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Theorem cdlemk11t 36143
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 1107 . . 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 1108 . . 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 35762 . . 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 661 . 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 1683 . . 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 2629 . . . . . 6  |-  F/_ b G
11 cdlemk5.x . . . . . . 7  |-  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 2848 . . . . . . . 8  |-  F/ b A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )
13 nfcv 2629 . . . . . . . 8  |-  F/_ b T
1412, 13nfriota 6265 . . . . . . 7  |-  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 2627 . . . . . 6  |-  F/_ b X
1610, 15nfcsb 3458 . . . . 5  |-  F/_ b [_ G  /  g ]_ X
17 nfcv 2629 . . . . 5  |-  F/_ b P
1816, 17nffv 5879 . . . 4  |-  F/_ b
( [_ G  /  g ]_ X `  P )
19 nfcv 2629 . . . 4  |-  F/_ b  .<_
20 nfcv 2629 . . . . . . 7  |-  F/_ b
I
2120, 15nfcsb 3458 . . . . . 6  |-  F/_ b [_ I  /  g ]_ X
2221, 17nffv 5879 . . . . 5  |-  F/_ b
( [_ I  /  g ]_ X `  P )
23 nfcv 2629 . . . . 5  |-  F/_ b  .\/
24 nfcv 2629 . . . . 5  |-  F/_ b
( R `  (
I  o.  `' G
) )
2522, 23, 24nfov 6318 . . . 4  |-  F/_ b
( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) )
2618, 19, 25nfbr 4497 . . 3  |-  F/ b ( [_ G  / 
g ]_ X `  P
)  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) )
27 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 ) )  /\  ( 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 ) ) ) )
28 simp12 1027 . . . . 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 ) ) )
29 simp2 997 . . . . 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
)
30 simp3l 1024 . . . . . 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 ) )
31 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 ) )  /\  ( 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 )
)
32 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 ) )  /\  ( 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 )
)
3330, 31, 323jca 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 ) )  /\  ( 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 ) ) )
34 simp13l 1111 . . . . . 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
)
35 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 ) )  /\  ( 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 ) )
36 simp3r3 1106 . . . . . 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 )
)
3734, 35, 363jca 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 ) )  /\  ( 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 )
) )
38 cdlemk5.l . . . . . 6  |-  .<_  =  ( le `  K )
39 cdlemk5.j . . . . . 6  |-  .\/  =  ( join `  K )
40 cdlemk5.m . . . . . 6  |-  ./\  =  ( meet `  K )
41 cdlemk5.a . . . . . 6  |-  A  =  ( Atoms `  K )
42 cdlemk5.z . . . . . 6  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
43 cdlemk5.y . . . . . 6  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
443, 38, 39, 40, 41, 4, 5, 6, 42, 43, 11cdlemk11tc 36142 . . . . 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 ) ) ) )
4527, 28, 29, 33, 37, 44syl113anc 1240 . . . 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 ) ) ) )
46453exp 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 ) )  /\  ( 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 ) ) ) ) ) )
479, 26, 46rexlimd 2951 . 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 ) ) ) ) )
488, 47mpd 15 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   [_csb 3440   class class class wbr 4453    _I cid 4796   `'ccnv 5004    |` cres 5007    o. ccom 5009   ` cfv 5594   iota_crio 6255  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Atomscatm 34461   HLchlt 34548   LHypclh 35181   LTrncltrn 35298   trLctrl 35355
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-riotaBAD 34157
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-undef 7014  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-llines 34695  df-lplanes 34696  df-lvols 34697  df-lines 34698  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356
This theorem is referenced by:  cdlemk45  36144
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