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Theorem heiborlem3 26412
Description: Lemma for heibor 26420. Using countable choice ax-cc 8271, we have fixed in advance a collection of finite  2 ^ -u n nets  ( F `  n ) for  X (note that an  r-net is a set of points in  X whose  r -balls cover  X). The set  G is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set  K). If the theorem was false, then  X would be in  K, and so some ball at each level would also be in  K. But we can say more than this; given a ball 
( y B n ) on level  n, since level  n  +  1 covers the space and thus also  (
y B n ), using heiborlem1 26410 there is a ball on the next level whose intersection with  ( y B n ) also has no finite subcover. Now since the set 
G is a countable union of finite sets, it is countable (which needs ax-cc 8271 via iunctb 8405), and so we can apply ax-cc 8271 to  G directly to get a function from  G to itself, which points from each ball in  K to a ball on the next level in  K, and such that the intersection between these balls is also in  K. (Contributed by Jeff Madsen, 18-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 ) )
Assertion
Ref Expression
heiborlem3  |-  ( ph  ->  E. g A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
Distinct variable groups:    x, n, y, u, F    x, g, G    ph, g, x    g, m, n, u, v, y, z, D, x    B, g, n, u, v, y   
g, J, m, n, u, v, x, y, z    U, g, n, u, v, x, y, z   
g, X, m, n, u, v, x, y, z    g, K, n, x, y, z    x, B
Allowed substitution hints:    ph( y, z, v, u, m, n)    B( z, m)    U( m)    F( z, v, g, m)    G( y, z, v, u, m, n)    K( v, u, m)

Proof of Theorem heiborlem3
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 nn0ex 10183 . . . . . 6  |-  NN0  e.  _V
2 fvex 5701 . . . . . . 7  |-  ( F `
 t )  e. 
_V
3 snex 4365 . . . . . . 7  |-  { t }  e.  _V
42, 3xpex 4949 . . . . . 6  |-  ( ( F `  t )  X.  { t } )  e.  _V
51, 4iunex 5950 . . . . 5  |-  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } )  e.  _V
6 heibor.4 . . . . . . . . 9  |-  G  =  { <. y ,  n >.  |  ( n  e. 
NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K
) }
76relopabi 4959 . . . . . . . 8  |-  Rel  G
8 1st2nd 6352 . . . . . . . 8  |-  ( ( Rel  G  /\  x  e.  G )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
97, 8mpan 652 . . . . . . 7  |-  ( x  e.  G  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
109eleq1d 2470 . . . . . . . . . . 11  |-  ( x  e.  G  ->  (
x  e.  G  <->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  G ) )
11 df-br 4173 . . . . . . . . . . 11  |-  ( ( 1st `  x ) G ( 2nd `  x
)  <->  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  G
)
1210, 11syl6bbr 255 . . . . . . . . . 10  |-  ( x  e.  G  ->  (
x  e.  G  <->  ( 1st `  x ) G ( 2nd `  x ) ) )
13 heibor.1 . . . . . . . . . . 11  |-  J  =  ( MetOpen `  D )
14 heibor.3 . . . . . . . . . . 11  |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin )
u  C_  U. v }
15 fvex 5701 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
16 fvex 5701 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
1713, 14, 6, 15, 16heiborlem2 26411 . . . . . . . . . 10  |-  ( ( 1st `  x ) G ( 2nd `  x
)  <->  ( ( 2nd `  x )  e.  NN0  /\  ( 1st `  x
)  e.  ( F `
 ( 2nd `  x
) )  /\  (
( 1st `  x
) B ( 2nd `  x ) )  e.  K ) )
1812, 17syl6bb 253 . . . . . . . . 9  |-  ( x  e.  G  ->  (
x  e.  G  <->  ( ( 2nd `  x )  e. 
NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) )  /\  ( ( 1st `  x
) B ( 2nd `  x ) )  e.  K ) ) )
1918ibi 233 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) )  /\  ( ( 1st `  x ) B ( 2nd `  x
) )  e.  K
) )
2016snid 3801 . . . . . . . . . . . 12  |-  ( 2nd `  x )  e.  {
( 2nd `  x
) }
21 opelxp 4867 . . . . . . . . . . . 12  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } )  <-> 
( ( 1st `  x
)  e.  ( F `
 ( 2nd `  x
) )  /\  ( 2nd `  x )  e. 
