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Theorem caublcls 19222
Description: The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
caubl.2  |-  ( ph  ->  D  e.  ( * Met `  X ) )
caubl.3  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
caubl.4  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) )
caublcls.6  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
caublcls  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  P  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( F `  A )
) ) )
Distinct variable groups:    D, n    n, F    n, X
Allowed substitution hints:    ph( n)    A( n)    P( n)    J( n)

Proof of Theorem caublcls
Dummy variables  k 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2412 . 2  |-  ( ZZ>= `  A )  =  (
ZZ>= `  A )
2 caubl.2 . . . 4  |-  ( ph  ->  D  e.  ( * Met `  X ) )
323ad2ant1 978 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  D  e.  ( * Met `  X ) )
4 caublcls.6 . . . 4  |-  J  =  ( MetOpen `  D )
54mopntopon 18430 . . 3  |-  ( D  e.  ( * Met `  X )  ->  J  e.  (TopOn `  X )
)
63, 5syl 16 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  J  e.  (TopOn `  X ) )
7 simp3 959 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  A  e.  NN )
87nnzd 10338 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  A  e.  ZZ )
9 simp2 958 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 1st  o.  F
) ( ~~> t `  J ) P )
10 fveq2 5695 . . . . . . . . 9  |-  ( r  =  A  ->  ( F `  r )  =  ( F `  A ) )
1110fveq2d 5699 . . . . . . . 8  |-  ( r  =  A  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  A ) ) )
1211sseq1d 3343 . . . . . . 7  |-  ( r  =  A  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  A ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
1312imbi2d 308 . . . . . 6  |-  ( r  =  A  ->  (
( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) ) )  <-> 
( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  C_  ( ( ball `  D
) `  ( F `  A ) ) ) ) )
14 fveq2 5695 . . . . . . . . 9  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
1514fveq2d 5699 . . . . . . . 8  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
1615sseq1d 3343 . . . . . . 7  |-  ( r  =  k  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
1716imbi2d 308 . . . . . 6  |-  ( r  =  k  ->  (
( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) ) )  <-> 
( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  k ) )  C_  ( ( ball `  D
) `  ( F `  A ) ) ) ) )
18 fveq2 5695 . . . . . . . . 9  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1918fveq2d 5699 . . . . . . . 8  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
2019sseq1d 3343 . . . . . . 7  |-  ( r  =  ( k  +  1 )  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
2120imbi2d 308 . . . . . 6  |-  ( r  =  ( k  +  1 )  ->  (
( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) ) )  <-> 
( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  A ) ) ) ) )
22 ssid 3335 . . . . . . 7  |-  ( (
ball `  D ) `  ( F `  A
) )  C_  (
( ball `  D ) `  ( F `  A
) )
2322a1ii 25 . . . . . 6  |-  ( A  e.  ZZ  ->  (
( ph  /\  A  e.  NN )  ->  (
( ball `  D ) `  ( F `  A
) )  C_  (
( ball `  D ) `  ( F `  A
) ) ) )
24 caubl.4 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) ) )
25 nnuz 10485 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
2625uztrn2 10467 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( ZZ>= `  A ) )  -> 
k  e.  NN )
27 oveq1 6055 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
2827fveq2d 5699 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2928fveq2d 5699 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
30 fveq2 5695 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3130fveq2d 5699 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
3229, 31sseq12d 3345 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
3332rspccva 3019 . . . . . . . . . . 11  |-  ( ( A. n  e.  NN  ( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  /\  k  e.  NN )  ->  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  k ) ) )
3424, 26, 33syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  NN  /\  k  e.  ( ZZ>= `  A )
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  k
) ) )
3534anassrs 630 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) )
36 sstr2 3323 . . . . . . . . 9  |-  ( ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  k ) )  -> 
( ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  A ) )  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
3735, 36syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( (
( ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  A
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  A
) ) ) )
3837expcom 425 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  A
)  ->  ( ( ph  /\  A  e.  NN )  ->  ( ( (
ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  A
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  A
) ) ) ) )
3938a2d 24 . . . . . 6  |-  ( k  e.  ( ZZ>= `  A
)  ->  ( (
( ph  /\  A  e.  NN )  ->  (
( ball `  D ) `  ( F `  k
) )  C_  (
( ball `  D ) `  ( F `  A
) ) )  -> 
( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  A ) ) ) ) )
4013, 17, 21, 17, 23, 39uzind4 10498 . . . . 5  |-  ( k  e.  ( ZZ>= `  A
)  ->  ( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
