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Theorem cdleme48d 34102
Description: TODO: fix comment. (Contributed by NM, 8-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46g.b  |-  B  =  ( Base `  K
)
cdlemef46g.l  |-  .<_  =  ( le `  K )
cdlemef46g.j  |-  .\/  =  ( join `  K )
cdlemef46g.m  |-  ./\  =  ( meet `  K )
cdlemef46g.a  |-  A  =  ( Atoms `  K )
cdlemef46g.h  |-  H  =  ( LHyp `  K
)
cdlemef46g.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdlemef46g.d  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
cdlemefs46g.e  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
cdlemef46g.f  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
cdlemef46.v  |-  V  =  ( ( Q  .\/  P )  ./\  W )
cdlemef46.n  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
cdlemefs46.o  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
cdlemef46.g  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B  A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
Assertion
Ref Expression
cdleme48d  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  ( F `  X
) )  =  X )
Distinct variable groups:    t, s, x, y, z, A    B, s, t, x, y, z    D, s, x, y, z   
x, E, y, z    H, s, t, x, y, z    .\/ , s, t, x, y, z    K, s, t, x, y, z    .<_ , s, t, x, y, z    ./\ , s, t, x, y, z    P, s, t, x, y, z    Q, s, t, x, y, z    U, s, t, x, y, z    W, s, t, x, y, z    S, s, t, x, y, z    a, b, c, u, v, A    B, a, b, c, u, v   
v, D    G, s,
t, x, y, z    H, a, b, c, u, v    .\/ , a, b, c, u, v    K, a, b, c, u, v    .<_ , a, b, c, u, v    ./\ , a, b, c, u, v    N, a, b, c    O, a, b, c    P, a, b, c, u, v    Q, a, b, c, u, v    S, a, b, c, u, v    V, a, b, c    W, a, b, c, u, v, x, y, z    u, N, x, y, z    x, O, y, z    v, t   
u, V    x, v,
y, z, V    D, a, b, c    E, a, b, c    F, a, b, c, u, v   
t, N    U, a,
b, c, v    t, V    s, a, t, b, c, x, y, z, u, v    X, a, c, s, t, u, v, x, z
Allowed substitution hints:    D( u, t)    U( u)    E( v, u, t, s)    F( x, y, z, t, s)    G( v, u, a, b, c)    N( v, s)    O( v, u, t, s)    V( s)    X( y, b)

Proof of Theorem cdleme48d
StepHypRef Expression
1 simp1 1008 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )
2 simp2l 1034 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  P  =/=  Q )
3 simp2rl 1077 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
4 cdlemef46g.b . . . . . 6  |-  B  =  ( Base `  K
)
5 cdlemef46g.l . . . . . 6  |-  .<_  =  ( le `  K )
6 cdlemef46g.j . . . . . 6  |-  .\/  =  ( join `  K )
7 cdlemef46g.m . . . . . 6  |-  ./\  =  ( meet `  K )
8 cdlemef46g.a . . . . . 6  |-  A  =  ( Atoms `  K )
9 cdlemef46g.h . . . . . 6  |-  H  =  ( LHyp `  K
)
10 cdlemef46g.u . . . . . 6  |-  U  =  ( ( P  .\/  Q )  ./\  W )
11 vex 3048 . . . . . . 7  |-  s  e. 
_V
12 cdlemef46g.d . . . . . . . 8  |-  D  =  ( ( t  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  t )  ./\  W
) ) )
13 eqid 2451 . . . . . . . 8  |-  ( ( s  .\/  U ) 
./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )  =  ( ( s  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s )  ./\  W
) ) )
1412, 13cdleme31sc 33951 . . . . . . 7  |-  ( s  e.  _V  ->  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) ) )
1511, 14ax-mp 5 . . . . . 6  |-  [_ s  /  t ]_ D  =  ( ( s 
.\/  U )  ./\  ( Q  .\/  ( ( P  .\/  s ) 
./\  W ) ) )
16 cdlemefs46g.e . . . . . 6  |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( s  .\/  t )  ./\  W
) ) )
17 eqid 2451 . . . . . 6  |-  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) )  =  (
iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) )
18 eqid 2451 . . . . . 6  |-  if ( s  .<_  ( P  .\/  Q ) ,  (
iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )  =  if ( s  .<_  ( P 
.\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P  .\/  Q
) )  ->  y  =  E ) ) , 
[_ s  /  t ]_ D )
19 eqid 2451 . . . . . 6  |-  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )  =  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s  .\/  (
x  ./\  W )
)  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) )
20 cdlemef46g.f . . . . . 6  |-  F  =  ( x  e.  B  |->  if ( ( P  =/=  Q  /\  -.  x  .<_  W ) ,  ( iota_ z  e.  B  A. s  e.  A  ( ( -.  s  .<_  W  /\  ( s 
.\/  ( x  ./\  W ) )  =  x )  ->  z  =  ( if ( s  .<_  ( P  .\/  Q ) ,  ( iota_ y  e.  B  A. t  e.  A  ( ( -.  t  .<_  W  /\  -.  t  .<_  ( P 
