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Theorem heiborlem3 26213
Description: Lemma for heibor 26221. Using countable choice ax-cc 8248, 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 26211 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 8248 via iunctb 8382), and so we can apply ax-cc 8248 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 10159 . . . . . 6  |-  NN0  e.  _V
2 fvex 5682 . . . . . . 7  |-  ( F `
 t )  e. 
_V
3 snex 4346 . . . . . . 7  |-  { t }  e.  _V
42, 3xpex 4930 . . . . . 6  |-  ( ( F `  t )  X.  { t } )  e.  _V
51, 4iunex 5930 . . . . 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 4940 . . . . . . . 8  |-  Rel  G
8 1st2nd 6332 . . . . . . . 8  |-  ( ( Rel  G  /\  x  e.  G )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
97, 8mpan 652 . . . . . . 7  |-  ( x  e.  G  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
109eleq1d 2453 . . . . . . . . . . 11  |-  ( x  e.  G  ->  (
x  e.  G  <->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  G ) )
11 df-br 4154 . . . . . . . . . . 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 5682 . . . . . . . . . . 11  |-  ( 1st `  x )  e.  _V
16 fvex 5682 . . . . . . . . . . 11  |-  ( 2nd `  x )  e.  _V
1713, 14, 6, 15, 16heiborlem2 26212 . . . . . . . . . 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 3784 . . . . . . . . . . . 12  |-  ( 2nd `  x )  e.  {
( 2nd `  x
) }
21 opelxp 4848 . . . . . . . . . . . 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 5668 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  ( F `  t )  =  ( F `  ( 2nd `  x ) ) )
24 sneq 3768 . . . . . . . . . . . . . 14  |-  ( t  =  ( 2nd `  x
)  ->  { t }  =  { ( 2nd `  x ) } )
2523, 24xpeq12d 4843 . . . . . . . . . . . . 13  |-  ( t  =  ( 2nd `  x
)  ->  ( ( F `  t )  X.  { t } )  =  ( ( F `
 ( 2nd `  x
) )  X.  {
( 2nd `  x
) } ) )
2625eleq2d 2454 . . . . . . . . . . . 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 2995 . . . . . . . . . . 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 4039 . . . . . . . . . 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 2461 . . . . . 6  |-  ( x  e.  G  ->  x  e.  U_ t  e.  NN0  ( ( F `  t )  X.  {
t } ) )
3433ssriv 3295 . . . . 5  |-  G  C_  U_ t  e.  NN0  (
( F `  t
)  X.  { t } )
35 ssdomg 7089 . . . . 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 11245 . . . . . . 7  |-  NN0  ~~  NN
38 nnenom 11246 . . . . . . 7  |-  NN  ~~  om
3937, 38entri 7097 . . . . . 6  |-  NN0  ~~  om
40 endom 7070 . . . . . 6  |-  ( NN0  ~~  om  ->  NN0  ~<_  om )
4139, 40ax-mp 8 . . . . 5  |-  NN0  ~<_  om
42 vex 2902 . . . . . . . 8  |-  t  e. 
_V
432, 42xpsnen 7128 . . . . . . 7  |-  ( ( F `  t )  X.  { t } )  ~~  ( F `
 t )
44 inss2 3505 . . . . . . . . 9  |-  ( ~P X  i^i  Fin )  C_ 
Fin
45 heibor.7 . . . . . . . . . 10  |-  ( ph  ->  F : NN0 --> ( ~P X  i^i  Fin )
)
4645ffvelrnda 5809 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  ( ~P X  i^i  Fin ) )
4744, 46sseldi 3289 . . . . . . . 8  |-  ( (
ph  /\  t  e.  NN0 )  ->  ( F `  t )  e.  Fin )
48 isfinite 7540 . . . . . . . . 9  |-  ( ( F `  t )  e.  Fin  <->  ( F `  t )  ~<  om )
49 sdomdom 7071 . . . . . . . . 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 7101 . . . . . . 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 2732 . . . . 5  |-  ( ph  ->  A. t  e.  NN0  ( ( F `  t )  X.  {
t } )  ~<_  om )
55 iunctb 8382 . . . . 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 7096 . . . 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 10192 . . . . . . . . 9  |-  ( ( 2nd `  x )  e.  NN0  ->  ( ( 2nd `  x )  +  1 )  e. 
