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Theorem cdlemk28-3 35921
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.)
Hypotheses
Ref Expression
cdlemk3.b  |-  B  =  ( Base `  K
)
cdlemk3.l  |-  .<_  =  ( le `  K )
cdlemk3.j  |-  .\/  =  ( join `  K )
cdlemk3.m  |-  ./\  =  ( meet `  K )
cdlemk3.a  |-  A  =  ( Atoms `  K )
cdlemk3.h  |-  H  =  ( LHyp `  K
)
cdlemk3.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk3.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk3.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk3.u1  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
Assertion
Ref Expression
cdlemk28-3  |-  ( ( ( 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 ) ) )  ->  E. z  e.  T  A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  z  =  ( b Y G ) ) )
Distinct variable groups:    e, d,
f, i,  ./\    .<_ , i    .\/ , d, e, f, i    A, i    j, d, e, f, i, F    G, d,
e, j    i, H    i, K    f, N, i    P, d, e, f, i    R, d, e, f, i    T, d, e, f, i    W, d, e, f, i, b    ./\ , j    .<_ , j    .\/ , j    A, j    j, F   
j, H    j, K    j, N    P, j    R, j   
b, d, S, e, j    T, j    j, W    F, d, e    .<_ , e    f, G, i    .<_ , b    A, b   
z, b, B    F, b, z    G, b, z    H, b    K, b    N, b    P, b    R, b, z    T, b, z    W, b, z    Y, b, z   
z, d, e, f, i, j
Allowed substitution hints:    A( z, e, f, d)    B( e, f, i, j, d)    P( z)    S( z, f, i)    H( z, e, f, d)    .\/ ( z, b)    K( z, e, f, d)    .<_ ( z, f, d)    ./\ ( z, b)    N( z, e, d)    Y( e, f, i, j, d)

Proof of Theorem cdlemk28-3
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 simp1 996 . . 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 ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp21l 1113 . . . 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 ) ) )  ->  F  e.  T )
3 simp21r 1114 . . . 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 ) ) )  ->  F  =/=  (  _I  |`  B ) )
4 simp23 1031 . . . 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 ) ) )  ->  N  e.  T )
52, 3, 43jca 1176 . . 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 ) ) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
) )
6 simp22l 1115 . . . 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 ) ) )  ->  G  e.  T )
7 simp22r 1116 . . . 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 ) ) )  ->  G  =/=  (  _I  |`  B ) )
8 simp3r 1025 . . . 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 ) ) )  ->  ( R `  F )  =  ( R `  N ) )
96, 7, 83jca 1176 . . 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 ) ) )  ->  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )
10 simp3l 1024 . . 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 ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
11 cdlemk3.b . . . 4  |-  B  =  ( Base `  K
)
12 cdlemk3.l . . . 4  |-  .<_  =  ( le `  K )
13 cdlemk3.j . . . 4  |-  .\/  =  ( join `  K )
14 cdlemk3.m . . . 4  |-  ./\  =  ( meet `  K )
15 cdlemk3.a . . . 4  |-  A  =  ( Atoms `  K )
16 cdlemk3.h . . . 4  |-  H  =  ( LHyp `  K
)
17 cdlemk3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
18 cdlemk3.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
19 cdlemk3.s . . . 4  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
20 cdlemk3.u1 . . . 4  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
2111, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk26b-3 35918 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B )  /\  ( R `  F )  =  ( R `  N ) ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( b Y G )  e.  T
) )
221, 5, 9, 10, 21syl31anc 1231 . 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 ) ) )  ->  E. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( b Y G )  e.  T
) )
23 simp11 1026 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2423ad2ant1 1017 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  F  e.  T )
25 simp2l 1022 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  b  e.  T )
26 simp123 1130 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  N  e.  T )
2724, 25, 263jca 1176 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( F  e.  T  /\  b  e.  T  /\  N  e.  T )
)
2863ad2ant1 1017 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  G  e.  T )
29 simp2r 1023 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  a  e.  T )
3028, 29jca 532 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( G  e.  T  /\  a  e.  T )
)
31 simp13l 1111 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
32 simp13r 1112 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
3333ad2ant1 1017 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
34 simp3l1 1101 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  b  =/=  (  _I  |`  B ) )
3532, 33, 343jca 1176 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  (
( R `  F
)  =  ( R `
 N )  /\  F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) ) )
