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Theorem cdlemk5u 30300
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
Hypotheses
Ref Expression
cdlemk1.b  |-  B  =  ( Base `  K
)
cdlemk1.l  |-  .<_  =  ( le `  K )
cdlemk1.j  |-  .\/  =  ( join `  K )
cdlemk1.m  |-  ./\  =  ( meet `  K )
cdlemk1.a  |-  A  =  ( Atoms `  K )
cdlemk1.h  |-  H  =  ( LHyp `  K
)
cdlemk1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk1.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk1.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk1.o  |-  O  =  ( S `  D
)
Assertion
Ref Expression
cdlemk5u  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' D ) ) ) )
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    D, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i
Allowed substitution hints:    A( f)    B( f, i)    S( f, i)    G( f, i)    H( f)    K( f)    .<_ ( f)    O( f, i)    X( f, i)

Proof of Theorem cdlemk5u
StepHypRef Expression
1 cdlemk1.b . 2  |-  B  =  ( Base `  K
)
2 cdlemk1.l . 2  |-  .<_  =  ( le `  K )
3 simp11l 1071 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  K  e.  HL )
4 hllat 28803 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 17 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  K  e.  Lat )
6 simp22l 1079 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  P  e.  A )
7 simp1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T ) )
8 simp211 1098 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  N  e.  T )
9 simp22 994 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
10 simp23 995 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( R `  F
)  =  ( R `
 N ) )
118, 9, 103jca 1137 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
12 simp3l1 1065 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  F  =/=  (  _I  |`  B ) )
13 simp3l2 1066 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  D  =/=  (  _I  |`  B ) )
14 simp3r1 1068 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( R `  D
)  =/=  ( R `
 F ) )
1512, 13, 143jca 1137 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F )
) )
16 cdlemk1.j . . . . . . 7  |-  .\/  =  ( join `  K )
17 cdlemk1.m . . . . . . 7  |-  ./\  =  ( meet `  K )
18 cdlemk1.a . . . . . . 7  |-  A  =  ( Atoms `  K )
19 cdlemk1.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
20 cdlemk1.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
21 cdlemk1.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
22 cdlemk1.s . . . . . . 7  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
23 cdlemk1.o . . . . . . 7  |-  O  =  ( S `  D
)
241, 2, 16, 17, 18, 19, 20, 21, 22, 23cdlemkoatnle 30290 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( ( O `
 P )  e.  A  /\  -.  ( O `  P )  .<_  W ) )
2524simpld 447 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( O `  P )  e.  A
)
267, 11, 15, 25syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( O `  P
)  e.  A )
271, 16, 18hlatjcl 28806 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  ( O `  P )  e.  A )  -> 
( P  .\/  ( O `  P )
)  e.  B )
283, 6, 26, 27syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( P  .\/  ( O `  P )
)  e.  B )
29 simp11 990 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
30 simp212 1099 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  G  e.  T )
312, 18, 19, 20ltrnat 29579 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  A
)  ->  ( G `  P )  e.  A
)
3229, 30, 6, 31syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( G `  P
)  e.  A )
33 simp13 992 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  D  e.  T )
34 simp3r2 1069 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( R `  G
)  =/=  ( R `
 D ) )
3518, 19, 20, 21trlcocnvat 30163 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  D  e.  T )  /\  ( R `  G )  =/=  ( R `  D
) )  ->  ( R `  ( G  o.  `' D ) )  e.  A )
3629, 30, 33, 34, 35syl121anc 1192 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( R `  ( G  o.  `' D
) )  e.  A
)
371, 16, 18hlatjcl 28806 . . . 4  |-  ( ( K  e.  HL  /\  ( G `  P )  e.  A  /\  ( R `  ( G  o.  `' D ) )  e.  A )  ->  (
( G `  P
)  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )
383, 32, 36, 37syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( G `  P )  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )
391, 17latmcl 14120 . . 3  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( O `
 P ) )  e.  B  /\  (
( G `  P
)  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )  -> 
( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  B )
405, 28, 38, 39syl3anc 1187 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  B )
412, 18, 19, 20ltrnat 29579 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T  /\  P  e.  A
)  ->  ( D `  P )  e.  A
)
4229, 33, 6, 41syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( D `  P
)  e.  A )
431, 18, 19, 20, 21trlnidat 29612 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  D  e.  T  /\  D  =/=  (  _I  |`  B ) )  ->  ( R `  D )  e.  A
)
4429, 33, 13, 43syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( R `  D
)  e.  A )
451, 16, 18hlatjcl 28806 . . . 4  |-  ( ( K  e.  HL  /\  ( D `  P )  e.  A  /\  ( R `  D )  e.  A )  ->  (
( D `  P
)  .\/  ( R `  D ) )  e.  B )
463, 42, 44, 45syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( D `  P )  .\/  ( R `  D )
)  e.  B )
471, 16, 18hlatjcl 28806 . . . 4  |-  ( ( K  e.  HL  /\  ( D `  P )  e.  A  /\  ( R `  ( G  o.  `' D ) )  e.  A )  ->  (
( D `  P
