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Theorem caublcls 19261
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 2436 . 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 18469 . . 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 10374 . 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 5728 . . . . . . . . 9  |-  ( r  =  A  ->  ( F `  r )  =  ( F `  A ) )
1110fveq2d 5732 . . . . . . . 8  |-  ( r  =  A  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  A ) ) )
1211sseq1d 3375 . . . . . . 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 5728 . . . . . . . . 9  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
1514fveq2d 5732 . . . . . . . 8  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
1615sseq1d 3375 . . . . . . 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 5728 . . . . . . . . 9  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1918fveq2d 5732 . . . . . . . 8  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
2019sseq1d 3375 . . . . . . 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 3367 . . . . . . 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 10521 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
2625uztrn2 10503 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( ZZ>= `  A ) )  -> 
k  e.  NN )
27 oveq1 6088 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
2827fveq2d 5732 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2928fveq2d 5732 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
30 fveq2 5728 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3130fveq2d 5732 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
3229, 31sseq12d 3377 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
3332rspccva 3051 . . . . . . . . . . 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 3355 . . . . . . . . 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 10534 . . . . 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 5871 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
49 xp1st 6376 . . . . . 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 6377 . . . . . 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 18443 . . . . 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 5800 . . . . 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 6386 . . . . . . 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 5732 . . . . 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 6084 . . . . 5  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
6159, 60syl6eqr 2486 . . . 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 2517 . . 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 3349 . 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 5870 . . . . . . 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 6386 . . . . . 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 5732 . . . 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 6084 . . . 4  |-  ( ( 1st `  ( F `
 A ) ) ( ball `  D
) ( 2nd `  ( F `  A )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  A
) ) ,  ( 2nd `  ( F `
 A ) )
>. )
7068, 69syl6eqr 2486 . . 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 6376 . . . . 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 6377 . . . . . 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 10649 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 2nd `  ( F `  A )
)  e.  RR* )
76 blssm 18448 . . . 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 3382 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  C_  X )
791, 6, 8, 9, 63, 78lmcls 17366 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 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   <.cop 3817   class class class wbr 4212    X. cxp 4876    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   1c1 8991    + caddc 8993   RR*cxr 9119   NNcn 10000   ZZcz 10282   ZZ>=cuz 10488   RR+crp 10612   * Metcxmt 16686   ballcbl 16688   MetOpencmopn 16691  TopOnctopon 16959   clsccl 17082   ~~> tclm 17290
This theorem is referenced by:  bcthlem3  19279  heiborlem8  26527
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cld 17083  df-ntr 17084  df-cls 17085  df-lm 17293
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