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Theorem heiborlem8 26645
Description: Lemma for heibor 26648. The previous lemmas establish that the sequence  M is Cauchy, so using completeness we now consider the convergent point 
Y. By assumption,  U is an open cover, so  Y is an element of some  Z  e.  U, and some ball centered at  Y is contained in  Z. But the sequence contains arbitrarily small balls close to  Y, so some element  ball ( M `  n ) of the sequence is contained in  Z. And finally we arrive at a contradiction, because  { Z } is a finite subcover of  U that covers  ball ( M `  n ), yet  ball ( M `  n )  e.  K. For convenience, we write this contradiction as 
ph  ->  ps where  ph is all the accumulated hypotheses and  ps is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)
Hypotheses
Ref Expression
heibor.1  |-  J  =  ( MetOpen `  D )
heibor.3  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
heibor.4  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
heibor.5  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
heibor.6  |-  ( ph  ->  D  e.  ( CMet `  X ) )
heibor.7  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
heibor.8  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
heibor.9  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
heibor.10  |-  ( ph  ->  C G 0 )
heibor.11  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
heibor.12  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
heibor.13  |-  ( ph  ->  U  C_  J )
heibor.14  |-  Y  e. 
_V
heibor.15  |-  ( ph  ->  Y  e.  Z )
heibor.16  |-  ( ph  ->  Z  e.  U )
heibor.17  |-  ( ph  ->  ( 1st  o.  M
) ( ~~> t `  J ) Y )
Assertion
Ref Expression
heiborlem8  |-  ( ph  ->  ps )
Distinct variable groups:    x, n, y, u, F    x, G    ph, x    m, n, u, v, x, y, z, D    m, M, u, x, y, z    T, m, n, x, y, z    B, n, u, v, y   
m, J, n, u, v, x, y, z    U, n, u, v, x, y, z    ps, y,
z    S, m, n, u, v, x, y, z   
m, X, n, u, v, x, y, z    C, m, n, u, v, y    n, K, x, y, z    x, Y   
v, Z, x    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    ps( x, v, u, m, n)    B( z, m)    C( x, z)    T( v, u)    U( m)    F( z, v, m)    G( y, z, v, u, m, n)    K( v, u, m)    M( v, n)    Y( y, z, v, u, m, n)    Z( y, z, u, m, n)

Proof of Theorem heiborlem8
Dummy variables  t 
k  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.6 . . . . 5  |-  ( ph  ->  D  e.  ( CMet `  X ) )
2 cmetmet 18728 . . . . 5  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
3 metxmet 17915 . . . . 5  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
41, 2, 33syl 18 . . . 4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
5 heibor.13 . . . . 5  |-  ( ph  ->  U  C_  J )
6 heibor.16 . . . . 5  |-  ( ph  ->  Z  e.  U )
75, 6sseldd 3194 . . . 4  |-  ( ph  ->  Z  e.  J )
8 heibor.15 . . . 4  |-  ( ph  ->  Y  e.  Z )
9 heibor.1 . . . . 5  |-  J  =  ( MetOpen `  D )
109mopni2 18055 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  Z  e.  J  /\  Y  e.  Z
)  ->  E. x  e.  RR+  ( Y (
ball `  D )
x )  C_  Z
)
114, 7, 8, 10syl3anc 1182 . . 3  |-  ( ph  ->  E. x  e.  RR+  ( Y ( ball `  D
) x )  C_  Z )
12 rphalfcl 10394 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  /  2 )  e.  RR+ )
13 breq2 4043 . . . . . . . . 9  |-  ( r  =  ( x  / 
2 )  ->  (
( 2nd `  ( M `  k )
)  <  r  <->  ( 2nd `  ( M `  k
) )  <  (
x  /  2 ) ) )
1413rexbidv 2577 . . . . . . . 8  |-  ( r  =  ( x  / 
2 )  ->  ( E. k  e.  NN  ( 2nd `  ( M `
 k ) )  <  r  <->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) ) )
15 heibor.3 . . . . . . . . 9  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
16 heibor.4 . . . . . . . . 9  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
17 heibor.5 . . . . . . . . 9  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
18 heibor.7 . . . . . . . . 9  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
19 heibor.8 . . . . . . . . 9  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
20 heibor.9 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( T `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
21 heibor.10 . . . . . . . . 9  |-  ( ph  ->  C G 0 )
22 heibor.11 . . . . . . . . 9  |-  S  =  seq  0 ( T ,  ( m  e. 
