlemg33b 31201, cdlemg33c 31202, cdlemg33d 31203, cdlemg33e 31204. TODO: Fix comment. (Contributed by NM, 30-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 )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
cdlemg33.o  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
Assertion
Ref Expression
cdlemg33  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
Distinct variable groups:    A, r    G, r    .\/ , r    .<_ , r    P, r    Q, r    W, r    F, r    z, A    z, F, r    H, r, z   
z,  .\/    K, r, z   
z,  .<_    N, r, z    z, P    z, Q    z, R    z, T    z, W    z,
v, r    z, G    z, O, r
Allowed substitution hints:    A( v)    P( v)    Q( v)    R( v, r)    T( v, r)    F( v)    G( v)    H( v)    .\/ ( v)    K( v)    .<_ ( v)    ./\ ( z,
v, r)    N( v)    O( v)    W( v)

Proof of Theorem cdlemg33
StepHypRef Expression
1 simp11 987 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 988 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp13 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp21 990 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( v  e.  A  /\  v  .<_  W ) )
5 simp22l 1076 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  F  e.  T
)
6 simp31 993 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  v  =/=  ( R `  F )
)
7 cdlemg12.l . . . 4  |-  .<_  =  ( le `  K )
8 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
9 cdlemg12.m . . . 4  |-  ./\  =  ( meet `  K )
10 cdlemg12.a . . . 4  |-  A  =  ( Atoms `  K )
11 cdlemg12.h . . . 4  |-  H  =  ( LHyp `  K
)
12 cdlemg12.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
13 cdlemg12b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
14 cdlemg31.n . . . 4  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
157, 8, 9, 10, 11, 12, 13, 14cdlemg31b0a 31189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( F  e.  T  /\  v  =/=  ( R `  F )
) )  ->  ( N  e.  A  \/  N  =  ( 0. `  K ) ) )
161, 2, 3, 4, 5, 6, 15syl132anc 1202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( N  e.  A  \/  N  =  ( 0. `  K
) ) )
17 simp22r 1077 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  G  e.  T
)
18 simp32 994 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  v  =/=  ( R `  G )
)
19 cdlemg33.o . . . 4  |-  O  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  G ) ) )
207, 8, 9, 10, 11, 12, 13, 19cdlemg31b0a 31189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( v  e.  A  /\  v  .<_  W ) )  /\  ( G  e.  T  /\  v  =/=  ( R `  G )
) )  ->  ( O  e.  A  \/  O  =  ( 0. `  K ) ) )
211, 2, 3, 4, 17, 18, 20syl132anc 1202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( O  e.  A  \/  O  =  ( 0. `  K
) ) )
22 simpl1 960 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
23 simpl21 1035 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
24 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  -> 
( N  e.  A  /\  O  e.  A
) )
25 simpl22 1036 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  -> 
( F  e.  T  /\  G  e.  T
) )
26 simpl23 1037 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  ->  P  =/=  Q )
27 simpl31 1038 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  -> 
v  =/=  ( R `
 F ) )
28 simpl33 1040 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
297, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33b 31201 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  e.  A  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
3022, 23, 24, 25, 26, 27, 28, 29syl133anc 1207 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  e.  A ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
3130ex 424 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( N  e.  A  /\  O  e.  A )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
) ) ) ) )
32 simpl1 960 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
33 simpl21 1035 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
34 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  -> 
( N  =  ( 0. `  K )  /\  O  e.  A
) )
35 simpl22 1036 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  -> 
( F  e.  T  /\  G  e.  T
) )
36 simpl23 1037 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  ->  P  =/=  Q )
37 simpl32 1039 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  -> 
v  =/=  ( R `
 G ) )
38 simpl33 1040 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
397, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33d 31203 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  e.  A )  /\  ( F  e.  T  /\  G  e.  T )
)  /\  ( P  =/=  Q  /\  v  =/=  ( R `  G
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
4032, 33, 34, 35, 36, 37, 38, 39syl133anc 1207 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  e.  A ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
4140ex 424 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( N  =  ( 0. `  K )  /\  O  e.  A )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
) ) ) ) )
42 simpl1 960 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
43 simpl21 1035 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
44 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  -> 
( N  e.  A  /\  O  =  ( 0. `  K ) ) )
45 simpl22 1036 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  -> 
( F  e.  T  /\  G  e.  T
) )
46 simpl23 1037 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  ->  P  =/=  Q )
47 simpl31 1038 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  -> 
v  =/=  ( R `
 F ) )
48 simpl33 1040 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) )
497, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33c 31202 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  e.  A  /\  O  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
5042, 43, 44, 45, 46, 47, 48, 49syl133anc 1207 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  e.  A  /\  O  =  ( 0. `  K
) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
5150ex 424 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( N  e.  A  /\  O  =  ( 0. `  K ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) ) )
52 simpl1 960 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
53 simpl21 1035 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  (
v  e.  A  /\  v  .<_  W ) )
54 simpr 448 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  ( N  =  ( 0. `  K )  /\  O  =  ( 0. `  K ) ) )
55 simpl22 1036 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  ( F  e.  T  /\  G  e.  T )
)
56 simpl23 1037 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  P  =/=  Q )
57 simpl31 1038 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  v  =/=  ( R `  F
) )
58 simpl33 1040 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) )
597, 8, 9, 10, 11, 12, 13, 14, 19cdlemg33e 31204 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( N  =  ( 0. `  K )  /\  O  =  ( 0. `  K ) )  /\  ( F  e.  T  /\  G  e.  T
) )  /\  ( P  =/=  Q  /\  v  =/=  ( R `  F
)  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
6052, 53, 54, 55, 56, 57, 58, 59syl133anc 1207 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  /\  ( N  =  ( 0. `  K
)  /\  O  =  ( 0. `  K ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
) ) ) )
6160ex 424 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( N  =  ( 0. `  K )  /\  O  =  ( 0. `  K ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) ) )
6231, 41, 51, 61ccased 914 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  ( ( ( N  e.  A  \/  N  =  ( 0. `  K ) )  /\  ( O  e.  A  \/  O  =  ( 0. `  K ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  (
z  =/=  N  /\  z  =/=  O  /\  z  .<_  ( P  .\/  v
) ) ) ) )
6316, 21, 62mp2and 661 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  ( F  e.  T  /\  G  e.  T )  /\  P  =/=  Q
)  /\  ( v  =/=  ( R `  F
)  /\  v  =/=  ( R `  G )  /\  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
.\/  r )  =  ( Q  .\/  r
) ) ) )  ->  E. z  e.  A  ( -.  z  .<_  W  /\  ( z  =/= 
N  /\  z  =/=  O  /\  z  .<_  ( P 
.\/  v ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   lecple 13499   joincjn 14364   meetcmee 14365   0.cp0 14429   Atomscatm 29758   HLchlt 29845   LHypclh 30478   LTrncltrn 30595   trLctrl 30652
This theorem is referenced by:  cdlemg34  31206
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-map 6987  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653
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