4140impcom 420 . . . 4  |-  ( ( ( ph  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) )
42413adantl2 1114 . . 3  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) )
433adantr 452 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  D  e.  ( * Met `  X
) )
44 simpl1 960 . . . . . . . 8  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ph )
45 caubl.3 . . . . . . . 8  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
4644, 45syl 16 . . . . . . 7  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  F : NN
--> ( X  X.  RR+ ) )
477, 26sylan 458 . . . . . . 7  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  k  e.  NN )
4846, 47ffvelrnd 5838 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
49 xp1st 6343 . . . . . 6  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  k
) )  e.  X
)
5048, 49syl 16 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( 1st `  ( F `  k
) )  e.  X
)
51 xp2nd 6344 . . . . . 6  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
5248, 51syl 16 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
53 blcntr 18404 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  ( 1st `  ( F `  k )
)  e.  X  /\  ( 2nd `  ( F `
 k ) )  e.  RR+ )  ->  ( 1st `  ( F `  k ) )  e.  ( ( 1st `  ( F `  k )
) ( ball `  D
) ( 2nd `  ( F `  k )
) ) )
5443, 50, 52, 53syl3anc 1184 . . . 4  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( 1st `  ( F `  k
) )  e.  ( ( 1st `  ( F `  k )
) ( ball `  D
) ( 2nd `  ( F `  k )
) ) )
55 fvco3 5767 . . . . 5  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  (
( 1st  o.  F
) `  k )  =  ( 1st `  ( F `  k )
) )
5646, 47, 55syl2anc 643 . . . 4  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( 1st  o.  F ) `  k )  =  ( 1st `  ( F `
 k ) ) )
57 1st2nd2 6353 . . . . . . 7  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
5848, 57syl 16 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
5958fveq2d 5699 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  =  ( (
ball `  D ) `  <. ( 1st `  ( F `  k )
) ,  ( 2nd `  ( F `  k
) ) >. )
)
60 df-ov 6051 . . . . 5  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
6159, 60syl6eqr 2462 . . . 4  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  =  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) ) )
6254, 56, 613eltr4d 2493 . . 3  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( 1st  o.  F ) `  k )  e.  ( ( ball `  D
) `  ( F `  k ) ) )
6342, 62sseldd 3317 . 2  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( 1st  o.  F ) `  k )  e.  ( ( ball `  D
) `  ( F `  A ) ) )
6445ffvelrnda 5837 . . . . . . 7  |-  ( (
ph  /\  A  e.  NN )  ->  ( F `
 A )  e.  ( X  X.  RR+ ) )
65643adant2 976 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( F `  A
)  e.  ( X  X.  RR+ ) )
66 1st2nd2 6353 . . . . . 6  |-  ( ( F `  A )  e.  ( X  X.  RR+ )  ->  ( F `  A )  =  <. ( 1st `  ( F `
 A ) ) ,  ( 2nd `  ( F `  A )
) >. )
6765, 66syl 16 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( F `  A
)  =  <. ( 1st `  ( F `  A ) ) ,  ( 2nd `  ( F `  A )
) >. )
6867fveq2d 5699 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  A
) ) ,  ( 2nd `  ( F `
 A ) )
>. ) )
69 df-ov 6051 . . . 4  |-  ( ( 1st `  ( F `
 A ) ) ( ball `  D
) ( 2nd `  ( F `  A )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  A
) ) ,  ( 2nd `  ( F `
 A ) )
>. )
7068, 69syl6eqr 2462 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  =  ( ( 1st `  ( F `  A )
) ( ball `  D
) ( 2nd `  ( F `  A )
) ) )
71 xp1st 6343 . . . . 5  |-  ( ( F `  A )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  A
) )  e.  X
)
7265, 71syl 16 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 1st `  ( F `  A )
)  e.  X )
73 xp2nd 6344 . . . . . 6  |-  ( ( F `  A )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  A
) )  e.  RR+ )
7465, 73syl 16 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 2nd `  ( F `  A )
)  e.  RR+ )
7574rpxrd 10613 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 2nd `  ( F `  A )
)  e.  RR* )
76 blssm 18409 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  ( 1st `  ( F `  A )
)  e.  X  /\  ( 2nd `  ( F `
 A ) )  e.  RR* )  ->  (
( 1st `  ( F `  A )
) ( ball `  D
) ( 2nd `  ( F `  A )
) )  C_  X
)
773, 72, 75, 76syl3anc 1184 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( 1st `  ( F `  A )
) ( ball `  D
) ( 2nd `  ( F `  A )
) )  C_  X
)
7870, 77eqsstrd 3350 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  C_  X )
791, 6, 8, 9, 63, 78lmcls 17328 1  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  P  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( F `  A )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674    C_ wss 3288   <.cop 3785   class class class wbr 4180    X. cxp 4843    o. ccom 4849   -->wf 5417   ` cfv 5421  (class class class)co 6048   1stc1st 6314   2ndc2nd 6315   1c1 8955    + caddc 8957   RR*cxr 9083   NNcn 9964   ZZcz 10246   ZZ>=cuz 10452   RR+crp 10576   * Metcxmt 16649   ballcbl 16651   MetOpencmopn 16654  TopOnctopon 16922   clsccl 17045   ~~> tclm 17252
This theorem is referenced by:  bcthlem3  19240  heiborlem8  26425
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929  df-cld 17046  df-ntr 17047  df-cls 17048  df-lm 17255
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