NN0  |->  if ( m  =  0 ,  C ,  ( m  - 
1 ) ) ) )
23 heibor.12 . . . . . . . . 9  |-  M  =  ( n  e.  NN  |->  <. ( S `  n
) ,  ( 3  /  ( 2 ^ n ) ) >.
)
249, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem7 26644 . . . . . . . 8  |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  r
2514, 24vtoclri 2871 . . . . . . 7  |-  ( ( x  /  2 )  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
2612, 25syl 15 . . . . . 6  |-  ( x  e.  RR+  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
2726adantl 452 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. k  e.  NN  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
28 nnnn0 9988 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  NN0 )
299, 15, 16, 17, 1, 18, 19, 20, 21, 22heiborlem4 26641 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( S `  k ) G k )
30 fvex 5555 . . . . . . . . . . . . 13  |-  ( S `
 k )  e. 
_V
31 vex 2804 . . . . . . . . . . . . 13  |-  k  e. 
_V
329, 15, 16, 30, 31heiborlem2 26639 . . . . . . . . . . . 12  |-  ( ( S `  k ) G k  <->  ( k  e.  NN0  /\  ( S `
 k )  e.  ( F `  k
)  /\  ( ( S `  k ) B k )  e.  K ) )
3332simp3bi 972 . . . . . . . . . . 11  |-  ( ( S `  k ) G k  ->  (
( S `  k
) B k )  e.  K )
3429, 33syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( ( S `  k ) B k )  e.  K )
3528, 34sylan2 460 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( S `  k ) B k )  e.  K )
3635ad2ant2r 727 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  e.  K
)
374ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  D  e.  ( * Met `  X
) )
389, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem5 26642 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M : NN --> ( X  X.  RR+ ) )
39 ffvelrn 5679 . . . . . . . . . . . . . . . 16  |-  ( ( M : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  ( M `  k )  e.  ( X  X.  RR+ ) )
4038, 39sylan 457 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  NN )  ->  ( M `
 k )  e.  ( X  X.  RR+ ) )
4140ad2ant2r 727 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( M `  k )  e.  ( X  X.  RR+ )
)
42 xp1st 6165 . . . . . . . . . . . . . 14  |-  ( ( M `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( M `  k
) )  e.  X
)
4341, 42syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1st `  ( M `  k )
)  e.  X )
44 2nn 9893 . . . . . . . . . . . . . . . . . 18  |-  2  e.  NN
45 nnexpcl 11132 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2  e.  NN  /\  k  e.  NN0 )  -> 
( 2 ^ k
)  e.  NN )
4644, 28, 45sylancr 644 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  NN )
4746nnrpd 10405 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  (
2 ^ k )  e.  RR+ )
4847rpreccld 10416 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  e.  RR+ )
4948ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  e.  RR+ )
5049rpxrd 10407 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  e.  RR* )
51 xp2nd 6166 . . . . . . . . . . . . . . 15  |-  ( ( M `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( M `  k
) )  e.  RR+ )
5241, 51syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  e.  RR+ )
5352rpxrd 10407 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  e.  RR* )
54 1re 8853 . . . . . . . . . . . . . . . . . 18  |-  1  e.  RR
55 3re 9833 . . . . . . . . . . . . . . . . . 18  |-  3  e.  RR
56 1lt3 9904 . . . . . . . . . . . . . . . . . 18  |-  1  <  3
5754, 55, 56ltleii 8957 . . . . . . . . . . . . . . . . 17  |-  1  <_  3
58 elrp 10372 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2 ^ k )  e.  RR+  <->  ( ( 2 ^ k )  e.  RR  /\  0  < 
