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Theorem caublcls 19134
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 2389 . 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 18361 . . 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 10308 . 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 5670 . . . . . . . . 9  |-  ( r  =  A  ->  ( F `  r )  =  ( F `  A ) )
1110fveq2d 5674 . . . . . . . 8  |-  ( r  =  A  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  A ) ) )
1211sseq1d 3320 . . . . . . 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 5670 . . . . . . . . 9  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
1514fveq2d 5674 . . . . . . . 8  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
1615sseq1d 3320 . . . . . . 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 5670 . . . . . . . . 9  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1918fveq2d 5674 . . . . . . . 8  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
2019sseq1d 3320 . . . . . . 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 3312 . . . . . . 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 10455 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
2625uztrn2 10437 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( ZZ>= `  A ) )  -> 
k  e.  NN )
27 oveq1 6029 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
2827fveq2d 5674 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2928fveq2d 5674 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
30 fveq2 5670 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3130fveq2d 5674 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
3229, 31sseq12d 3322 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
3332rspccva 2996 . . . . . . . . . . 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 3300 . . . . . . . . 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 10468 . . . . 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 5812 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
49 xp1st 6317 . . . . . 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 6318 . . . . . 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 18340 . . . . 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 5741 . . . . 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 6327 . . . . . . 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 5674 . . . . 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 6025 . . . . 5  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
6159, 60syl6eqr 2439 . . . 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 2470 . . 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 3294 . 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 5811 . . . . . . 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 6327 . . . . . 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 5674 . . . 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 6025 . . . 4  |-  ( ( 1st `  ( F `
 A ) ) ( ball `  D
) ( 2nd `  ( F `  A )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  A
) ) ,  ( 2nd `  ( F `
 A ) )
>. )
7068, 69syl6eqr 2439 . . 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 6317 . . . . 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 6318 . . . . . 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 10583 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 2nd `  ( F `  A )
)  e.  RR* )
76 blssm 18344 . . . 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 3327 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  C_  X )
791, 6, 8, 9, 63, 78lmcls 17290 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 1717   A.wral 2651    C_ wss 3265   <.cop 3762   class class class wbr 4155    X. cxp 4818    o. ccom 4824   -->wf 5392   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   1c1 8926    + caddc 8928   RR*cxr 9054   NNcn 9934   ZZcz 10216   ZZ>=cuz 10422   RR+crp 10546   * Metcxmt 16614   ballcbl 16616   MetOpencmopn 16619  TopOnctopon 16884   clsccl 17007   ~~> tclm 17214
This theorem is referenced by:  bcthlem3  19150  heiborlem8  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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-topgen 13596  df-xmet 16621  df-bl 16623  df-mopn 16624  df-top 16888  df-bases 16890  df-topon 16891  df-cld 17008  df-ntr 17009  df-cls 17010  df-lm 17217
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