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Theorem cdlemk11t 31135
Description: Part of proof of Lemma K of [Crawley] p. 118. Eq. 5, line 36, p. 119.  G,  I stand for g, h.  X represents tau. (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 ) ) ) )
cdlemk5.x  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
Assertion
Ref Expression
cdlemk11t  |-  ( ( ( ( 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ G  /  g ]_ X `  P ) 
.<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) ) )
Distinct variable groups:    ./\ , g    .\/ , g    B, g    P, g    R, g    T, g    g, Z    g, b, G, z    ./\ , b, z    .<_ , b    z,
g,  .<_    .\/ , b, z    A, b, g, z    B, b, z    F, b, g, z   
z, G    H, b,
g, z    K, b,
g, z    N, b,
g, z    P, b,
z    R, b, z    T, b, z    W, b, g, z    z, Y    G, b    I, b, g, z
Allowed substitution hints:    X( z, g, b)    Y( g, b)    Z( z, b)

Proof of Theorem cdlemk11t
StepHypRef Expression
1 simp11l 1066 . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  K  e.  HL )
2 simp11r 1067 . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  W  e.  H )
3 cdlemk5.b . . . 4  |-  B  =  ( Base `  K
)
4 cdlemk5.h . . . 4  |-  H  =  ( LHyp `  K
)
5 cdlemk5.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
6 cdlemk5.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
73, 4, 5, 6cdlemftr3 30754 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G )  /\  ( R `  b )  =/=  ( R `  I ) ) ) )
81, 2, 7syl2anc 642 . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( ( R `
 b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b
)  =/=  ( R `
 I ) ) ) )
9 nfv 1605 . . 3  |-  F/ b ( ( ( 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )
10 nfcv 2419 . . . . . . 7  |-  F/_ b G
11 cdlemk5.x . . . . . . . 8  |-  X  =  ( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
12 nfra1 2593 . . . . . . . . 9  |-  F/ b A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )
13 nfcv 2419 . . . . . . . . 9  |-  F/_ b T
1412, 13nfriota 6314 . . . . . . . 8  |-  F/_ b
( iota_ z  e.  T A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
1511, 14nfcxfr 2416 . . . . . . 7  |-  F/_ b X
1610, 15nfcsb 3115 . . . . . 6  |-  F/_ b [_ G  /  g ]_ X
17 nfcv 2419 . . . . . 6  |-  F/_ b P
1816, 17nffv 5532 . . . . 5  |-  F/_ b
( [_ G  /  g ]_ X `  P )
19 nfcv 2419 . . . . 5  |-  F/_ b  .<_
20 nfcv 2419 . . . . . . . 8  |-  F/_ b
I
2120, 15nfcsb 3115 . . . . . . 7  |-  F/_ b [_ I  /  g ]_ X
2221, 17nffv 5532 . . . . . 6  |-  F/_ b
( [_ I  /  g ]_ X `  P )
23 nfcv 2419 . . . . . 6  |-  F/_ b  .\/
24 nfcv 2419 . . . . . 6  |-  F/_ b
( R `  (
I  o.  `' G
) )
2522, 23, 24nfov 5881 . . . . 5  |-  F/_ b
( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) )
2618, 19, 25nfbr 4067 . . . 4  |-  F/ b ( [_ G  / 
g ]_ X `  P
)  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) )
2726a1i 10 . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  F/ b ( [_ G  /  g ]_ X `  P )  .<_  ( (
[_ I  /  g ]_ X `  P ) 
.\/  ( R `  ( I  o.  `' G ) ) ) )
28 simp11 985 . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) ) )
29 simp12 986 . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( N  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )
30 simp2 956 . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  b  e.  T
)
31 simp3l 983 . . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  b  =/=  (  _I  |`  B ) )
32 simp3r1 1063 . . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( R `  b )  =/=  ( R `  F )
)
33 simp3r2 1064 . . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( R `  b )  =/=  ( R `  G )
)
3431, 32, 333jca 1132 . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G ) ) )
35 simp13l 1070 . . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  I  e.  T
)
36 simp13r 1071 . . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  I  =/=  (  _I  |`  B ) )
37 simp3r3 1065 . . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( R `  b )  =/=  ( R `  I )
)
3835, 36, 373jca 1132 . . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( I  e.  T  /\  I  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  I )
) )
39 cdlemk5.l . . . . . 6  |-  .<_  =  ( le `  K )
40 cdlemk5.j . . . . . 6  |-  .\/  =  ( join `  K )
41 cdlemk5.m . . . . . 6  |-  ./\  =  ( meet `  K )
42 cdlemk5.a . . . . . 6  |-  A  =  ( Atoms `  K )
43 cdlemk5.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
44 cdlemk5.y . . . . . 6  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
453, 39, 40, 41, 42, 4, 5, 6, 43, 44, 11cdlemk11tc 31134 . . . . 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 )
) ) )  -> 
( [_ G  /  g ]_ X `  P ) 
.<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) ) )
4628, 29, 30, 34, 38, 45syl113anc 1194 . . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  /\  b  e.  T  /\  (
b  =/=  (  _I  |`  B )  /\  (
( R `  b
)  =/=  ( R `
 F )  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b )  =/=  ( R `  I
) ) ) )  ->  ( [_ G  /  g ]_ X `  P )  .<_  ( (
[_ I  /  g ]_ X `  P ) 
.\/  ( R `  ( I  o.  `' G ) ) ) )
47463exp 1150 . . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  (
b  e.  T  -> 
( ( b  =/=  (  _I  |`  B )  /\  ( ( R `
 b )  =/=  ( R `  F
)  /\  ( R `  b )  =/=  ( R `  G )  /\  ( R `  b
)  =/=  ( R `
 I ) ) )  ->  ( [_ G  /  g ]_ X `  P )  .<_  ( (
[_ I  /  g ]_ X `  P ) 
.\/  ( R `  ( I  o.  `' G ) ) ) ) ) )
489, 27, 47rexlimd2 2665 . 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( E. b  e.  T  ( b  =/=  (  _I  |`  B )  /\  ( ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 G )  /\  ( R `  b )  =/=  ( R `  I ) ) )  ->  ( [_ G  /  g ]_ X `  P )  .<_  ( (
[_ I  /  g ]_ X `  P ) 
.\/  ( R `  ( I  o.  `' G ) ) ) ) )
498, 48mpd 14 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 ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )  ->  ( [_ G  /  g ]_ X `  P ) 
.<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1531    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   [_csb 3081   class class class wbr 4023    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemk45  31136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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