3673ad2ant1 1017 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
37 simp3r1 1104 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  a  =/=  (  _I  |`  B ) )
3836, 37jca 532 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( G  =/=  (  _I  |`  B )  /\  a  =/=  (  _I  |`  B ) ) )
39 simp3r3 1106 . . . . . . 7  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  a )  =/=  ( R `  G
) )
4039necomd 2738 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  G )  =/=  ( R `  a
) )
41 simp3r2 1105 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  a )  =/=  ( R `  F
) )
42 simp3l2 1102 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  b )  =/=  ( R `  F
) )
4340, 41, 423jca 1176 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  (
( R `  G
)  =/=  ( R `
 a )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  F
) ) )
44 simp3l3 1103 . . . . . 6  |-  ( ( ( ( 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  b )  =/=  ( R `  G
) )
4544necomd 2738 . . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  G )  =/=  ( R `  b
) )
4611, 12, 13, 14, 15, 16, 17, 18, 19, 20cdlemk27-3 35920 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T )  /\  ( G  e.  T  /\  a  e.  T
) )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( ( R `
 F )  =  ( R `  N
)  /\  F  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( G  =/=  (  _I  |`  B )  /\  a  =/=  (  _I  |`  B ) ) )  /\  ( ( ( R `  G
)  =/=  ( R `
 a )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  F
) )  /\  ( R `  G )  =/=  ( R `  b
) ) )  -> 
( b Y G )  =  ( a Y G ) )
4723, 27, 30, 31, 35, 38, 43, 45, 46syl332anc 1259 . . . 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  /\  a  e.  T )  /\  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )  ->  (
b Y G )  =  ( a Y G ) )
48473exp 1195 . . 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  /\  a  e.  T
)  ->  ( (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  /\  (
a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F
)  /\  ( R `  a )  =/=  ( R `  G )
) )  ->  (
b Y G )  =  ( a Y G ) ) ) )
4948ralrimivv 2884 . 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 ) ) )  ->  A. b  e.  T  A. a  e.  T  ( (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  /\  (
a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F
)  /\  ( R `  a )  =/=  ( R `  G )
) )  ->  (
b Y G )  =  ( a Y G ) ) )
50 neeq1 2748 . . . . 5  |-  ( b  =  a  ->  (
b  =/=  (  _I  |`  B )  <->  a  =/=  (  _I  |`  B ) ) )
51 fveq2 5866 . . . . . 6  |-  ( b  =  a  ->  ( R `  b )  =  ( R `  a ) )
5251neeq1d 2744 . . . . 5  |-  ( b  =  a  ->  (
( R `  b
)  =/=  ( R `
 F )  <->  ( R `  a )  =/=  ( R `  F )
) )
5351neeq1d 2744 . . . . 5  |-  ( b  =  a  ->  (
( R `  b
)  =/=  ( R `
 G )  <->  ( R `  a )  =/=  ( R `  G )
) )
5450, 52, 533anbi123d 1299 . . . 4  |-  ( b  =  a  ->  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  <->  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) ) )
55 oveq1 6292 . . . 4  |-  ( b  =  a  ->  (
b Y G )  =  ( a Y G ) )
5654, 55reusv3 4655 . . 3  |-  ( E. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  /\  (
b Y G )  e.  T )  -> 
( A. b  e.  T  A. a  e.  T  ( ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) )  ->  ( b Y G )  =  ( a Y G ) )  <->  E. z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  ( b Y G ) ) ) )
5756biimpd 207 . 2  |-  ( E. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  /\  (
b Y G )  e.  T )  -> 
( A. b  e.  T  A. a  e.  T  ( ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( a  =/=  (  _I  |`  B )  /\  ( R `  a )  =/=  ( R `  F )  /\  ( R `  a
)  =/=  ( R `
 G ) ) )  ->  ( b Y G )  =  ( a Y G ) )  ->  E. z  e.  T  A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  z  =  ( b Y G ) ) ) )
5822, 49, 57sylc 60 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 ) ) )  ->  E. z  e.  T  A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  z  =  ( b Y G ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   class class class wbr 4447    |-> cmpt 4505    _I cid 4790   `'ccnv 4998    |` cres 5001    o. ccom 5003   ` cfv 5588   iota_crio 6245  (class class class)co 6285    |-> cmpt2 6287   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   Atomscatm 34277   HLchlt 34364   LHypclh 34997   LTrncltrn 35114   trLctrl 35171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-riotaBAD 33973
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-undef 7003  df-map 7423  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-p1 15530  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-llines 34511  df-lplanes 34512  df-lvols 34513  df-lines 34514  df-psubsp 34516  df-pmap 34517  df-padd 34809  df-lhyp 35001  df-laut 35002  df-ldil 35117  df-ltrn 35118  df-trl 35172
This theorem is referenced by:  cdlemk29-3  35924
  Copyright terms: Public domain W3C validator