Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemk11u Structured version   Unicode version

Theorem cdlemk11u 34147
Description: Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma1 ( U) case. (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
)
cdlemk1.u  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
cdlemk1.v  |-  V  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' D ) )  .\/  ( R `  ( X  o.  `' D ) ) ) )
Assertion
Ref Expression
cdlemk11u  |-  ( ( ( ( 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  .<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
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    ./\ , e    .\/ , e    D, e, j    e, G, j   
e, O    P, e    R, e    T, e    e, W    ./\ , j    .<_ , j    .\/ , j    A, j    D, j    j, F   
j, H    j, K    j, N    j, O    P, j    R, j    T, j   
j, W    e, F    e, X, j
Allowed substitution hints:    A( e, f)    B( e, f, i, j)    S( e, f, i, j)    U( e, f, i, j)    G( f, i)    H( e, f)    K( e, f)    .<_ ( e, f)    N( e)    O( f, i)    V( e, f, i, j)    X( f, i)

Proof of Theorem cdlemk11u
StepHypRef Expression
1 cdlemk1.b . 2  |-  B  =  ( Base `  K
)
2 cdlemk1.l . 2  |-  .<_  =  ( le `  K )
3 simp11l 1116 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  K  e.  HL )
4 hllat 32638 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  K  e.  Lat )
6 simp11r 1117 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  W  e.  H )
73, 6jca 534 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
8 simp23 1040 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
9 simp212 1144 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  G  e.  T )
10 simp12 1036 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  F  e.  T )
11 simp13 1037 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  D  e.  T )
12 simp211 1143 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  N  e.  T )
13 simp331 1158 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  D )  =/=  ( R `  F
) )
14 simp332 1159 . . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  G )  =/=  ( R `  D
) )
1514necomd 2693 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  D )  =/=  ( R `  G
) )
1613, 15jca 534 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  G ) ) )
17 simp311 1152 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
18 simp313 1154 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
19 simp312 1153 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  D  =/=  (  _I  |`  B ) )
2017, 18, 193jca 1185 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
21 simp22 1039 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
22 cdlemk1.j . . . . 5  |-  .\/  =  ( join `  K )
23 cdlemk1.m . . . . 5  |-  ./\  =  ( meet `  K )
24 cdlemk1.a . . . . 5  |-  A  =  ( Atoms `  K )
25 cdlemk1.h . . . . 5  |-  H  =  ( LHyp `  K
)
26 cdlemk1.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
27 cdlemk1.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
28 cdlemk1.s . . . . 5  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
29 cdlemk1.o . . . . 5  |-  O  =  ( S `  D
)
30 cdlemk1.u . . . . 5  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
311, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30cdlemkuat 34142 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  e.  A
)
327, 8, 9, 10, 11, 12, 16, 20, 21, 31syl333anc 1296 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  e.  A )
331, 24atbase 32564 . . 3  |-  ( ( ( U `  G
) `  P )  e.  A  ->  ( ( U `  G ) `
 P )  e.  B )
3432, 33syl 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  e.  B )
35 simp213 1145 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  X  e.  T )
36 simp333 1160 . . . . . . 7  |-  ( ( ( ( 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  X )  =/=  ( R `  D
) )
3736necomd 2693 . . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  D )  =/=  ( R `  X
) )
3813, 37jca 534 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( R `  D
)  =/=  ( R `
 F )  /\  ( R `  D )  =/=  ( R `  X ) ) )
39 simp32 1042 . . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  X  =/=  (  _I  |`  B ) )
4017, 39, 193jca 1185 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) ) )
411, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30cdlemkuat 34142 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  X  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  X )