( 2 ^ k
) ) )
59 lediv1 9637 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1  e.  RR  /\  3  e.  RR  /\  (
( 2 ^ k
)  e.  RR  /\  0  <  ( 2 ^ k ) ) )  ->  ( 1  <_ 
3  <->  ( 1  / 
( 2 ^ k
) )  <_  (
3  /  ( 2 ^ k ) ) ) )
6054, 55, 59mp3an12 1267 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2 ^ k
)  e.  RR  /\  0  <  ( 2 ^ k ) )  -> 
( 1  <_  3  <->  ( 1  /  ( 2 ^ k ) )  <_  ( 3  / 
( 2 ^ k
) ) ) )
6158, 60sylbi 187 . . . . . . . . . . . . . . . . 17  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 1  <_  3  <->  ( 1  /  ( 2 ^ k ) )  <_ 
( 3  /  (
2 ^ k ) ) ) )
6257, 61mpbii 202 . . . . . . . . . . . . . . . 16  |-  ( ( 2 ^ k )  e.  RR+  ->  ( 1  /  ( 2 ^ k ) )  <_ 
( 3  /  (
2 ^ k ) ) )
6347, 62syl 15 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
1  /  ( 2 ^ k ) )  <_  ( 3  / 
( 2 ^ k
) ) )
6463ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  <_  (
3  /  ( 2 ^ k ) ) )
65 fveq2 5541 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  ( S `  n )  =  ( S `  k ) )
66 oveq2 5882 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  k  ->  (
2 ^ n )  =  ( 2 ^ k ) )
6766oveq2d 5890 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  k  ->  (
3  /  ( 2 ^ n ) )  =  ( 3  / 
( 2 ^ k
) ) )
6865, 67opeq12d 3820 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  k  ->  <. ( S `  n ) ,  ( 3  / 
( 2 ^ n
) ) >.  =  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
69 opex 4253 . . . . . . . . . . . . . . . . . 18  |-  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >.  e.  _V
7068, 23, 69fvmpt 5618 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  ( M `  k )  =  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )
7170fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 2nd `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
72 ovex 5899 . . . . . . . . . . . . . . . . 17  |-  ( 3  /  ( 2 ^ k ) )  e. 
_V
7330, 72op2nd 6145 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  (
3  /  ( 2 ^ k ) )
7471, 73syl6eq 2344 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( 2nd `  ( M `  k ) )  =  ( 3  /  (
2 ^ k ) ) )
7574ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  =  ( 3  /  ( 2 ^ k ) ) )
7664, 75breqtrrd 4065 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1  / 
( 2 ^ k
) )  <_  ( 2nd `  ( M `  k ) ) )
77 ssbl 17987 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  ( * Met `  X
)  /\  ( 1st `  ( M `  k
) )  e.  X
)  /\  ( (
1  /  ( 2 ^ k ) )  e.  RR*  /\  ( 2nd `  ( M `  k ) )  e. 
RR* )  /\  (
1  /  ( 2 ^ k ) )  <_  ( 2nd `  ( M `  k )
) )  ->  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) ) )
7837, 43, 50, 53, 76, 77syl221anc 1193 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 1  /  ( 2 ^ k ) ) ) 
C_  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 2nd `  ( M `  k
) ) ) )
7928ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  k  e.  NN0 )
80 oveq1 5881 . . . . . . . . . . . . . . 15  |-  ( z  =  ( 1st `  ( M `  k )
)  ->  ( z
( ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) ) )
81 oveq2 5882 . . . . . . . . . . . . . . . . 17  |-  ( m  =  k  ->  (
2 ^ m )  =  ( 2 ^ k ) )
8281oveq2d 5890 . . . . . . . . . . . . . . . 16  |-  ( m  =  k  ->  (
1  /  ( 2 ^ m ) )  =  ( 1  / 
( 2 ^ k
) ) )
8382oveq2d 5890 . . . . . . . . . . . . . . 15  |-  ( m  =  k  ->  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) )  =  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) ) )
84 ovex 5899 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) )  e. 