{ ( 2nd `  x
) } ) )
2220, 21mpbiran2 886 . . . . . . . . . . 11  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } )  <-> 
( 1st `  x
)  e.  ( F `
 ( 2nd `  x
) ) )
23 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  ( F `  t )  =  ( F `  ( 2nd `  x ) ) )
24 sneq 3785 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  { t }  =  { ( 2nd `  x ) } )
2523, 24xpeq12d 4862 . . . . . . . . . . . . 13  |-  ( t  =  ( 2nd `  x
)  ->  ( ( F `  t )  X.  { t } )  =  ( ( F `
 ( 2nd `  x
) )  X.  {
( 2nd `  x
) } ) )
2625eleq2d 2471 . . . . . . . . . . . 12  |-  ( t  =  ( 2nd `  x
)  ->  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  ( ( F `  t )  X.  { t } )  <->  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } ) ) )
2726rspcev 3012 . . . . . . . . . . 11  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  ( 2nd `  x ) )  X.  { ( 2nd `  x ) } ) )  ->  E. t  e.  NN0  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  t
)  X.  { t } ) )
2822, 27sylan2br 463 . . . . . . . . . 10  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )  ->  E. t  e.  NN0  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  t
)  X.  { t } ) )
29 eliun 4057 . . . . . . . . . 10  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  U_ t  e.  NN0  ( ( F `  t )  X.  { t } )  <->  E. t  e.  NN0  <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  ( ( F `  t
)  X.  { t } ) )
3028, 29sylibr 204 . . . . . . . . 9  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } ) )
31303adant3 977 . . . . . . . 8  |-  ( ( ( 2nd `  x
)  e.  NN0  /\  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) )  /\  ( ( 1st `  x ) B ( 2nd `  x
) )  e.  K
)  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  U_ t  e. 
NN0  ( ( F `
 t )  X. 
{ t } ) )
3219, 31syl 16 . . . . . . 7  |-  ( x  e.  G  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } ) )
339, 32eqeltrd 2478 . . . . . 6  |-  ( x  e.  G  ->  x  e.  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } ) )
3433ssriv 3312 . . . . 5  |-  G  C_  U_ t  e.  NN0  (
( F `  t
)  X.  { t } )
35 ssdomg 7112 . . . . 5  |-  ( U_ t  e.  NN0  ( ( F `  t )  X.  { t } )  e.  _V  ->  ( G  C_  U_ t  e. 
NN0  ( ( F `
 t )  X. 
{ t } )  ->  G  ~<_  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } ) ) )
365, 34, 35mp2 9 . . . 4  |-  G  ~<_  U_ t  e.  NN0  ( ( F `  t )  X.  { t } )
37 nn0ennn 11273 . . . . . . 7  |-  NN0  ~~  NN
38 nnenom 11274 . . . . . . 7  |-  NN  ~~  om
3937, 38entri 7120 . . . . . 6  |-  NN0  ~~  om
40 endom 7093 . . . . . 6  |-  ( NN0  ~~  om  ->  NN0  ~<_  om )
4139, 40ax-mp 8 . . . . 5  |-  NN0  ~<_  om
42 vex 2919 . . . . . . . 8  |-  t  e. 
_V
432, 42xpsnen 7151 . . . . . . 7  |-  ( ( F `  t )  X.  { t } )  ~~  ( F `
 t )
44 inss2 3522 . . . . . . . . 9  |-  ( ~P X  i^i  Fin )  C_ 
Fin
45 heibor.7 . . . . . . . . . 10  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
4645ffvelrnda 5829 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  ( ~P X  i^i  Fin ) )
4744, 46sseldi 3306 . . . . . . . 8  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  Fin )
48 isfinite 7563 . . . . . . . . 9  |-  ( ( F `  t )  e.  Fin  <->  ( F `  t )  ~<  om )
49 sdomdom 7094 . . . . . . . . 9  |-  ( ( F `  t ) 
~<  om  ->  ( F `  t )  ~<_  om )
5048, 49sylbi 188 . . . . . . . 8  |-  ( ( F `  t )  e.  Fin  ->  ( F `  t )  ~<_  om )
5147, 50syl 16 . . . . . . 7  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  ~<_  om )
52 endomtr 7124 . . . . . . 7  |-  ( ( ( ( F `  t )  X.  {
t } )  ~~  ( F `  t )  /\  ( F `  t )  ~<_  om )  ->  ( ( F `  t )  X.  {
t } )  ~<_  om )
5343, 51, 52sylancr 645 . . . . . 6  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( ( F `  t )  X.  { t } )  ~<_  om )
5453ralrimiva 2749 . . . . 5  |-  ( ph  ->  A. t  e.  NN0  ( ( F `  t )  X.  {
t } )  ~<_  om )
55 iunctb 8405 . . . . 5  |-  ( ( NN0  ~<_  om  /\  A. t  e.  NN0  ( ( F `
 t )  X. 
