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Theorem cdlemk33N 35705
Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. TODO: not needed, is embodied in cdlemk34 35706. (Contributed by NM, 18-Jul-2013.) (New usage is discouraged.)
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 ) ) ) ) ) )
cdlemk3.x  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  ( b Y G ) ) )
Assertion
Ref Expression
cdlemk33N  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  ( ( b Y G ) `  P ) ) ) )
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, z    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    z,  .<_    z, A    z, H    z, K    z, N    z, P
Allowed substitution hints:    A( e, f, d)    B( e, f, i, j, d)    S( z, f, i)    H( e, f, d)    .\/ ( z, b)    K( e, f, d)    .<_ ( f, d)    ./\ ( z, b)    N( e, d)    X( z, e, f, i, j, b, d)    Y( e, f, i, j, d)

Proof of Theorem cdlemk33N
StepHypRef Expression
1 cdlemk3.x . 2  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  ( b Y G ) ) )
2 fveq1 5863 . . . . . . . . 9  |-  ( z  =  ( b Y G )  ->  (
z `  P )  =  ( ( b Y G ) `  P ) )
3 simpl11 1071 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  K  e.  HL )
4 simpl12 1072 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  W  e.  H )
53, 4jca 532 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simpl31 1077 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  z  e.  T )
7 simp11 1026 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  K  e.  HL )
8 simp12 1027 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  W  e.  H )
97, 8jca 532 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
10 simp13 1028 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  F )  =  ( R `  N ) )
11 simp22l 1115 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  G  e.  T )
129, 10, 113jca 1176 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T
) )
1312adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T ) )
14 simp211 1134 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  F  e.  T )
15 simp32 1033 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  b  e.  T )
16 simp213 1136 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  N  e.  T )
1714, 15, 163jca 1176 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( F  e.  T  /\  b  e.  T  /\  N  e.  T )
)
1817adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  ( F  e.  T  /\  b  e.  T  /\  N  e.  T ) )
19 simp332 1150 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  b )  =/=  ( R `  F
) )
20 simp333 1151 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( R `  b )  =/=  ( R `  G
) )
2119, 20jca 532 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G ) ) )
22 simp212 1135 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  F  =/=  (  _I  |`  B ) )
23 simp22r 1116 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  G  =/=  (  _I  |`  B ) )
24 simp331 1149 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  b  =/=  (  _I  |`  B ) )
2522, 23, 243jca 1176 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) ) )
26 simp23 1031 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
2721, 25, 263jca 1176 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )
2827adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  ( (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G ) )  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )
29 cdlemk3.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  K
)
30 cdlemk3.l . . . . . . . . . . . . 13  |-  .<_  =  ( le `  K )
31 cdlemk3.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
32 cdlemk3.m . . . . . . . . . . . . 13  |-  ./\  =  ( meet `  K )
33 cdlemk3.a . . . . . . . . . . . . 13  |-  A  =  ( Atoms `  K )
34 cdlemk3.h . . . . . . . . . . . . 13  |-  H  =  ( LHyp `  K
)
35 cdlemk3.t . . . . . . . . . . . . 13  |-  T  =  ( ( LTrn `  K
) `  W )
36 cdlemk3.r . . . . . . . . . . . . 13  |-  R  =  ( ( trL `  K
) `  W )
37 cdlemk3.s . . . . . . . . . . . . 13  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T  ( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
38 cdlemk3.u1 . . . . . . . . . . . . 13  |-  Y  =  ( d  e.  T ,  e  e.  T  |->  ( iota_ j  e.  T  ( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( ( S `  d ) `
 P )  .\/  ( R `  ( e  o.  `' d ) ) ) ) ) )
3929, 30, 31, 32, 33, 34, 35, 36, 37, 38cdlemkuel-3 35694 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  b  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  b  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( b Y G )  e.  T
)
4013, 18, 28, 39syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  ( b Y G )  e.  T
)
41 simpl23 1076 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
42 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  ( z `  P )  =  ( ( b Y G ) `  P ) )
4330, 33, 34, 35cdlemd 35003 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  z  e.  T  /\  (
b Y G )  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  z  =  ( b Y G ) )
445, 6, 40, 41, 42, 43syl311anc 1242 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) ) )  /\  ( z `  P )  =  ( ( b Y G ) `  P ) )  ->  z  =  ( b Y G ) )
4544ex 434 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
( z `  P
)  =  ( ( b Y G ) `
 P )  -> 
z  =  ( b Y G ) ) )
462, 45impbid2 204 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) ) )  ->  (
z  =  ( b Y G )  <->  ( z `  P )  =  ( ( b Y G ) `  P ) ) )
47463expia 1198 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( (
z  e.  T  /\  b  e.  T  /\  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) ) )  -> 
( z  =  ( b Y G )  <-> 
( z `  P
)  =  ( ( b Y G ) `
 P ) ) ) )
48473expd 1213 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( z  e.  T  ->  ( b  e.  T  ->  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  (
z  =  ( b Y G )  <->  ( z `  P )  =  ( ( b Y G ) `  P ) ) ) ) ) )
4948imp31 432 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  /\  z  e.  T )  /\  b  e.  T )  ->  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  (
z  =  ( b Y G )  <->  ( z `  P )  =  ( ( b Y G ) `  P ) ) ) )
5049pm5.74d 247 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  /\  z  e.  T )  /\  b  e.  T )  ->  (
( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  z  =  ( b Y G ) )  <->  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  ( ( b Y G ) `  P ) ) ) )
5150ralbidva 2900 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `
 F )  =  ( R `  N
) )  /\  (
( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  /\  z  e.  T
)  ->  ( A. b  e.  T  (
( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b )  =/=  ( R `  G
) )  ->  z  =  ( b Y G ) )  <->  A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  ( z `  P )  =  ( ( b Y G ) `  P ) ) ) )
5251riotabidva 6260 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( iota_ z  e.  T  A. b  e.  T  ( (
b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )
)  ->  z  =  ( b Y G ) ) )  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  ( ( b Y G ) `  P ) ) ) )
531, 52syl5eq 2520 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  ( R `  F )  =  ( R `  N ) )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B )  /\  N  e.  T
)  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) )  ->  ( z `  P )  =  ( ( b Y G ) `  P ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   class class class wbr 4447    |-> cmpt 4505    _I cid 4790   `'ccnv 4998    |` cres 5001    o. ccom 5003   ` cfv 5586   iota_crio 6242  (class class class)co 6282    |-> cmpt2 6284   Basecbs 14486   lecple 14558   joincjn 15427   meetcmee 15428   Atomscatm 34060   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   trLctrl 34954
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 6574  ax-riotaBAD 33756
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955
This theorem is referenced by: (None)
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