_V
8580, 83, 17, 84ovmpt2 5999 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( M `  k )
)  e.  X  /\  k  e.  NN0 )  -> 
( ( 1st `  ( M `  k )
) B k )  =  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 1  /  ( 2 ^ k ) ) ) )
8643, 79, 85syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) B k )  =  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 1  / 
( 2 ^ k
) ) ) )
8770fveq2d 5545 . . . . . . . . . . . . . . . 16  |-  ( k  e.  NN  ->  ( 1st `  ( M `  k ) )  =  ( 1st `  <. ( S `  k ) ,  ( 3  / 
( 2 ^ k
) ) >. )
)
8830, 72op1st 6144 . . . . . . . . . . . . . . . 16  |-  ( 1st `  <. ( S `  k ) ,  ( 3  /  ( 2 ^ k ) )
>. )  =  ( S `  k )
8987, 88syl6eq 2344 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  ( 1st `  ( M `  k ) )  =  ( S `  k
) )
9089ad2antrl 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 1st `  ( M `  k )
)  =  ( S `
 k ) )
9190oveq1d 5889 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) B k )  =  ( ( S `  k ) B k ) )
9286, 91eqtr3d 2330 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 1  /  ( 2 ^ k ) ) )  =  ( ( S `
 k ) B k ) )
93 1st2nd2 6175 . . . . . . . . . . . . . . 15  |-  ( ( M `  k )  e.  ( X  X.  RR+ )  ->  ( M `  k )  =  <. ( 1st `  ( M `
 k ) ) ,  ( 2nd `  ( M `  k )
) >. )
9441, 93syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( M `  k )  =  <. ( 1st `  ( M `
 k ) ) ,  ( 2nd `  ( M `  k )
) >. )
9594fveq2d 5545 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  =  ( (
ball `  D ) `  <. ( 1st `  ( M `  k )
) ,  ( 2nd `  ( M `  k
) ) >. )
)
96 df-ov 5877 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( M `
 k ) ) ( ball `  D
) ( 2nd `  ( M `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( M `  k
) ) ,  ( 2nd `  ( M `
 k ) )
>. )
9795, 96syl6reqr 2347 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 2nd `  ( M `  k
) ) )  =  ( ( ball `  D
) `  ( M `  k ) ) )
9878, 92, 973sstr3d 3233 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  C_  (
( ball `  D ) `  ( M `  k
) ) )
999mopntop 18002 . . . . . . . . . . . . . 14  |-  ( D  e.  ( * Met `  X )  ->  J  e.  Top )
10037, 99syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  J  e.  Top )
101 blssm 17984 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( * Met `  X )  /\  ( 1st `  ( M `  k )
)  e.  X  /\  ( 2nd `  ( M `
 k ) )  e.  RR* )  ->  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) )  C_  X
)
10237, 43, 53, 101syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( 2nd `  ( M `  k
) ) )  C_  X )
1039mopnuni 18003 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
10437, 103syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  X  =  U. J )
105102, 97, 1043sstr3d 3233 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  U. J )
106 eqid 2296 . . . . . . . . . . . . . 14  |-  U. J  =  U. J
107106sscls 16809 . . . . . . . . . . . . 13  |-  ( ( J  e.  Top  /\  ( ( ball `  D
) `  ( M `  k ) )  C_  U. J )  ->  (
( ball `  D ) `  ( M `  k
) )  C_  (
( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
108100, 105, 107syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  ( ( cls `  J ) `  ( ( ball `  D
) `  ( M `  k ) ) ) )
10997fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) ) )  =  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
11012ad2antlr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( x  / 
2 )  e.  RR+ )
111110rpxrd 10407 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( x  / 
2 )  e.  RR* )
112 simprr 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( 2nd `  ( M `  k )
)  <  ( x  /  2 ) )
1139blsscls 18069 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( * Met `  X
)  /\  ( 1st `  ( M `  k
) )  e.  X
)  /\  ( ( 2nd `  ( M `  k ) )  e. 
RR*  /\  ( x  /  2 )  e. 