{ t } )  ~<_  om )  ->  U_ t  e.  NN0  ( ( F `
 t )  X. 
{ t } )  ~<_  om )
5641, 54, 55sylancr 645 . . . 4  |-  ( ph  ->  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } )  ~<_  om )
57 domtr 7119 . . . 4  |-  ( ( G  ~<_  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } )  /\  U_ t  e.  NN0  (
( F `  t
)  X.  { t } )  ~<_  om )  ->  G  ~<_  om )
5836, 56, 57sylancr 645 . . 3  |-  ( ph  ->  G  ~<_  om )
5919simp1d 969 . . . . . . . . 9  |-  ( x  e.  G  ->  ( 2nd `  x )  e. 
NN0 )
60 peano2nn0 10216 . . . . . . . . 9  |-  ( ( 2nd `  x )  e.  NN0  ->  ( ( 2nd `  x )  +  1 )  e. 
NN0 )
6159, 60syl 16 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 2nd `  x
)  +  1 )  e.  NN0 )
62 ffvelrn 5827 . . . . . . . 8  |-  ( ( F : NN0 --> ( ~P X  i^i  Fin )  /\  ( ( 2nd `  x
)  +  1 )  e.  NN0 )  -> 
( F `  (
( 2nd `  x
)  +  1 ) )  e.  ( ~P X  i^i  Fin )
)
6345, 61, 62syl2an 464 . . . . . . 7  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e.  ( ~P X  i^i  Fin ) )
6444, 63sseldi 3306 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
Fin )
65 iunin2 4115 . . . . . . . 8  |-  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( ( B `  x
)  i^i  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
66 heibor.8 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n ) )
67 oveq1 6047 . . . . . . . . . . . . . . . 16  |-  ( y  =  t  ->  (
y B n )  =  ( t B n ) )
6867cbviunv 4090 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  n ) ( t B n )
69 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( F `  n )  =  ( F `  ( ( 2nd `  x )  +  1 ) ) )
7069iuneq1d 4076 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ t  e.  ( F `  n
) ( t B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B n ) )
7168, 70syl5eq 2448 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B n ) )
72 oveq2 6048 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( t B n )  =  ( t B ( ( 2nd `  x
)  +  1 ) ) )
7372iuneq2d 4078 . . . . . . . . . . . . . 14  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7471, 73eqtrd 2436 . . . . . . . . . . . . 13  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7574eqeq2d 2415 . . . . . . . . . . . 12  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( X  =  U_ y  e.  ( F `  n ) ( y B n )  <->  X  =  U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) ) )
7675rspccva 3011 . . . . . . . . . . 11  |-  ( ( A. n  e.  NN0  X  =  U_ y  e.  ( F `  n
) ( y B n )  /\  (
( 2nd `  x
)  +  1 )  e.  NN0 )  ->  X  =  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7766, 61, 76syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  X  =  U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )
7877ineq2d 3502 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  G )  ->  (
( B `  x
)  i^i  X )  =  ( ( B `
 x )  i^i  U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( t B ( ( 2nd `  x )  +  1 ) ) ) )
799fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( x  e.  G  ->  ( B `  x )  =  ( B `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
80 df-ov 6043 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) B ( 2nd `  x
) )  =  ( B `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
8179, 80syl6eqr 2454 . . . . . . . . . . . . 13  |-  ( x  e.  G  ->  ( B `  x )  =  ( ( 1st `  x ) B ( 2nd `  x ) ) )
8281adantl 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  =  ( ( 1st `  x ) B ( 2nd `  x ) ) )
83 inss1 3521 . . . . . . . . . . . . . . . 16  |-  ( ~P X  i^i  Fin )  C_ 
~P X
84 ffvelrn 5827 . . . . . . . . . . . . . . . . 17  |-  ( ( F : NN0 --> ( ~P X  i^i  Fin )  /\  ( 2nd `  x
)  e.  NN0 )  ->  ( F `  ( 2nd `  x ) )  e.  ( ~P X  i^i  Fin ) )
8545, 59, 84syl2an 464 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  e.  ( ~P X  i^i  Fin ) )
8683, 85sseldi 3306 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  e. 