)  /\  ( F  =/=  (  _I  |`  B )  /\  X  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  X ) `  P )  e.  A
)
427, 8, 35, 10, 11, 12, 38, 40, 21, 41syl333anc 1296 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  X
) `  P )  e.  A )
431, 24atbase 32564 . . . 4  |-  ( ( ( U `  X
) `  P )  e.  A  ->  ( ( U `  X ) `
 P )  e.  B )
4442, 43syl 17 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  X
) `  P )  e.  B )
45 simp22l 1124 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  P  e.  A )
46 cdlemk1.v . . . . 5  |-  V  =  ( ( ( G `
 P )  .\/  ( X `  P ) )  ./\  ( ( R `  ( G  o.  `' D ) )  .\/  ( R `  ( X  o.  `' D ) ) ) )
471, 2, 22, 24, 25, 26, 27, 23, 46cdlemkvcl 34118 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  P  e.  A )  ->  V  e.  B )
483, 6, 11, 9, 35, 45, 47syl231anc 1284 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  V  e.  B )
491, 22latjcl 16241 . . 3  |-  ( ( K  e.  Lat  /\  ( ( U `  X ) `  P
)  e.  B  /\  V  e.  B )  ->  ( ( ( U `
 X ) `  P )  .\/  V
)  e.  B )
505, 44, 48, 49syl3anc 1264 . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( ( U `  X ) `  P
)  .\/  V )  e.  B )
5125, 26ltrncnv 33420 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  `' G  e.  T )
527, 9, 51syl2anc 665 . . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  `' G  e.  T )
5325, 26ltrnco 33995 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  T  /\  `' G  e.  T
)  ->  ( X  o.  `' G )  e.  T
)
547, 35, 52, 53syl3anc 1264 . . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( X  o.  `' G
)  e.  T )
551, 25, 26, 27trlcl 33439 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  o.  `' G )  e.  T
)  ->  ( R `  ( X  o.  `' G ) )  e.  B )
567, 54, 55syl2anc 665 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( R `  ( X  o.  `' G ) )  e.  B )
571, 22latjcl 16241 . . 3  |-  ( ( K  e.  Lat  /\  ( ( U `  X ) `  P
)  e.  B  /\  ( R `  ( X  o.  `' G ) )  e.  B )  ->  ( ( ( U `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) )  e.  B
)
585, 44, 56, 57syl3anc 1264 . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( ( U `  X ) `  P
)  .\/  ( R `  ( X  o.  `' G ) ) )  e.  B )
591, 2, 22, 23, 24, 25, 26, 27, 28, 29, 30, 46cdlemk7u 34146 . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  .<_  ( ( ( U `
 X ) `  P )  .\/  V
) )
601, 2, 22, 24, 25, 26, 27, 23, 46cdlemk10 34119 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( D  e.  T  /\  G  e.  T  /\  X  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  V  .<_  ( R `  ( X  o.  `' G ) ) )
613, 6, 11, 9, 35, 21, 60syl231anc 1284 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  V  .<_  ( R `  ( X  o.  `' G
) ) )
621, 2, 22latjlej2 16256 . . . 4  |-  ( ( K  e.  Lat  /\  ( V  e.  B  /\  ( R `  ( X  o.  `' G
) )  e.  B  /\  ( ( U `  X ) `  P
)  e.  B ) )  ->  ( V  .<_  ( R `  ( X  o.  `' G
) )  ->  (
( ( U `  X ) `  P
)  .\/  V )  .<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
635, 48, 56, 44, 62syl13anc 1266 . . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  ( V  .<_  ( R `  ( X  o.  `' G ) )  -> 
( ( ( U `
 X ) `  P )  .\/  V
)  .<_  ( ( ( U `  X ) `
 P )  .\/  ( R `  ( X  o.  `' G ) ) ) ) )
6461, 63mpd 15 . 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( ( U `  X ) `  P
)  .\/  V )  .<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
651, 2, 5, 34, 50, 58, 59, 64lattrd 16248 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 ) )  /\  X  =/=  (  _I  |`  B )  /\  ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  D )  /\  ( R `  X
)  =/=  ( R `
 D ) ) ) )  ->  (
( U `  G
) `  P )  .<_  ( ( ( U `
 X ) `  P )  .\/  ( R `  ( X  o.  `' G ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   class class class wbr 4417    |-> cmpt 4475    _I cid 4755   `'ccnv 4844    |` cres 4847    o. ccom 4849   ` cfv 5592   iota_crio 6257  (class class class)co 6296   Basecbs 15073   lecple 15149   joincjn 16133   meetcmee 16134   Latclat 16235   Atomscatm 32538   HLchlt 32625   LHypclh 33258   LTrncltrn 33375   trLctrl 33433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-riotaBAD 32234
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-undef 7019  df-map 7473  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-p1 16230  df-lat 16236  df-clat 16298  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434
This theorem is referenced by:  cdlemk12u  34148  cdlemk11u-2N  34165  cdlemk11ta  34205
  Copyright terms: Public domain W3C validator