RR*  /\  ( 2nd `  ( M `  k
) )  <  (
x  /  2 ) ) )  ->  (
( cls `  J
) `  ( ( 1st `  ( M `  k ) ) (
ball `  D )
( 2nd `  ( M `  k )
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
11437, 43, 53, 111, 112, 113syl23anc 1189 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( 1st `  ( M `  k )
) ( ball `  D
) ( 2nd `  ( M `  k )
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
115109, 114eqsstr3d 3226 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( ball `  D ) `  ( M `  k
) ) )  C_  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
116 rpre 10376 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  x  e.  RR )
117116ad2antlr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  x  e.  RR )
118 heibor.17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 1st  o.  M
) ( ~~> t `  J ) Y )
1199, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem6 26643 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  A. t  e.  NN  ( ( ball `  D
) `  ( M `  ( t  +  1 ) ) )  C_  ( ( ball `  D
) `  ( M `  t ) ) )
1204, 38, 119, 9caublcls 18750 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( 1st  o.  M ) ( ~~> t `  J ) Y  /\  k  e.  NN )  ->  Y  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( M `  k )
) ) )
1211203expia 1153 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( 1st  o.  M ) ( ~~> t `  J ) Y )  ->  ( k  e.  NN  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) ) )
122118, 121mpdan 649 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( k  e.  NN  ->  Y  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( M `  k )
) ) ) )
123122imp 418 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  NN )  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
124123ad2ant2r 727 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  Y  e.  ( ( cls `  J
) `  ( ( ball `  D ) `  ( M `  k ) ) ) )
125115, 124sseldd 3194 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  Y  e.  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) )
126 blhalf 17976 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  ( * Met `  X
)  /\  ( 1st `  ( M `  k
) )  e.  X
)  /\  ( x  e.  RR  /\  Y  e.  ( ( 1st `  ( M `  k )
) ( ball `  D
) ( x  / 
2 ) ) ) )  ->  ( ( 1st `  ( M `  k ) ) (
ball `  D )
( x  /  2
) )  C_  ( Y ( ball `  D
) x ) )
12737, 43, 117, 125, 126syl22anc 1183 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( 1st `  ( M `  k
) ) ( ball `  D ) ( x  /  2 ) ) 
C_  ( Y (
ball `  D )
x ) )
128115, 127sstrd 3202 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( cls `  J ) `  (
( ball `  D ) `  ( M `  k
) ) )  C_  ( Y ( ball `  D
) x ) )
129108, 128sstrd 3202 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ball `  D ) `  ( M `  k )
)  C_  ( Y
( ball `  D )
x ) )
13098, 129sstrd 3202 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( S `
 k ) B k )  C_  ( Y ( ball `  D
) x ) )
131 sstr2 3199 . . . . . . . . . 10  |-  ( ( ( S `  k
) B k ) 
C_  ( Y (
ball `  D )
x )  ->  (
( Y ( ball `  D ) x ) 
C_  Z  ->  (
( S `  k
) B k ) 
C_  Z ) )
132130, 131syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( Y ( ball `  D
) x )  C_  Z  ->  ( ( S `
 k ) B k )  C_  Z
) )
133 unisng 3860 . . . . . . . . . . . . . . . 16  |-  ( Z  e.  U  ->  U. { Z }  =  Z
)
1346, 133syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U. { Z }  =  Z )
135134sseq2d 3219 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( S `
 k ) B k )  C_  U. { Z }  <->  ( ( S `
 k ) B k )  C_  Z
) )
136135biimpar 471 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  (
( S `  k
) B k ) 
C_  U. { Z }
)
1376snssd 3776 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  { Z }  C_  U )
138 snex 4232 . . . . . . . . . . . . . . . . 17  |-  { Z }  e.  _V
139138elpw 3644 . . . . . . . . . . . . . . . 16  |-  ( { Z }  e.  ~P U 
<->  { Z }  C_  U )
140137, 139sylibr 203 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { Z }  e.  ~P U )
141 snfi 6957 . . . . . . . . . . . . . . . 16  |-  { Z }  e.  Fin
142141a1i 10 . . . . . . . . . . . . . . 15  |-  ( ph  ->  { Z }  e.  Fin )
143 elin 3371 . . . . . . . . . . . . . . 15  |-  ( { Z }  e.  ( ~P U  i^i  Fin ) 
<->  ( { Z }  e.  ~P U  /\  { Z }  e.  Fin ) )
144140, 142, 143sylanbrc 645 . . . . . . . . . . . . . 14  |-  ( ph  ->  { Z }  e.  ( ~P U  i^i  Fin ) )
145 unieq 3852 . . . . . . . . . . . . . . . 16  |-  ( v  =  { Z }  ->  U. v  =  U. { Z } )
146145sseq2d 3219 . . . . . . . . . . . . . . 15  |-  ( v  =  { Z }  ->  ( ( ( S `
 k ) B k )  C_  U. v  <->  ( ( S `  k
) B k ) 
C_  U. { Z }
) )
147146rspcev 2897 . . . . . . . . . . . . . 14  |-  ( ( { Z }  e.  ( ~P U  i^i  Fin )  /\  ( ( S `
 k ) B k )  C_  U. { Z } )  ->  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
148144, 147sylan 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  U. { Z } )  ->  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
149136, 148syldan 456 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
150 ovex 5899 . . . . . . . . . . . . . 14  |-  ( ( S `  k ) B k )  e. 