~P X )
8786elpwid 3768 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  C_  X )
8819simp2d 970 . . . . . . . . . . . . . . 15  |-  ( x  e.  G  ->  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )
8988adantl 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  ( 1st `  x )  e.  ( F `  ( 2nd `  x ) ) )
9087, 89sseldd 3309 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 1st `  x )  e.  X )
9159adantl 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 2nd `  x )  e. 
NN0 )
92 oveq1 6047 . . . . . . . . . . . . . 14  |-  ( z  =  ( 1st `  x
)  ->  ( z
( ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) ) )
93 oveq2 6048 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( 2nd `  x
)  ->  ( 2 ^ m )  =  ( 2 ^ ( 2nd `  x ) ) )
9493oveq2d 6056 . . . . . . . . . . . . . . 15  |-  ( m  =  ( 2nd `  x
)  ->  ( 1  /  ( 2 ^ m ) )  =  ( 1  /  (
2 ^ ( 2nd `  x ) ) ) )
9594oveq2d 6056 . . . . . . . . . . . . . 14  |-  ( m  =  ( 2nd `  x
)  ->  ( ( 1st `  x ) (
ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) ) )
96 heibor.5 . . . . . . . . . . . . . 14  |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D ) ( 1  /  ( 2 ^ m ) ) ) )
97 ovex 6065 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  e.  _V
9892, 95, 96, 97ovmpt2 6168 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  x
)  e.  X  /\  ( 2nd `  x )  e.  NN0 )  -> 
( ( 1st `  x
) B ( 2nd `  x ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) ) )
9990, 91, 98syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  (
( 1st `  x
) B ( 2nd `  x ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) ) )
10082, 99eqtrd 2436 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  =  ( ( 1st `  x ) ( ball `  D ) ( 1  /  ( 2 ^ ( 2nd `  x
) ) ) ) )
101 heibor.6 . . . . . . . . . . . . . . 15  |-  ( ph  ->  D  e.  ( CMet `  X ) )
102 cmetmet 19192 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
103101, 102syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  ( Met `  X ) )
104 metxmet 18317 . . . . . . . . . . . . . 14  |-  ( D  e.  ( Met `  X
)  ->  D  e.  ( * Met `  X
) )
105103, 104syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  D  e.  ( * Met `  X ) )
106105adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  D  e.  ( * Met `  X
) )
107 2nn 10089 . . . . . . . . . . . . . . . 16  |-  2  e.  NN
108 nnexpcl 11349 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  NN  /\  ( 2nd `  x )  e.  NN0 )  -> 
( 2 ^ ( 2nd `  x ) )  e.  NN )
109107, 91, 108sylancr 645 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  G )  ->  (
2 ^ ( 2nd `  x ) )  e.  NN )
110109nnrpd 10603 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  (
2 ^ ( 2nd `  x ) )  e.  RR+ )
111110rpreccld 10614 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e.  RR+ )
112111rpxrd 10605 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e. 
RR* )
113 blssm 18401 . . . . . . . . . . . 12  |-  ( ( D  e.  ( * Met `  X )  /\  ( 1st `  x
)  e.  X  /\  ( 1  /  (
2 ^ ( 2nd `  x ) ) )  e.  RR* )  ->  (
( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  C_  X
)
114106, 90, 112, 113syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  (
( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  C_  X
)
115100, 114eqsstrd 3342 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  C_  X )
116 df-ss 3294 . . . . . . . . . 10  |-  ( ( B `  x ) 
C_  X  <->  ( ( B `  x )  i^i  X )  =  ( B `  x ) )
117115, 116sylib 189 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  G )  ->  (
( B `  x
)  i^i  X )  =  ( B `  x ) )
11878, 117eqtr3d 2438 . . . . . . . 8  |-  ( (
ph  /\  x  e.  G )  ->  (
( B `  x
)  i^i  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( B `  x ) )
11965, 118syl5eq 2448 . . . . . . 7  |-  ( (
ph  /\  x  e.  G )  ->  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( B `  x ) )
120 eqimss2 3361 . . . . . . 7  |-  ( U_ t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  =  ( B `  x )  ->  ( B `  x )  C_  U_ t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) ) )
121119, 120syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  C_ 
U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) ) )
12219simp3d 971 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 1st `  x
) B ( 2nd `  x ) )  e.  K )
12381, 122eqeltrd 2478 . . . . . . 7  |-  ( x  e.  G  ->  ( B `  x )  e.  K )
124123adantl 453 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  e.  K )
125 fvex 5701 . . . . . . . 8  |-  ( B `
 x )  e. 