.\/  Q ) )  ->  y  =  E ) ) ,  [_ s  /  t ]_ D
)  .\/  ( x  ./\ 
W ) ) ) ) ,  x ) )
214, 5, 6, 7, 8, 9, 10, 15, 12, 16, 17, 18, 19, 20cdleme32fvcl 34007 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  X  e.  B )  ->  ( F `  X
)  e.  B )
221, 3, 21syl2anc 667 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  e.  B
)
234, 5, 6, 7, 8, 9, 10, 12, 16, 20cdleme48bw 34069 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  ( F `  X )  .<_  W )
2422, 23jca 535 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( F `  X )  e.  B  /\  -.  ( F `  X )  .<_  W ) )
25 simp3l 1036 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
264, 5, 6, 7, 8, 9, 10, 12, 16, 20cdleme46fvaw 34068 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  ->  ( ( F `  S )  e.  A  /\  -.  ( F `  S )  .<_  W ) )
271, 25, 26syl2anc 667 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( F `  S )  e.  A  /\  -.  ( F `  S )  .<_  W ) )
284, 5, 6, 7, 8, 9, 10, 12, 16, 20cdleme48b 34070 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( F `  X )  ./\  W )  =  ( X  ./\  W )
)
2928oveq2d 6306 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( F `  S )  .\/  ( ( F `  X )  ./\  W
) )  =  ( ( F `  S
)  .\/  ( X  ./\ 
W ) ) )
304, 5, 6, 7, 8, 9, 10, 12, 16, 20cdleme48fv 34066 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( F `  X )  =  ( ( F `  S
)  .\/  ( X  ./\ 
W ) ) )
3129, 30eqtr4d 2488 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( F `  S )  .\/  ( ( F `  X )  ./\  W
) )  =  ( F `  X ) )
32 cdlemef46.v . . . 4  |-  V  =  ( ( Q  .\/  P )  ./\  W )
33 cdlemef46.n . . . 4  |-  N  =  ( ( v  .\/  V )  ./\  ( P  .\/  ( ( Q  .\/  v )  ./\  W
) ) )
34 cdlemefs46.o . . . 4  |-  O  =  ( ( Q  .\/  P )  ./\  ( N  .\/  ( ( u  .\/  v )  ./\  W
) ) )
35 cdlemef46.g . . . 4  |-  G  =  ( a  e.  B  |->  if ( ( Q  =/=  P  /\  -.  a  .<_  W ) ,  ( iota_ c  e.  B  A. u  e.  A  ( ( -.  u  .<_  W  /\  ( u 
.\/  ( a  ./\  W ) )  =  a )  ->  c  =  ( if ( u  .<_  ( Q  .\/  P ) ,  ( iota_ b  e.  B  A. v  e.  A  ( ( -.  v  .<_  W  /\  -.  v  .<_  ( Q 
.\/  P ) )  ->  b  =  O ) ) ,  [_ u  /  v ]_ N
)  .\/  ( a  ./\  W ) ) ) ) ,  a ) )
364, 5, 6, 7, 8, 9, 32, 33, 34, 35cdleme4gfv 34074 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( ( F `  X )  e.  B  /\  -.  ( F `  X )  .<_  W ) )  /\  ( ( ( F `  S
)  e.  A  /\  -.  ( F `  S
)  .<_  W )  /\  ( ( F `  S )  .\/  (
( F `  X
)  ./\  W )
)  =  ( F `
 X ) ) )  ->  ( G `  ( F `  X
) )  =  ( ( G `  ( F `  S )
)  .\/  ( ( F `  X )  ./\  W ) ) )
371, 2, 24, 27, 31, 36syl122anc 1277 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  ( F `  X
) )  =  ( ( G `  ( F `  S )
)  .\/  ( ( F `  X )  ./\  W ) ) )
384, 5, 6, 7, 8, 9, 10, 12, 16, 20, 32, 33, 34, 35cdlemeg46gf 34100 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) )  ->  ( G `  ( F `  S ) )  =  S )
391, 2, 25, 38syl12anc 1266 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  ( F `  S
) )  =  S )
4039, 28oveq12d 6308 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( G `  ( F `  S ) )  .\/  ( ( F `  X )  ./\  W
) )  =  ( S  .\/  ( X 
./\  W ) ) )
41 simp3r 1037 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( S  .\/  ( X  ./\  W
) )  =  X )
4237, 40, 413eqtrd 2489 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( P  =/=  Q  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( S  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( G `  ( F `  X
) )  =  X )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   _Vcvv 3045   [_csb 3363   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   ` cfv 5582   iota_crio 6251  (class class class)co 6290   Basecbs 15121   lecple 15197   joincjn 16189   meetcmee 16190   Atomscatm 32829   HLchlt 32916   LHypclh 33549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-riotaBAD 32525
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-undef 7020  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32742  df-ol 32744  df-oml 32745  df-covers 32832  df-ats 32833  df-atl 32864  df-cvlat 32888  df-hlat 32917  df-llines 33063  df-lplanes 33064  df-lvols 33065  df-lines 33066  df-psubsp 33068  df-pmap 33069  df-padd 33361  df-lhyp 33553
This theorem is referenced by:  cdleme48gfv1  34103
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