NN0 )
6159, 60syl 16 . . . . . . . 8  |-  ( x  e.  G  ->  (
( 2nd `  x
)  +  1 )  e.  NN0 )
62 ffvelrn 5807 . . . . . . . 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 3289 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
Fin )
65 iunin2 4096 . . . . . . . 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 6027 . . . . . . . . . . . . . . . 16  |-  ( y  =  t  ->  (
y B n )  =  ( t B n ) )
6867cbviunv 4071 . . . . . . . . . . . . . . 15  |-  U_ y  e.  ( F `  n
) ( y B n )  =  U_ t  e.  ( F `  n ) ( t B n )
69 fveq2 5668 . . . . . . . . . . . . . . . 16  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( F `  n )  =  ( F `  ( ( 2nd `  x )  +  1 ) ) )
7069iuneq1d 4058 . . . . . . . . . . . . . . 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 2431 . . . . . . . . . . . . . 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 6028 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( 2nd `  x )  +  1 )  ->  ( t B n )  =  ( t B ( ( 2nd `  x
)  +  1 ) ) )
7372iuneq2d 4060 . . . . . . . . . . . . . 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 2419 . . . . . . . . . . . . 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 2398 . . . . . . . . . . . 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 2994 . . . . . . . . . . 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 3485 . . . . . . . . 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 5672 . . . . . . . . . . . . . 14  |-  ( x  e.  G  ->  ( B `  x )  =  ( B `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
80 df-ov 6023 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) B ( 2nd `  x
) )  =  ( B `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )
8179, 80syl6eqr 2437 . . . . . . . . . . . . 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 3504 . . . . . . . . . . . . . . . 16  |-  ( ~P X  i^i  Fin )  C_ 
~P X
84 ffvelrn 5807 . . . . . . . . . . . . . . . . 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 3289 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( 2nd `  x ) )  e. 
~P X )
8786elpwid 3751 . . . . . . . . . . . . . 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 3292 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 1st `  x )  e.  X )
9159adantl 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  ( 2nd `  x )  e. 
NN0 )
92 oveq1 6027 . . . . . . . . . . . . . 14  |-  ( z  =  ( 1st `  x
)  ->  ( z
( ball `  D )
( 1  /  (
2 ^ m ) ) )  =  ( ( 1st `  x
) ( ball `  D
) ( 1  / 
( 2 ^ m
) ) ) )
93 oveq2 6028 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( 2nd `  x
)  ->  ( 2 ^ m )  =  ( 2 ^ ( 2nd `  x ) ) )
9493oveq2d 6036 . . . . . . . . . . . . . . 15  |-  ( m  =  ( 2nd `  x
)  ->  ( 1  /  ( 2 ^ m ) )  =  ( 1  /  (
2 ^ ( 2nd `  x ) ) ) )
9594oveq2d 6036 . . . . . . . . . . . . . 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 6045 . . . . . . . . . . . . . 14  |-  ( ( 1st `  x ) ( ball `  D
) ( 1  / 
( 2 ^ ( 2nd `  x ) ) ) )  e.  _V
9892, 95, 96, 97ovmpt2 6148 . . . . . . . . . . . . 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 2419 . . . . . . . . . . 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 19110 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( CMet `  X
)  ->  D  e.  ( Met `  X ) )
103101, 102syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  D  e.  ( Met `  X ) )
104 metxmet 18273 . . . . . . . . . . . . . 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 10065 . . . . . . . . . . . . . . . 16  |-  2  e.  NN
108 nnexpcl 11321 . . . . . . . . . . . . . . . 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 10579 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  G )  ->  (
2 ^ ( 2nd `  x ) )  e.  RR+ )
111110rpreccld 10590 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e.  RR+ )
112111rpxrd 10581 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  (
1  /  ( 2 ^ ( 2nd `  x
) ) )  e. 
RR* )
113 blssm 18342 . . . . . . . . . . . 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 3325 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  C_  X )
116 df-ss 3277 . . . . . . . . . 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 2421 . . . . . . . 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 2431 . . . . . . 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 3344 . . . . . . 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 2461 . . . . . . 7  |-  ( x  e.  G  ->  ( B `  x )  e.  K )
124123adantl 453 . . . . . 6  |-  ( (
ph  /\  x  e.  G )  ->  ( B `  x )  e.  K )
125 fvex 5682 . . . . . . . 8  |-  ( B `
 x )  e. 
_V
126125inex1 4285 . . . . . . 7  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  e.  _V
12713, 14, 126heiborlem1 26211 . . . . . 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 3289 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  e. 