_V
126125inex1 4304 . . . . . . 7  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  _V
12713, 14, 126heiborlem1 26410 . . . . . 6  |-  ( ( ( F `  (
( 2nd `  x
)  +  1 ) )  e.  Fin  /\  ( B `  x ) 
C_  U_ t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  /\  ( B `
 x )  e.  K )  ->  E. t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)
12864, 121, 124, 127syl3anc 1184 . . . . 5  |-  ( (
ph  /\  x  e.  G )  ->  E. t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)
12983, 63sseldi 3306 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
~P X )
130129elpwid 3768 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  X )
13113mopnuni 18424 . . . . . . . . . . . . 13  |-  ( D  e.  ( * Met `  X )  ->  X  =  U. J )
132105, 131syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  X  =  U. J
)
133132adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  X  =  U. J )
134130, 133sseqtrd 3344 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  U. J )
135134sselda 3308 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  G )  /\  t  e.  ( F `  (
( 2nd `  x
)  +  1 ) ) )  ->  t  e.  U. J )
136135adantrr 698 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
t  e.  U. J
)
13761adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  (
( 2nd `  x
)  +  1 )  e.  NN0 )
138 id 20 . . . . . . . . . 10  |-  ( t  e.  ( F `  ( ( 2nd `  x
)  +  1 ) )  ->  t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) )
139 snfi 7146 . . . . . . . . . . . 12  |-  { ( t B ( ( 2nd `  x )  +  1 ) ) }  e.  Fin
140 inss2 3522 . . . . . . . . . . . . 13  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  (
t B ( ( 2nd `  x )  +  1 ) )
141 ovex 6065 . . . . . . . . . . . . . . 15  |-  ( t B ( ( 2nd `  x )  +  1 ) )  e.  _V
142141unisn 3991 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  ( t B ( ( 2nd `  x )  +  1 ) )
143 uniiun 4104 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
144142, 143eqtr3i 2426 . . . . . . . . . . . . 13  |-  ( t B ( ( 2nd `  x )  +  1 ) )  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
145140, 144sseqtri 3340 . . . . . . . . . . . 12  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  U_ g  e.  { ( t B ( ( 2nd `  x
)  +  1 ) ) } g
146 vex 2919 . . . . . . . . . . . . 13  |-  g  e. 
_V
14713, 14, 146heiborlem1 26410 . . . . . . . . . . . 12  |-  ( ( { ( t B ( ( 2nd `  x
)  +  1 ) ) }  e.  Fin  /\  ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  C_  U_ g  e. 
{ ( t B ( ( 2nd `  x
)  +  1 ) ) } g  /\  ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K )  ->  E. g  e.  {
( t B ( ( 2nd `  x
)  +  1 ) ) } g  e.  K )
148139, 145, 147mp3an12 1269 . . . . . . . . . . 11  |-  ( ( ( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  ->  E. g  e.  { ( t B ( ( 2nd `  x
)  +  1 ) ) } g  e.  K )
149 eleq1 2464 . . . . . . . . . . . 12  |-  ( g  =  ( t B ( ( 2nd `  x
)  +  1 ) )  ->  ( g  e.  K  <->  ( t B ( ( 2nd `  x
)  +  1 ) )  e.  K ) )
150141, 149rexsn 3810 . . . . . . . . . . 11  |-  ( E. g  e.  { ( t B ( ( 2nd `  x )  +  1 ) ) } g  e.  K  <->  ( t B ( ( 2nd `  x )  +  1 ) )  e.  K )
151148, 150sylib 189 . . . . . . . . . 10  |-  ( ( ( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  ->  ( t B ( ( 2nd `  x )  +  1 ) )  e.  K
)
152 ovex 6065 . . . . . . . . . . . 12  |-  ( ( 2nd `  x )  +  1 )  e. 
_V
15313, 14, 6, 42, 152heiborlem2 26411 . . . . . . . . . . 11  |-  ( t G ( ( 2nd `  x )  +  1 )  <->  ( ( ( 2nd `  x )  +  1 )  e. 
