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Theorem cdlemk11ta 35600
Description: Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119.  G,  I stand for g, h. TODO: fix comment. (Contributed by NM, 21-Jul-2013.)
Hypotheses
Ref Expression
cdlemk5.b  |-  B  =  ( Base `  K
)
cdlemk5.l  |-  .<_  =  ( le `  K )
cdlemk5.j  |-  .\/  =  ( join `  K )
cdlemk5.m  |-  ./\  =  ( meet `  K )
cdlemk5.a  |-  A  =  ( Atoms `  K )
cdlemk5.h  |-  H  =  ( LHyp `  K
)
cdlemk5.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk5.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk5.z  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
cdlemk5.y  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
cdlemk5c.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk5a.u2  |-  C  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  b ) `
 P )  .\/  ( R `  ( e  o.  `' b ) ) ) ) ) )
Assertion
Ref Expression
cdlemk11ta  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  [_ G  /  g ]_ Y  .<_  ( [_ I  /  g ]_ Y  .\/  ( R `  (
I  o.  `' G
) ) ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b    g, G, e    f, g, i, j, e,  ./\    .<_ , i, j    .\/ , e, f, i, j    A, i, j    f, F, i, j    e, G, j    i, H, j   
i, K, j    f, N, i, j    P, e, f, i, j    R, e, f, i, j    e,
b, j, S    T, e, f, i, j    e, W, f, i, j    f,
b, i    e, F    e, I, g, j
Allowed substitution hints:    A( e, f, g, b)    B( e, f, i, j, b)    C( e, f, g, i, j, b)    P( b)    R( b)    S( f, g, i)    T( b)    F( g, b)    G( f, i, b)    H( e, f, g, b)    I( f, i, b)    .\/ ( b)    K( e, f, g, b)    .<_ ( e, f, g, b)    ./\ ( b)    N( e, g, b)    W( g, b)    Y( e, f, g, i, j, b)    Z( e, f, i, j, b)

Proof of Theorem cdlemk11ta
StepHypRef Expression
1 simp11 1021 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
2 simp12l 1104 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  F  e.  T )
3 simp31 1027 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
b  e.  T )
4 simp21 1024 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  N  e.  T )
5 simp13l 1106 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  G  e.  T )
6 simp331 1144 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  I  e.  T )
74, 5, 63jca 1171 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( N  e.  T  /\  G  e.  T  /\  I  e.  T
) )
8 simp22 1025 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
9 simp23 1026 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( R `  F
)  =  ( R `
 N ) )
10 simp12r 1105 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  F  =/=  (  _I  |`  B ) )
11 simp321 1141 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
b  =/=  (  _I  |`  B ) )
12 simp13r 1107 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  G  =/=  (  _I  |`  B ) )
1310, 11, 123jca 1171 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) ) )
14 simp332 1145 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  I  =/=  (  _I  |`  B ) )
15 simp322 1142 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( R `  b
)  =/=  ( R `
 F ) )
16 simp323 1143 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( R `  b
)  =/=  ( R `
 G ) )
1716necomd 2731 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( R `  G
)  =/=  ( R `
 b ) )
18 simp333 1146 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( R `  b
)  =/=  ( R `
 I ) )
1918necomd 2731 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( R `  I
)  =/=  ( R `
 b ) )
2015, 17, 193jca 1171 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( ( R `  b )  =/=  ( R `  F )  /\  ( R `  G
)  =/=  ( R `
 b )  /\  ( R `  I )  =/=  ( R `  b ) ) )
21 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
22 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
23 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
24 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
25 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
26 cdlemk5.h . . . 4  |-  H  =  ( LHyp `  K
)
27 cdlemk5.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
28 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
29 cdlemk5c.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
30 eqid 2460 . . . 4  |-  ( S `
 b )  =  ( S `  b
)
31 cdlemk5a.u2 . . . 4  |-  C  =  ( e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  b ) `
 P )  .\/  ( R `  ( e  o.  `' b ) ) ) ) ) )
32 eqid 2460 . . . 4  |-  ( ( ( G `  P
)  .\/  ( I `  P ) )  ./\  ( ( R `  ( G  o.  `' b ) )  .\/  ( R `  ( I  o.  `' b ) ) ) )  =  ( ( ( G `
 P )  .\/  ( I `  P
) )  ./\  (
( R `  ( G  o.  `' b
) )  .\/  ( R `  ( I  o.  `' b ) ) ) )
3321, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32cdlemk11u 35542 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  b  e.  T )  /\  (
( N  e.  T  /\  G  e.  T  /\  I  e.  T
)  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B ) )  /\  I  =/=  (  _I  |`  B )  /\  ( ( R `
 b )  =/=  ( R `  F
)  /\  ( R `  G )  =/=  ( R `  b )  /\  ( R `  I
)  =/=  ( R `
 b ) ) ) )  ->  (
( C `  G
) `  P )  .<_  ( ( ( C `
 I ) `  P )  .\/  ( R `  ( I  o.  `' G ) ) ) )
341, 2, 3, 7, 8, 9, 13, 14, 20, 33syl333anc 1255 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( ( C `  G ) `  P
)  .<_  ( ( ( C `  I ) `
 P )  .\/  ( R `  ( I  o.  `' G ) ) ) )
35 simp32 1028 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) )
363, 35jca 532 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )
37 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
38 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
3921, 22, 23, 24, 25, 26, 27, 28, 37, 38, 29, 31cdlemkyuu 35599 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
) ) )  ->  [_ G  /  g ]_ Y  =  (
( C `  G
) `  P )
)
4036, 39syld3an3 1268 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  [_ G  /  g ]_ Y  =  (
( C `  G
) `  P )
)
41 simp12 1022 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
426, 14jca 532 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )
43 simp2 992 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
4411, 15, 183jca 1171 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  I
) ) )
4521, 22, 23, 24, 25, 26, 27, 28, 37, 38, 29, 31cdlemkyuu 35599 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  [_ I  /  g ]_ Y  =  (
( C `  I
) `  P )
)
461, 41, 42, 43, 3, 44, 45syl312anc 1244 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  [_ I  /  g ]_ Y  =  (
( C `  I
) `  P )
)
4746oveq1d 6290 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  -> 
( [_ I  /  g ]_ Y  .\/  ( R `
 ( I  o.  `' G ) ) )  =  ( ( ( C `  I ) `
 P )  .\/  ( R `  ( I  o.  `' G ) ) ) )
4834, 40, 473brtr4d 4470 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )  /\  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) )  /\  ( b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) ) )  ->  [_ G  /  g ]_ Y  .<_  ( [_ I  /  g ]_ Y  .\/  ( R `  (
I  o.  `' G
) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   [_csb 3428   class class class wbr 4440    |-> cmpt 4498    _I cid 4783   `'ccnv 4991    |` cres 4994    o. ccom 4996   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-undef 6992  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830
This theorem is referenced by:  cdlemk11tb  35602
  Copyright terms: Public domain W3C validator