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Theorem caublcls 21916
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 2454 . 2  |-  ( ZZ>= `  A )  =  (
ZZ>= `  A )
2 caubl.2 . . . 4  |-  ( ph  ->  D  e.  ( *Met `  X ) )
323ad2ant1 1015 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  D  e.  ( *Met `  X ) )
4 caublcls.6 . . . 4  |-  J  =  ( MetOpen `  D )
54mopntopon 21111 . . 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 996 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  A  e.  NN )
87nnzd 10964 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  A  e.  ZZ )
9 simp2 995 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 1st  o.  F
) ( ~~> t `  J ) P )
10 fveq2 5848 . . . . . . . . 9  |-  ( r  =  A  ->  ( F `  r )  =  ( F `  A ) )
1110fveq2d 5852 . . . . . . . 8  |-  ( r  =  A  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  A ) ) )
1211sseq1d 3516 . . . . . . 7  |-  ( r  =  A  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  A ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
1312imbi2d 314 . . . . . 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 5848 . . . . . . . . 9  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
1514fveq2d 5852 . . . . . . . 8  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
1615sseq1d 3516 . . . . . . 7  |-  ( r  =  k  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
1716imbi2d 314 . . . . . 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 5848 . . . . . . . . 9  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1918fveq2d 5852 . . . . . . . 8  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
2019sseq1d 3516 . . . . . . 7  |-  ( r  =  ( k  +  1 )  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
2120imbi2d 314 . . . . . 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 3508 . . . . . . 7  |-  ( (
ball `  D ) `  ( F `  A
) )  C_  (
( ball `  D ) `  ( F `  A
) )
2322a1ii 27 . . . . . 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 eluznn 11153 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( ZZ>= `  A ) )  -> 
k  e.  NN )
26 oveq1 6277 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
2726fveq2d 5852 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2827fveq2d 5852 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
29 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3029fveq2d 5852 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
3128, 30sseq12d 3518 . . . . . . . . . . . 12  |-  ( n  =  k  ->  (
( ( ball `  D
) `  ( F `  ( n  +  1 ) ) )  C_  ( ( ball `  D
) `  ( F `  n ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) ) )
3231rspccva 3206 . . . . . . . . . . 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 ) ) )
3324, 25, 32syl2an 475 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  NN  /\  k  e.  ( ZZ>= `  A )
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  k
) ) )
3433anassrs 646 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) )
35 sstr2 3496 . . . . . . . . 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 ) ) ) )
3634, 35syl 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
) ) ) )
3736expcom 433 . . . . . . 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
) ) ) ) )
3837a2d 26 . . . . . 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 ) ) ) ) )
3913, 17, 21, 17, 23, 38uzind4 11140 . . . . 5  |-  ( k  e.  ( ZZ>= `  A
)  ->  ( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
4039impcom 428 . . . 4  |-  ( ( ( ph  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) )
41403adantl2 1151 . . 3  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) )
423adantr 463 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  D  e.  ( *Met `  X
) )
43 simpl1 997 . . . . . . . 8  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ph )
44 caubl.3 . . . . . . . 8  |-  ( ph  ->  F : NN --> ( X  X.  RR+ ) )
4543, 44syl 16 . . . . . . 7  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  F : NN
--> ( X  X.  RR+ ) )
46253ad2antl3 1158 . . . . . . 7  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  k  e.  NN )
4745, 46ffvelrnd 6008 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