)  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )
483, 42, 36, 47syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( D `  P )  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )
491, 17latmcl 14120 . . 3  |-  ( ( K  e.  Lat  /\  ( ( D `  P )  .\/  ( R `  D )
)  e.  B  /\  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B )  -> 
( ( ( D `
 P )  .\/  ( R `  D ) )  ./\  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  B )
505, 46, 48, 49syl3anc 1187 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( ( D `
 P )  .\/  ( R `  D ) )  ./\  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  e.  B )
51 simp213 1100 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  X  e.  T )
522, 18, 19, 20ltrnat 29579 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  P  e.  A
)  ->  ( X `  P )  e.  A
)
5329, 51, 6, 52syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( X `  P
)  e.  A )
54 simp3r3 1070 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( R `  X
)  =/=  ( R `
 D ) )
5518, 19, 20, 21trlcocnvat 30163 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  T  /\  D  e.  T )  /\  ( R `  X )  =/=  ( R `  D
) )  ->  ( R `  ( X  o.  `' D ) )  e.  A )
5629, 51, 33, 54, 55syl121anc 1192 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( R `  ( X  o.  `' D
) )  e.  A
)
571, 16, 18hlatjcl 28806 . . 3  |-  ( ( K  e.  HL  /\  ( X `  P )  e.  A  /\  ( R `  ( X  o.  `' D ) )  e.  A )  ->  (
( X `  P
)  .\/  ( R `  ( X  o.  `' D ) ) )  e.  B )
583, 53, 56, 57syl3anc 1187 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( X `  P )  .\/  ( R `  ( X  o.  `' D ) ) )  e.  B )
591, 2, 16, 17, 18, 19, 20, 21, 22, 23cdlemk1u 30298 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  ( R `  D )  =/=  ( R `  F ) ) )  ->  ( P  .\/  ( O `  P ) )  .<_  ( ( D `  P )  .\/  ( R `  D
) ) )
607, 11, 15, 59syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( P  .\/  ( O `  P )
)  .<_  ( ( D `
 P )  .\/  ( R `  D ) ) )
611, 2, 17latmlem1 14150 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  ( O `  P ) )  e.  B  /\  ( ( D `  P )  .\/  ( R `  D )
)  e.  B  /\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D ) ) )  e.  B ) )  ->  ( ( P 
.\/  ( O `  P ) )  .<_  ( ( D `  P )  .\/  ( R `  D )
)  ->  ( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  .<_  ( (
( D `  P
)  .\/  ( R `  D ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) ) )
625, 28, 46, 38, 61syl13anc 1189 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( P  .\/  ( O `  P ) )  .<_  ( ( D `  P )  .\/  ( R `  D
) )  ->  (
( P  .\/  ( O `  P )
)  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( ( ( D `
 P )  .\/  ( R `  D ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) ) )
6360, 62mpd 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( ( ( D `
 P )  .\/  ( R `  D ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
64 simp11r 1072 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  W  e.  H )
651, 2, 16, 18, 19, 20, 21cdlemk2 30271 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) )  =  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
663, 64, 33, 30, 9, 65syl221anc 1198 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( G `  P )  .\/  ( R `  ( G  o.  `' D ) ) )  =  ( ( D `
 P )  .\/  ( R `  ( G  o.  `' D ) ) ) )
6766oveq2d 5808 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( ( D `
 P )  .\/  ( R `  D ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) )  =  ( ( ( D `  P ) 
.\/  ( R `  D ) )  ./\  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) ) )
6863, 67breqtrd 4021 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( ( ( D `
 P )  .\/  ( R `  D ) )  ./\  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) )
69 simp3l3 1067 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  ->  G  =/=  (  _I  |`  B ) )
7013, 69jca 520 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
711, 2, 16, 18, 19, 20, 21, 17cdlemk5a 30274 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  (
( R `  G
)  =/=  ( R `
 D )  /\  ( D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
( D `  P
)  .\/  ( R `  D ) )  ./\  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D ) ) ) )  .<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' D
) ) ) )
723, 64, 33, 30, 51, 34, 70, 9, 71syl233anc 1216 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( ( D `
 P )  .\/  ( R `  D ) )  ./\  ( ( D `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' D ) ) ) )
731, 2, 5, 40, 50, 58, 68, 72lattrd 14127 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  D  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  ( ( R `  D )  =/=  ( R `  F )  /\  ( R `  G )  =/=  ( R `  D
)  /\  ( R `  X )  =/=  ( R `  D )
) ) )  -> 
( ( P  .\/  ( O `  P ) )  ./\  ( ( G `  P )  .\/  ( R `  ( G  o.  `' D
) ) ) ) 
.<_  ( ( X `  P )  .\/  ( R `  ( X  o.  `' D ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997    e. cmpt 4051    _I cid 4276   `'ccnv 4660    |` cres 4663    o. ccom 4665   ` cfv 4673  (class class class)co 5792   iota_crio 6263   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   Latclat 14114   Atomscatm 28703   HLchlt 28790   LHypclh 29423   LTrncltrn 29540   trLctrl 29597
This theorem is referenced by:  cdlemk6u  30301
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-llines 28937  df-lplanes 28938  df-lvols 28939  df-lines 28940  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544  df-trl 29598
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