_V
151 sseq1 3212 . . . . . . . . . . . . . . . 16  |-  ( u  =  ( ( S `
 k ) B k )  ->  (
u  C_  U. v  <->  ( ( S `  k
) B k ) 
C_  U. v ) )
152151rexbidv 2577 . . . . . . . . . . . . . . 15  |-  ( u  =  ( ( S `
 k ) B k )  ->  ( E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v  <->  E. v  e.  ( ~P U  i^i  Fin )
( ( S `  k ) B k )  C_  U. v
) )
153152notbid 285 . . . . . . . . . . . . . 14  |-  ( u  =  ( ( S `
 k ) B k )  ->  ( -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v  <->  -. 
E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
) )
154150, 153, 15elab2 2930 . . . . . . . . . . . . 13  |-  ( ( ( S `  k
) B k )  e.  K  <->  -.  E. v  e.  ( ~P U  i^i  Fin ) ( ( S `
 k ) B k )  C_  U. v
)
155154con2bii 322 . . . . . . . . . . . 12  |-  ( E. v  e.  ( ~P U  i^i  Fin )
( ( S `  k ) B k )  C_  U. v  <->  -.  ( ( S `  k ) B k )  e.  K )
156149, 155sylib 188 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( S `  k ) B k )  C_  Z )  ->  -.  ( ( S `  k ) B k )  e.  K )
157156ex 423 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( S `
 k ) B k )  C_  Z  ->  -.  ( ( S `
 k ) B k )  e.  K
) )
158157ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( ( S `  k ) B k )  C_  Z  ->  -.  ( ( S `  k ) B k )  e.  K ) )
159132, 158syld 40 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  ( ( Y ( ball `  D
) x )  C_  Z  ->  -.  ( ( S `  k ) B k )  e.  K ) )
16036, 159mt2d 109 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR+ )  /\  (
k  e.  NN  /\  ( 2nd `  ( M `
 k ) )  <  ( x  / 
2 ) ) )  ->  -.  ( Y
( ball `  D )
x )  C_  Z
)
161160exp32 588 . . . . . 6  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( k  e.  NN  ->  ( ( 2nd `  ( M `  k ) )  < 
( x  /  2
)  ->  -.  ( Y ( ball `  D
) x )  C_  Z ) ) )
162161rexlimdv 2679 . . . . 5  |-  ( (
ph  /\  x  e.  RR+ )  ->  ( E. k  e.  NN  ( 2nd `  ( M `  k ) )  < 
( x  /  2
)  ->  -.  ( Y ( ball `  D
) x )  C_  Z ) )
16327, 162mpd 14 . . . 4  |-  ( (
ph  /\  x  e.  RR+ )  ->  -.  ( Y ( ball `  D
) x )  C_  Z )
164163nrexdv 2659 . . 3  |-  ( ph  ->  -.  E. x  e.  RR+  ( Y ( ball `  D ) x ) 
C_  Z )
16511, 164pm2.65i 165 . 2  |-  -.  ph
166165pm2.21i 123 1  |-  ( ph  ->  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   ifcif 3578   ~Pcpw 3638   {csn 3653   <.cop 3656   U.cuni 3843   U_ciun 3921   class class class wbr 4039   {copab 4092    e. cmpt 4093    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Fincfn 6879   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   3c3 9812   NN0cn0 9981   RR+crp 10370    seq cseq 11062   ^cexp 11120   * Metcxmt 16385   Metcme 16386   ballcbl 16387   MetOpencmopn 16388   Topctop 16647   clsccl 16771   ~~> tclm 16972   CMetcms 18696
This theorem is referenced by:  heiborlem9  26646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-fl 10941  df-seq 11063  df-exp 11121  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-ntr 16773  df-cls 16774  df-lm 16975  df-cmet 18699
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