NN0  /\  t  e.  ( F `  ( ( 2nd `  x )  +  1 ) )  /\  ( t B ( ( 2nd `  x
)  +  1 ) )  e.  K ) )
154153biimpri 198 . . . . . . . . . 10  |-  ( ( ( ( 2nd `  x
)  +  1 )  e.  NN0  /\  t  e.  ( F `  (
( 2nd `  x
)  +  1 ) )  /\  ( t B ( ( 2nd `  x )  +  1 ) )  e.  K
)  ->  t G
( ( 2nd `  x
)  +  1 ) )
155137, 138, 151, 154syl3an 1226 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  G )  /\  t  e.  ( F `  (
( 2nd `  x
)  +  1 ) )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  ->  t G
( ( 2nd `  x
)  +  1 ) )
1561553expb 1154 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
t G ( ( 2nd `  x )  +  1 ) )
157 simprr 734 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K )
158136, 156, 157jca32 522 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  G )  /\  (
t  e.  ( F `
 ( ( 2nd `  x )  +  1 ) )  /\  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  -> 
( t  e.  U. J  /\  ( t G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
159158ex 424 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  (
( t  e.  ( F `  ( ( 2nd `  x )  +  1 ) )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  ->  ( t  e.  U. J  /\  (
t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) ) ) )
160159reximdv2 2775 . . . . 5  |-  ( (
ph  /\  x  e.  G )  ->  ( E. t  e.  ( F `  ( ( 2nd `  x )  +  1 ) ) ( ( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  ->  E. t  e.  U. J ( t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
161128, 160mpd 15 . . . 4  |-  ( (
ph  /\  x  e.  G )  ->  E. t  e.  U. J ( t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
162161ralrimiva 2749 . . 3  |-  ( ph  ->  A. x  e.  G  E. t  e.  U. J
( t G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
163 fvex 5701 . . . . . 6  |-  ( MetOpen `  D )  e.  _V
16413, 163eqeltri 2474 . . . . 5  |-  J  e. 
_V
165164uniex 4664 . . . 4  |-  U. J  e.  _V
166 breq1 4175 . . . . 5  |-  ( t  =  ( g `  x )  ->  (
t G ( ( 2nd `  x )  +  1 )  <->  ( g `  x ) G ( ( 2nd `  x
)  +  1 ) ) )
167 oveq1 6047 . . . . . . 7  |-  ( t  =  ( g `  x )  ->  (
t B ( ( 2nd `  x )  +  1 ) )  =  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )
168167ineq2d 3502 . . . . . 6  |-  ( t  =  ( g `  x )  ->  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  =  ( ( B `  x )  i^i  (
( g `  x
) B ( ( 2nd `  x )  +  1 ) ) ) )
169168eleq1d 2470 . . . . 5  |-  ( t  =  ( g `  x )  ->  (
( ( B `  x )  i^i  (
t B ( ( 2nd `  x )  +  1 ) ) )  e.  K  <->  ( ( B `  x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
170166, 169anbi12d 692 . . . 4  |-  ( t  =  ( g `  x )  ->  (
( t G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
)  <->  ( ( g `
 x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
171165, 170axcc4dom 8277 . . 3  |-  ( ( G  ~<_  om  /\  A. x  e.  G  E. t  e.  U. J ( t G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )  ->  E. g
( g : G --> U. J  /\  A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
17258, 162, 171syl2anc 643 . 2  |-  ( ph  ->  E. g ( g : G --> U. J  /\  A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) ) )
173 simpr 448 . . 3  |-  ( ( g : G --> U. J  /\  A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )  ->  A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
174173eximi 1582 . 2  |-  ( E. g ( g : G --> U. J  /\  A. x  e.  G  (
( g `  x
) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  (
( g `  x
) B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )  ->  E. g A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x
)  +  1 )  /\  ( ( B `
 x )  i^i  ( ( g `  x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
175172, 174syl 16 1  |-  ( ph  ->  E. g A. x  e.  G  ( (
g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )  e.  K
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   {csn 3774   <.cop 3777   U.cuni 3975   U_ciun 4053   class class class wbr 4172   {copab 4225   omcom 4804    X. cxp 4835   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   Fincfn 7068   1c1 8947    + caddc 8949   RR*cxr 9075    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ^cexp 11337   * Metcxmt 16641   Metcme 16642   ballcbl 16643   MetOpencmopn 16646   CMetcms 19160
This theorem is referenced by:  heiborlem10  26419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-seq 11279  df-exp 11338  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cmet 19163
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