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Theorem cdlemk11 31038
Description: Part of proof of Lemma K of [Crawley] p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-Jun-2013.)
Hypotheses
Ref Expression
cdlemk.b  |-  B  =  ( Base `  K
)
cdlemk.l  |-  .<_  =  ( le `  K )
cdlemk.j  |-  .\/  =  ( join `  K )
cdlemk.a  |-  A  =  ( Atoms `  K )
cdlemk.h  |-  H  =  ( LHyp `  K
)
cdlemk.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk.m  |-  ./\  =  ( meet `  K )
cdlemk.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk.v  |-  V  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )
Assertion
Ref Expression
cdlemk11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  .<_  ( ( ( S `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
Distinct variable groups:    ./\ , f    .\/ , f    f, F, i    f, G, i    f, N    P, f    R, f    T, f   
f, W    ./\ , i    .<_ , i    .\/ , i    A, i    i, F   
i, H    i, K    i, N    P, i    R, i    T, i    i, W    f, X, i
Allowed substitution hints:    A( f)    B( f, i)    S( f, i)    H( f)    K( f)    .<_ ( f)    V( f, i)

Proof of Theorem cdlemk11
StepHypRef Expression
1 cdlemk.b . 2  |-  B  =  ( Base `  K
)
2 cdlemk.l . 2  |-  .<_  =  ( le `  K )
3 simp11l 1066 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  K  e.  HL )
4 hllat 29553 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 15 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  K  e.  Lat )
6 simp1 955 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T
) )
7 simp21l 1072 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  N  e.  T )
8 simp22 989 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
9 simp23 990 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  F )  =  ( R `  N ) )
10 simp311 1102 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  F  =/=  (  _I  |`  B ) )
11 simp312 1103 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  G  =/=  (  _I  |`  B ) )
12 simp32 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  G )  =/=  ( R `  F
) )
13 cdlemk.j . . . . 5  |-  .\/  =  ( join `  K )
14 cdlemk.a . . . . 5  |-  A  =  ( Atoms `  K )
15 cdlemk.h . . . . 5  |-  H  =  ( LHyp `  K
)
16 cdlemk.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
17 cdlemk.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
18 cdlemk.m . . . . 5  |-  ./\  =  ( meet `  K )
19 cdlemk.s . . . . 5  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
201, 2, 13, 14, 15, 16, 17, 18, 19cdlemksat 31035 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  ( R `  G )  =/=  ( R `  F ) ) )  ->  ( ( S `
 G ) `  P )  e.  A
)
216, 7, 8, 9, 10, 11, 12, 20syl133anc 1205 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  e.  A )
221, 14atbase 29479 . . 3  |-  ( ( ( S `  G
) `  P )  e.  A  ->  ( ( S `  G ) `
 P )  e.  B )
2321, 22syl 15 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  e.  B )
24 simp11 985 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
25 simp12 986 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  F  e.  T )
26 simp21r 1073 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  X  e.  T )
27 simp313 1104 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  X  =/=  (  _I  |`  B ) )
28 simp33 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  X )  =/=  ( R `  F
) )
291, 2, 13, 14, 15, 16, 17, 18, 19cdlemksat 31035 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  X  e.  T )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  ( R `  X )  =/=  ( R `  F ) ) )  ->  ( ( S `
 X ) `  P )  e.  A
)
3024, 25, 26, 7, 8, 9, 10, 27, 28, 29syl333anc 1214 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  X
) `  P )  e.  A )
311, 14atbase 29479 . . . 4  |-  ( ( ( S `  X
) `  P )  e.  A  ->  ( ( S `  X ) `
 P )  e.  B )
3230, 31syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  X
) `  P )  e.  B )
33 simp11r 1067 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  W  e.  H )
34 simp13 987 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  G  e.  T )
35 simp22l 1074 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  P  e.  A )
36 cdlemk.v . . . . 5  |-  V  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' F ) )  .\/  ( R `  ( X  o.  `' F ) ) ) )
371, 2, 13, 14, 15, 16, 17, 18, 36cdlemkvcl 31031 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  P  e.  A )  ->  V  e.  B )
383, 33, 25, 34, 26, 35, 37syl231anc 1202 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  V  e.  B )
391, 13latjcl 14156 . . 3  |-  ( ( K  e.  Lat  /\  ( ( S `  X ) `  P
)  e.  B  /\  V  e.  B )  ->  ( ( ( S `
 X ) `  P )  .\/  V
)  e.  B )
405, 32, 38, 39syl3anc 1182 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( ( S `  X ) `  P
)  .\/  V )  e.  B )
4115, 16ltrncnv 30335 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  `' G  e.  T )
4224, 34, 41syl2anc 642 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  `' G  e.  T )
4315, 16ltrnco 30908 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  `' G  e.  T
)  ->  ( X  o.  `' G )  e.  T
)
4424, 26, 42, 43syl3anc 1182 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( X  o.  `' G
)  e.  T )
451, 15, 16, 17trlcl 30353 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  o.  `' G )  e.  T
)  ->  ( R `  ( X  o.  `' G ) )  e.  B )
4624, 44, 45syl2anc 642 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( R `  ( X  o.  `' G ) )  e.  B )
471, 13latjcl 14156 . . 3  |-  ( ( K  e.  Lat  /\  ( ( S `  X ) `  P
)  e.  B  /\  ( R `  ( X  o.  `' G ) )  e.  B )  ->  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) )  e.  B
)
485, 32, 46, 47syl3anc 1182 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( ( S `  X ) `  P
)  .\/  ( R `  ( X  o.  `' G ) ) )  e.  B )
491, 2, 13, 14, 15, 16, 17, 18, 19, 36cdlemk7 31037 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  .<_  ( ( ( S `
 X ) `  P )  .\/  V
) )
501, 2, 13, 14, 15, 16, 17, 18, 36cdlemk10 31032 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  V  .<_  ( R `  ( X  o.  `' G ) ) )
513, 33, 25, 34, 26, 8, 50syl231anc 1202 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  V  .<_  ( R `  ( X  o.  `' G
) ) )
521, 2, 13latjlej2 14172 . . . 4  |-  ( ( K  e.  Lat  /\  ( V  e.  B  /\  ( R `  ( X  o.  `' G
) )  e.  B  /\  ( ( S `  X ) `  P
)  e.  B ) )  ->  ( V  .<_  ( R `  ( X  o.  `' G
) )  ->  (
( ( S `  X ) `  P
)  .\/  V )  .<_  ( ( ( S `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
535, 38, 46, 32, 52syl13anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  ( V  .<_  ( R `  ( X  o.  `' G ) )  -> 
( ( ( S `
 X ) `  P )  .\/  V
)  .<_  ( ( ( S `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
5451, 53mpd 14 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( ( S `  X ) `  P
)  .\/  V )  .<_  ( ( ( S `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
551, 2, 5, 23, 40, 48, 49, 54lattrd 14164 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  G  e.  T )  /\  (
( N  e.  T  /\  X  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B ) )  /\  ( R `
 G )  =/=  ( R `  F
)  /\  ( R `  X )  =/=  ( R `  F )
) )  ->  (
( S `  G
) `  P )  .<_  ( ( ( S `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    e. cmpt 4077    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemk12  31039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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