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Theorem cdlemk11t 31582
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 1068 . . 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 1069 . . 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 31201 . . 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 643 . 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 1629 . . 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 2571 . . . . . 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 2748 . . . . . . . 8  |-  F/ b A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y )
13 nfcv 2571 . . . . . . . 8  |-  F/_ b T
1412, 13nfriota 6550 . . . . . . 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 2568 . . . . . 6  |-  F/_ b X
1610, 15nfcsb 3277 . . . . 5  |-  F/_ b [_ G  /  g ]_ X
17 nfcv 2571 . . . . 5  |-  F/_ b P
1816, 17nffv 5726 . . . 4  |-  F/_ b
( [_ G  /  g ]_ X `  P )
19 nfcv 2571 . . . 4  |-  F/_ b  .<_
20 nfcv 2571 . . . . . . 7  |-  F/_ b
I
2120, 15nfcsb 3277 . . . . . 6  |-  F/_ b [_ I  /  g ]_ X
2221, 17nffv 5726 . . . . 5  |-  F/_ b
( [_ I  /  g ]_ X `  P )
23 nfcv 2571 . . . . 5  |-  F/_ b  .\/
24 nfcv 2571 . . . . 5  |-  F/_ b
( R `  (
I  o.  `' G
) )
2522, 23, 24nfov 6095 . . . 4  |-  F/_ b
( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) )
2618, 19, 25nfbr 4248 . . 3  |-  F/ b ( [_ G  / 
g ]_ X `  P
)  .<_  ( ( [_ I  /  g ]_ X `  P )  .\/  ( R `  ( I  o.  `' G ) ) )
27 simp11 987 . . . . 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 988 . . . . 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 958 . . . . 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 985 . . . . . 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 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 `  F )
)
32 simp3r2 1066 . . . . . 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 1134 . . . . 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 1072 . . . . . 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 1073 . . . . . 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 1067 . . . . . 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 1134 . . . . 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 31581 . . . . 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 1196 . . . 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 1152 . . 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 2819 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   [_csb 3243   class class class wbr 4204    _I cid 4485   `'ccnv 4868    |` cres 4871    o. ccom 4873   ` cfv 5445  (class class class)co 6072   iota_crio 6533   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737   trLctrl 30794
This theorem is referenced by:  cdlemk45  31583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795
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