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Theorem heiborlem3 26513
Description: Lemma for heibor 26521. Using countable choice ax-cc 8307, 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 26511 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 8307 via iunctb 8441), and so we can apply ax-cc 8307 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 10219 . . . . . 6  |-  NN0  e.  _V
2 fvex 5734 . . . . . . 7  |-  ( F `
 t )  e. 
_V
3 snex 4397 . . . . . . 7  |-  { t }  e.  _V
42, 3xpex 4982 . . . . . 6  |-  ( ( F `  t )  X.  { t } )  e.  _V
51, 4iunex 5983 . . . . 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 4992 . . . . . . . 8  |-  Rel  G
8 1st2nd 6385 . . . . . . . 8  |-  ( ( Rel  G  /\  x  e.  G )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
97, 8mpan 652 . . . . . . 7  |-  ( x  e.  G  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
109eleq1d 2501 . . . . . . . . . . 11  |-  ( x  e.  G  ->  (
x  e.  G  <->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  G ) )
11 df-br 4205 . . . . . . . . . . 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 5734 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
16 fvex 5734 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
1713, 14, 6, 15, 16heiborlem2 26512 . . . . . . . . . 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 3833 . . . . . . . . . . . 12  |-  ( 2nd `  x )  e.  {
( 2nd `  x
) }
21 opelxp 4900 . . . . . . . . . . . 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 5720 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  ( F `  t )  =  ( F `  ( 2nd `  x ) ) )
24 sneq 3817 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  { t }  =  { ( 2nd `  x ) } )
2523, 24xpeq12d 4895 . . . . . . . . . . . . 13  |-  ( t  =  ( 2nd `  x
)  ->  ( ( F `  t )  X.  { t } )  =  ( ( F `
 ( 2nd `  x
) )  X.  {
( 2nd `  x
) } ) )
2625eleq2d 2502 . . . . . . . . . . . 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 3044 . . . . . . . . . . 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 4089 . . . . . . . . . 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 2509 . . . . . 6  |-  ( x  e.  G  ->  x  e.  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } ) )
3433ssriv 3344 . . . . 5  |-  G  C_  U_ t  e.  NN0  (
( F `  t
)  X.  { t } )
35 ssdomg 7145 . . . . 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 11310 . . . . . . 7  |-  NN0  ~~  NN
38 nnenom 11311 . . . . . . 7  |-  NN  ~~  om
3937, 38entri 7153 . . . . . 6  |-  NN0  ~~  om
40 endom 7126 . . . . . 6  |-  ( NN0  ~~  om  ->  NN0  ~<_  om )
4139, 40ax-mp 8 . . . . 5  |-  NN0  ~<_  om
42 vex 2951 . . . . . . . 8  |-  t  e. 
_V
432, 42xpsnen 7184 . . . . . . 7  |-  ( ( F `  t )  X.  { t } )  ~~  ( F `
 t )
44 inss2 3554 . . . . . . . . 9  |-  ( ~P X  i^i  Fin )  C_ 
Fin
45 heibor.7 . . . . . . . . . 10  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
4645ffvelrnda 5862 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  ( ~P X  i^i  Fin ) )
4744, 46sseldi 3338 . . . . . . . 8  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  Fin )
48 isfinite 7599 . . . . . . . . 9  |-  ( ( F `  t )  e.  Fin  <->  ( F `  t )  ~<  om )
49 sdomdom 7127 . . . . . . . . 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 7157 . . . . . . 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 2781 . . . . 5  |-  ( ph  ->  A. t  e.  NN0  ( ( F `  t )  X.  {
t } )  ~<_  om )
55 iunctb 8441 . . . . 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 7152 . . . 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 10252 . . . . . . . . 9  |-  ( ( 2nd `  x )  e.  NN0  ->  ( ( 2nd `  x )  +  1 )  e. 
NN0 )
6159, 60syl 16 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 2nd `  x
)  +  1 )  e.  NN0 )
62 ffvelrn 5860 . . . . . . . 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 3338 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
Fin )
65 iunin2 4147 . . . . . . . 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 6080 . . . . . . . . . . . . . . . 16  |-  ( y  =  t  ->  (
y B n )  =  ( t B n ) )
6867cbviunv 4122 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  n ) ( t B n )
69 fveq2 5720 . . . . . . . . . . . . . . . 16  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( F `  n )  =  ( F `  ( ( 2nd `  x )  +  1 ) ) )
7069iuneq1d 4108 . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . 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 6081 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( t B n )  =  ( t B ( ( 2nd `  x
)  +  1 ) ) )
7372iuneq2d 4110 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . 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 2446 . . . . . . . . . . . 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 3043 . . . . . . . . . . 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 3534 . . . . . . . . 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 5724 . . . . . . . . . . . . . 14  |-  ( x  e.  G  ->  ( B `  x )  =  ( B `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
80 df-ov 6076 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) B ( 2nd `  x
) )  =  ( B `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
8179, 80syl6eqr 2485 . . . . . . . . . . . . 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 3553 . . . . . . . . . . . . . . . 16  |-  ( ~P X  i^i  Fin )  C_ 
~P X
84 ffvelrn 5860 . . . . . . . . . . . . . . . . 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 3338 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  e. 
~P X )
8786elpwid 3800 . . . . . . . . . . . . . 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 3341 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 1st `  x )  e.  X )
9159adantl 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 2nd `  x )  e. 