48 xp1st 6803 . . . . . 6  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  k
) )  e.  X
)
4947, 48syl 16 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( 1st `  ( F `  k
) )  e.  X
)
50 xp2nd 6804 . . . . . 6  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
5147, 50syl 16 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( 2nd `  ( F `  k
) )  e.  RR+ )
52 blcntr 21085 . . . . 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 )
) ) )
5342, 49, 51, 52syl3anc 1226 . . . 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 )
) ) )
54 fvco3 5925 . . . . 5  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  (
( 1st  o.  F
) `  k )  =  ( 1st `  ( F `  k )
) )
5545, 46, 54syl2anc 659 . . . 4  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( 1st  o.  F ) `  k )  =  ( 1st `  ( F `
 k ) ) )
56 1st2nd2 6810 . . . . . . 7  |-  ( ( F `  k )  e.  ( X  X.  RR+ )  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
5747, 56syl 16 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( F `  k )  =  <. ( 1st `  ( F `
 k ) ) ,  ( 2nd `  ( F `  k )
) >. )
5857fveq2d 5852 . . . . 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
) ) >. )
)
59 df-ov 6273 . . . . 5  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
6058, 59syl6eqr 2513 . . . 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 )
) ) )
6153, 55, 603eltr4d 2557 . . 3  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( 1st  o.  F ) `  k )  e.  ( ( ball `  D
) `  ( F `  k ) ) )
6241, 61sseldd 3490 . 2  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( 1st  o.  F ) `  k )  e.  ( ( ball `  D
) `  ( F `  A ) ) )
6344ffvelrnda 6007 . . . . . . 7  |-  ( (
ph  /\  A  e.  NN )  ->  ( F `
 A )  e.  ( X  X.  RR+ ) )
64633adant2 1013 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( F `  A
)  e.  ( X  X.  RR+ ) )
65 1st2nd2 6810 . . . . . 6  |-  ( ( F `  A )  e.  ( X  X.  RR+ )  ->  ( F `  A )  =  <. ( 1st `  ( F `
 A ) ) ,  ( 2nd `  ( F `  A )
) >. )
6664, 65syl 16 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( F `  A
)  =  <. ( 1st `  ( F `  A ) ) ,  ( 2nd `  ( F `  A )
) >. )
6766fveq2d 5852 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  A
) ) ,  ( 2nd `  ( F `
 A ) )
>. ) )
68 df-ov 6273 . . . 4  |-  ( ( 1st `  ( F `
 A ) ) ( ball `  D
) ( 2nd `  ( F `  A )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  A
) ) ,  ( 2nd `  ( F `
 A ) )
>. )
6967, 68syl6eqr 2513 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  =  ( ( 1st `  ( F `  A )
) ( ball `  D
) ( 2nd `  ( F `  A )
) ) )
70 xp1st 6803 . . . . 5  |-  ( ( F `  A )  e.  ( X  X.  RR+ )  ->  ( 1st `  ( F `  A
) )  e.  X
)
7164, 70syl 16 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 1st `  ( F `  A )
)  e.  X )
72 xp2nd 6804 . . . . . 6  |-  ( ( F `  A )  e.  ( X  X.  RR+ )  ->  ( 2nd `  ( F `  A
) )  e.  RR+ )
7364, 72syl 16 . . . . 5  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 2nd `  ( F `  A )
)  e.  RR+ )
7473rpxrd 11260 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 2nd `  ( F `  A )
)  e.  RR* )
75 blssm 21090 . . . 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
)
763, 71, 74, 75syl3anc 1226 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( 1st `  ( F `  A )
) ( ball `  D
) ( 2nd `  ( F `  A )
) )  C_  X
)
7769, 76eqsstrd 3523 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  C_  X )
781, 6, 8, 9, 62, 77lmcls 19973 1  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  P  e.  ( ( cls `  J ) `
 ( ( ball `  D ) `  ( F `  A )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   <.cop 4022   class class class wbr 4439    X. cxp 4986    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   1c1 9482    + caddc 9484   RR*cxr 9616   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   RR+crp 11221   *Metcxmt 18601   ballcbl 18603   MetOpencmopn 18606  TopOnctopon 19565   clsccl 19689   ~~> tclm 19897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-topgen 14936  df-psmet 18609  df-xmet 18610  df-bl 18612  df-mopn 18613  df-top 19569  df-bases 19571  df-topon 19572  df-cld 19690  df-ntr 19691  df-cls 19692  df-lm 19900
This theorem is referenced by:  bcthlem3  21934  heiborlem8  30557
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