~P X )
130129elpwid 3751 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  X )
13113mopnuni 18361 . . . . . . . . . . . . 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 3327 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  G )  ->  ( F `  ( ( 2nd `  x )  +  1 ) )  C_  U. J )
135134sselda 3291 . . . . . . . . 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 7123 . . . . . . . . . . . 12  |-  { ( t B ( ( 2nd `  x )  +  1 ) ) }  e.  Fin
140 inss2 3505 . . . . . . . . . . . . 13  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  (
t B ( ( 2nd `  x )  +  1 ) )
141 ovex 6045 . . . . . . . . . . . . . . 15  |-  ( t B ( ( 2nd `  x )  +  1 ) )  e.  _V
142141unisn 3973 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  ( t B ( ( 2nd `  x )  +  1 ) )
143 uniiun 4085 . . . . . . . . . . . . . 14  |-  U. {
( t B ( ( 2nd `  x
)  +  1 ) ) }  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
144142, 143eqtr3i 2409 . . . . . . . . . . . . 13  |-  ( t B ( ( 2nd `  x )  +  1 ) )  =  U_ g  e.  { (
t B ( ( 2nd `  x )  +  1 ) ) } g
145140, 144sseqtri 3323 . . . . . . . . . . . 12  |-  ( ( B `  x )  i^i  ( t B ( ( 2nd `  x
)  +  1 ) ) )  C_  U_ g  e.  { ( t B ( ( 2nd `  x
)  +  1 ) ) } g
146 vex 2902 . . . . . . . . . . . . 13  |-  g  e. 
_V
14713, 14, 146heiborlem1 26211 . . . . . . . . . . . 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 2447 . . . . . . . . . . . 12  |-  ( g  =  ( t B ( ( 2nd `  x
)  +  1 ) )  ->  ( g  e.  K  <->  ( t B ( ( 2nd `  x
)  +  1 ) )  e.  K ) )
150141, 149rexsn 3793 . . . . . . . . . . 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 6045 . . . . . . . . . . . 12  |-  ( ( 2nd `  x )  +  1 )  e. 
_V
15313, 14, 6, 42, 152heiborlem2 26212 . . . . . . . . . . 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 2758 . . . . 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 2732 . . 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 5682 . . . . . 6  |-  ( MetOpen `  D )  e.  _V
16413, 163eqeltri 2457 . . . . 5  |-  J  e. 
_V
165164uniex 4645 . . . 4  |-  U. J  e.  _V
166 breq1 4156 . . . . 5  |-  ( t  =  ( g `  x )  ->  (
t G ( ( 2nd `  x )  +  1 )  <->  ( g `  x ) G ( ( 2nd `  x
)  +  1 ) ) )
167 oveq1 6027 . . . . . . 7  |-  ( t  =  ( g `  x )  ->  (
t B ( ( 2nd `  x )  +  1 ) )  =  ( ( g `
 x ) B ( ( 2nd `  x
)  +  1 ) ) )
168167ineq2d 3485 . . . . . 6  |-  ( t  =  ( g `  x )  ->  (
( B `  x
)  i^i  ( t B ( ( 2nd `  x )  +  1 ) ) )  =  ( ( B `  x )  i^i  (
( g `  x
) B ( ( 2nd `  x )  +  1 ) ) ) )
169168eleq1d 2453 . . . . 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 8254 . . 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 1717   {cab 2373   A.wral 2649   E.wrex 2650   _Vcvv 2899    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   {csn 3757   <.cop 3760   U.cuni 3957   U_ciun 4035   class class class wbr 4153   {copab 4206   omcom 4785    X. cxp 4816   Rel wrel 4823   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022   1stc1st 6286   2ndc2nd 6287    ~~ cen 7042    ~<_ cdom 7043    ~< csdm 7044   Fincfn 7045   1c1 8924    + caddc 8926   RR*cxr 9052    / cdiv 9609   NNcn 9932   2c2 9981   NN0cn0 10153   ^cexp 11309   * Metcxmt 16612   Metcme 16613   ballcbl 16614   MetOpencmopn 16617   CMetcms 19078
This theorem is referenced by:  heiborlem10  26220
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cc 8248  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-acn 7762  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-seq 11251  df-exp 11310  df-topgen 13594  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-top 16886  df-bases 16888  df-topon 16889  df-cmet 19081
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