NN0 )
92 oveq1 6080 . . . . . . . . . . . . . 14  |-  ( z  =  ( 1st `  x
)  ->  ( z
( ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) ) )
93 oveq2 6081 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( 2nd `  x
)  ->  ( 2 ^ m )  =  ( 2 ^ ( 2nd `  x ) ) )
9493oveq2d 6089 . . . . . . . . . . . . . . 15  |-  ( m  =  ( 2nd `  x
)  ->  ( 1  /  ( 2 ^ m ) )  =  ( 1  /  (
2 ^ ( 2nd `  x ) ) ) )
9594oveq2d 6089 . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  e.  _V
9892, 95, 96, 97ovmpt2 6201 . . . . . . . . . . . . 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 2467 . . . . . . . . . . 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 19231 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
103101, 102syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  ( Met `  X ) )
104 metxmet 18356 . . . . . . . . . . . . . 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 10125 . . . . . . . . . . . . . . . 16  |-  2  e.  NN
108 nnexpcl 11386 . . . . . . . . . . . . . . . 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 10639 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  (
2 ^ ( 2nd `  x ) )  e.  RR+ )
111110rpreccld 10650 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e.  RR+ )
112111rpxrd 10641 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e. 
RR* )
113 blssm 18440 . . . . . . . . . . . 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 3374 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  C_  X )
116 df-ss 3326 . . . . . . . . . 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 2469 . . . . . . . 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 2479 . . . . . . 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 3393 . . . . . . 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 2509 . . . . . . 7  |-  ( x  e.  G  ->  ( B `  x )  e.  K )
124123adantl 453 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  e.  K )
125 fvex 5734 . . . . . . . 8  |-  ( B `
 x )  e. 
_V
126125inex1 4336 . . . . . . 7  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  _V
12713, 14, 126heiborlem1 26511 . . . . . 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 3338 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
~P X )
130129elpwid 3800 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  X )
13113mopnuni 18463 . . . . . . . . . . . . 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 3376 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  U. J )
135134sselda 3340 . . . . . . . . 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 7179 . . . . . . . . . . . 12  |-  { ( t B ( ( 2nd `  x )  +  1 ) ) }  e.  Fin
140 inss2 3554 . . . . . . . . . . . . 13  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  (
t B ( ( 2nd `  x )  +  1 ) )
141 ovex 6098 . . . . . . . . . . . . . . 15  |-  ( t B ( ( 2nd `  x )  +  1 ) )  e.  _V
142141unisn 4023 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  ( t B ( ( 2nd `  x )  +  1 ) )
143 uniiun 4136 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
144142, 143eqtr3i 2457 . . . . . . . . . . . . 13  |-  ( t B ( ( 2nd `  x )  +  1 ) )  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
145140, 144sseqtri 3372 . . . . . . . . . . . 12  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  U_ g  e.  { ( t B ( ( 2nd `  x
)  +  1 ) ) } g
146 vex 2951 . . . . . . . . . . . . 13  |-  g  e. 
_V
14713, 14, 146heiborlem1 26511 . . . . . . . . . . . 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 2495 . . . . . . . . . . . 12  |-  ( g  =  ( t B ( ( 2nd `  x
)  +  1 ) )  ->  ( g  e.  K  <->  ( t B ( ( 2nd `  x
)  +  1 ) )  e.  K ) )
150141, 149rexsn 3842 . . . . . . . . . . 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 6098 . . . . . . . . . . . 12  |-  ( ( 2nd `  x )  +  1 )  e. 
_V
15313, 14, 6, 42, 152heiborlem2 26512 . . . . . . . . . . 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 2807 . . . . 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 2781 . . 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 5734 . . . . . 6  |-  ( MetOpen `  D )  e.  _V
16413, 163eqeltri 2505 . . . . 5  |-  J  e. 
_V
165164uniex 4697 . . . 4  |-  U. J  e.  _V
166 breq1 4207 . . . . 5  |-  ( t  =  ( g `  x )  ->  (
t G ( ( 2nd `  x )  +  1 )  <->  ( g `  x ) G ( ( 2nd `  x
)  +  1 ) ) )
167 oveq1 6080 . . . . . . 7  |-  ( t  =  ( g `  x )  ->  (
t B ( ( 2nd `  x )  +  1 ) )  =  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )
168167ineq2d 3534 . . . . . 6  |-  ( t  =  ( g `  x )  ->  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  =  ( ( B `  x )  i^i  (
( g `  x
) B ( ( 2nd `  x )  +  1 ) ) ) )
169168eleq1d 2501 . . . . 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 8313 . . 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 1585 . 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   <.cop 3809   U.cuni 4007   U_ciun 4085   class class class wbr 4204   {copab 4257   omcom 4837    X. cxp 4868   Rel wrel 4875   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100   Fincfn 7101   1c1 8983    + caddc 8985   RR*cxr 9111    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ^cexp 11374   * Metcxmt 16678   Metcme 16679   ballcbl 16680   MetOpencmopn 16683   CMetcms 19199
This theorem is referenced by:  heiborlem10  26520
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 4693  ax-inf2 7588  ax-cc 8307  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-seq 11316  df-exp 11375  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-top 16955  df-bases 16957  df-topon 16958  df-cmet 19202
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