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Theorem cdlemg33 31582
Description: Combine cdlemg33b 31578, cdlemg33c 31579, cdlemg33d 31580, cdlemg33e 31581. 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 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
) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp12 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
) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
3 simp13 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
) ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4 simp21 991 . . 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 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
) ) ) )  ->  F  e.  T
)
6 simp31 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 `  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 31566 . . 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 1203 . 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 1078 . . 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 995 . . 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 31566 . . 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 1203 . 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 961 . . . . 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 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 ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
24 simpr 449 . . . . 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 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 ) )  -> 
( F  e.  T  /\  G  e.  T
) )
26 simpl23 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 ) )  ->  P  =/=  Q )
27 simpl31 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  e.  A  /\  O  e.  A ) )  -> 
v  =/=  ( R `
 F ) )
28 simpl33 1041 . . . . 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 31578 . . . . 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 1208 . . . 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 425 . . 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 961 . . . . 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 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 ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
34 simpr 449 . . . . 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 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 ) )  -> 
( F  e.  T  /\  G  e.  T
) )
36 simpl23 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  e.  A ) )  ->  P  =/=  Q )
37 simpl32 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 ) )  -> 
v  =/=  ( R `
 G ) )
38 simpl33 1041 . . . . 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 31580 . . . . 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 1208 . . . 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 425 . . 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 961 . . . . 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 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
) ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
44 simpr 449 . . . . 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 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
) ) )  -> 
( F  e.  T  /\  G  e.  T
) )
46 simpl23 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
) ) )  ->  P  =/=  Q )
47 simpl31 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  e.  A  /\  O  =  ( 0. `  K
) ) )  -> 
v  =/=  ( R `
 F ) )
48 simpl33 1041 . . . . 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 31579 . . . . 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 1208 . . . 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 425 . . 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 961 . . . . 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 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 ) ) )  ->  (
v  e.  A  /\  v  .<_  W ) )
54 simpr 449 . . . . 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 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 ) ) )  ->  ( F  e.  T  /\  G  e.  T )
)
56 simpl23 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 ) ) )  ->  P  =/=  Q )
57 simpl31 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  =  ( 0. `  K ) ) )  ->  v  =/=  ( R `  F
) )
58 simpl33 1041 . . . . 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 31581 . . . . 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 1208 . . . 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 425 . . 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 915 . 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 662 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 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   E.wrex 2708   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   meetcmee 14407   0.cp0 14471   Atomscatm 30135   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029
This theorem is referenced by:  cdlemg34  31583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030
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