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Theorem cdlemk53a 34918
Description: Lemma for cdlemk53 34920. (Contributed by NM, 26-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
cdlemk53a  |-  ( ( ( ( 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ ( G  o.  I )  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
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 cdlemk53a
StepHypRef Expression
1 simp11l 1099 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  K  e.  HL )
2 simp11r 1100 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  W  e.  H )
31, 2jca 532 . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4 simp12 1019 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )
5 simp13l 1103 . . . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  G  e.  T )
6 simp31 1024 . . . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  I  e.  T )
7 cdlemk5.h . . . . . 6  |-  H  =  ( LHyp `  K
)
8 cdlemk5.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
97, 8ltrnco 34682 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  I  e.  T
)  ->  ( G  o.  I )  e.  T
)
101, 2, 5, 6, 9syl211anc 1225 . . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( G  o.  I )  e.  T )
11 simp33 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  G )  =/=  ( R `  I
) )
12 cdlemk5.b . . . . . 6  |-  B  =  ( Base `  K
)
13 cdlemk5.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
1412, 7, 8, 13trlconid 34688 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( G  e.  T  /\  I  e.  T )  /\  ( R `  G )  =/=  ( R `  I
) )  ->  ( G  o.  I )  =/=  (  _I  |`  B ) )
153, 5, 6, 11, 14syl121anc 1224 . . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( G  o.  I )  =/=  (  _I  |`  B ) )
1610, 15jca 532 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( G  o.  I
)  e.  T  /\  ( G  o.  I
)  =/=  (  _I  |`  B ) ) )
17 simp21 1021 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  N  e.  T )
18 simp22 1022 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
19 simp23 1023 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( R `  F )  =  ( R `  N ) )
20 cdlemk5.l . . . 4  |-  .<_  =  ( le `  K )
21 cdlemk5.j . . . 4  |-  .\/  =  ( join `  K )
22 cdlemk5.m . . . 4  |-  ./\  =  ( meet `  K )
23 cdlemk5.a . . . 4  |-  A  =  ( Atoms `  K )
24 cdlemk5.z . . . 4  |-  Z  =  ( ( P  .\/  ( R `  b ) )  ./\  ( ( N `  P )  .\/  ( R `  (
b  o.  `' F
) ) ) )
25 cdlemk5.y . . . 4  |-  Y  =  ( ( P  .\/  ( R `  g ) )  ./\  ( Z  .\/  ( R `  (
g  o.  `' b ) ) ) )
26 cdlemk5.x . . . 4  |-  X  =  ( iota_ z  e.  T  A. b  e.  T  ( ( b  =/=  (  _I  |`  B )  /\  ( R `  b )  =/=  ( R `  F )  /\  ( R `  b
)  =/=  ( R `
 g ) )  ->  ( z `  P )  =  Y ) )
2712, 20, 21, 22, 23, 7, 8, 13, 24, 25, 26cdlemk35s 34900 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( ( G  o.  I )  e.  T  /\  ( G  o.  I )  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ ( G  o.  I )  /  g ]_ X  e.  T )
283, 4, 16, 17, 18, 19, 27syl132anc 1237 . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ ( G  o.  I )  /  g ]_ X  e.  T )
29 simp13 1020 . . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( G  e.  T  /\  G  =/=  (  _I  |`  B ) ) )
3012, 20, 21, 22, 23, 7, 8, 13, 24, 25, 26cdlemk35s 34900 . . . 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  /  g ]_ X  e.  T )
313, 4, 29, 17, 18, 19, 30syl132anc 1237 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ G  /  g ]_ X  e.  T )
32 simp32 1025 . . . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  I  =/=  (  _I  |`  B ) )
336, 32jca 532 . . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
I  e.  T  /\  I  =/=  (  _I  |`  B ) ) )
3412, 20, 21, 22, 23, 7, 8, 13, 24, 25, 26cdlemk35s 34900 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( F  e.  T  /\  F  =/=  (  _I  |`  B ) )  /\  ( I  e.  T  /\  I  =/=  (  _I  |`  B ) )  /\  N  e.  T )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  ( R `  F )  =  ( R `  N ) ) )  ->  [_ I  /  g ]_ X  e.  T )
353, 4, 33, 17, 18, 19, 34syl132anc 1237 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ I  /  g ]_ X  e.  T )
367, 8ltrnco 34682 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ G  / 
g ]_ X  e.  T  /\  [_ I  /  g ]_ X  e.  T
)  ->  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X )  e.  T
)
371, 2, 31, 35, 36syl211anc 1225 . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
)  e.  T )
3812, 20, 21, 22, 23, 7, 8, 13, 24, 25, 26cdlemk52 34917 . . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  (
( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
) `  P )  =  ( [_ ( G  o.  I )  /  g ]_ X `  P ) )
3938eqcomd 2460 . 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  ( [_ ( G  o.  I
)  /  g ]_ X `  P )  =  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
) )
4020, 23, 7, 8cdlemd 34170 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  [_ ( G  o.  I )  /  g ]_ X  e.  T  /\  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X
)  e.  T )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( [_ ( G  o.  I
)  /  g ]_ X `  P )  =  ( ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) `  P
) )  ->  [_ ( G  o.  I )  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
413, 28, 37, 18, 39, 40syl311anc 1233 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 )  /\  ( R `  G )  =/=  ( R `  I )
) )  ->  [_ ( G  o.  I )  /  g ]_ X  =  ( [_ G  /  g ]_ X  o.  [_ I  /  g ]_ X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   A.wral 2796   [_csb 3390   class class class wbr 4395    _I cid 4734   `'ccnv 4942    |` cres 4945    o. ccom 4947   ` cfv 5521   iota_crio 6155  (class class class)co 6195   Basecbs 14287   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-riotaBAD 32923
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-undef 6897  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122
This theorem is referenced by:  cdlemk53b  34919
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