MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caublcls Structured version   Unicode version

Theorem caublcls 21475
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 2460 . 2  |-  ( ZZ>= `  A )  =  (
ZZ>= `  A )
2 caubl.2 . . . 4  |-  ( ph  ->  D  e.  ( *Met `  X ) )
323ad2ant1 1012 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  D  e.  ( *Met `  X ) )
4 caublcls.6 . . . 4  |-  J  =  ( MetOpen `  D )
54mopntopon 20670 . . 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 993 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  A  e.  NN )
87nnzd 10954 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  A  e.  ZZ )
9 simp2 992 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 1st  o.  F
) ( ~~> t `  J ) P )
10 fveq2 5857 . . . . . . . . 9  |-  ( r  =  A  ->  ( F `  r )  =  ( F `  A ) )
1110fveq2d 5861 . . . . . . . 8  |-  ( r  =  A  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  A ) ) )
1211sseq1d 3524 . . . . . . 7  |-  ( r  =  A  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  A ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
1312imbi2d 316 . . . . . 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 5857 . . . . . . . . 9  |-  ( r  =  k  ->  ( F `  r )  =  ( F `  k ) )
1514fveq2d 5861 . . . . . . . 8  |-  ( r  =  k  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
1615sseq1d 3524 . . . . . . 7  |-  ( r  =  k  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
1716imbi2d 316 . . . . . 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 5857 . . . . . . . . 9  |-  ( r  =  ( k  +  1 )  ->  ( F `  r )  =  ( F `  ( k  +  1 ) ) )
1918fveq2d 5861 . . . . . . . 8  |-  ( r  =  ( k  +  1 )  ->  (
( ball `  D ) `  ( F `  r
) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
2019sseq1d 3524 . . . . . . 7  |-  ( r  =  ( k  +  1 )  ->  (
( ( ball `  D
) `  ( F `  r ) )  C_  ( ( ball `  D
) `  ( F `  A ) )  <->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
2120imbi2d 316 . . . . . 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 3516 . . . . . . 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 11141 . . . . . . . . . . 11  |-  ( ( A  e.  NN  /\  k  e.  ( ZZ>= `  A ) )  -> 
k  e.  NN )
26 oveq1 6282 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  (
n  +  1 )  =  ( k  +  1 ) )
2726fveq2d 5861 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  ( n  +  1 ) )  =  ( F `  ( k  +  1 ) ) )
2827fveq2d 5861 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  (
n  +  1 ) ) )  =  ( ( ball `  D
) `  ( F `  ( k  +  1 ) ) ) )
29 fveq2 5857 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
3029fveq2d 5861 . . . . . . . . . . . . 13  |-  ( n  =  k  ->  (
( ball `  D ) `  ( F `  n
) )  =  ( ( ball `  D
) `  ( F `  k ) ) )
3128, 30sseq12d 3526 . . . . . . . . . . . 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 477 . . . . . . . . . 10  |-  ( (
ph  /\  ( A  e.  NN  /\  k  e.  ( ZZ>= `  A )
) )  ->  (
( ball `  D ) `  ( F `  (
k  +  1 ) ) )  C_  (
( ball `  D ) `  ( F `  k
) ) )
3433anassrs 648 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  ( k  +  1 ) ) )  C_  ( ( ball `  D ) `  ( F `  k ) ) )
35 sstr2 3504 . . . . . . . . 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 435 . . . . . . 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 11128 . . . . 5  |-  ( k  e.  ( ZZ>= `  A
)  ->  ( ( ph  /\  A  e.  NN )  ->  ( ( ball `  D ) `  ( F `  k )
)  C_  ( ( ball `  D ) `  ( F `  A ) ) ) )
4039impcom 430 . . . 4  |-  ( ( ( ph  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( ball `  D ) `  ( F `  k ) )  C_  ( ( ball `  D ) `  ( F `  A ) ) )
41403adantl2 1148 . . 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 465 . . . . 5  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  D  e.  ( *Met `  X
) )
43 simpl1 994 . . . . . . . 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+ ) )
467, 25sylan 471 . . . . . . 7  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  k  e.  NN )
4745, 46ffvelrnd 6013 . . . . . 6  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( F `  k )  e.  ( X  X.  RR+ )
)
48 xp1st 6804 . . . . . 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 6805 . . . . . 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 20644 . . . . 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 1223 . . . 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 5935 . . . . 5  |-  ( ( F : NN --> ( X  X.  RR+ )  /\  k  e.  NN )  ->  (
( 1st  o.  F
) `  k )  =  ( 1st `  ( F `  k )
) )
5545, 46, 54syl2anc 661 . . . 4  |-  ( ( ( ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  /\  k  e.  ( ZZ>= `  A )
)  ->  ( ( 1st  o.  F ) `  k )  =  ( 1st `  ( F `
 k ) ) )
56 1st2nd2 6811 . . . . . . 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 5861 . . . . 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 6278 . . . . 5  |-  ( ( 1st `  ( F `
 k ) ) ( ball `  D
) ( 2nd `  ( F `  k )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  k
) ) ,  ( 2nd `  ( F `
 k ) )
>. )
6058, 59syl6eqr 2519 . . . 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 2563 . . 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 3498 . 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 6012 . . . . . . 7  |-  ( (
ph  /\  A  e.  NN )  ->  ( F `
 A )  e.  ( X  X.  RR+ ) )
64633adant2 1010 . . . . . 6  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( F `  A
)  e.  ( X  X.  RR+ ) )
65 1st2nd2 6811 . . . . . 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 5861 . . . 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 6278 . . . 4  |-  ( ( 1st `  ( F `
 A ) ) ( ball `  D
) ( 2nd `  ( F `  A )
) )  =  ( ( ball `  D
) `  <. ( 1st `  ( F `  A
) ) ,  ( 2nd `  ( F `
 A ) )
>. )
6967, 68syl6eqr 2519 . . 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 6804 . . . . 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 6805 . . . . . 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 11246 . . . 4  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( 2nd `  ( F `  A )
)  e.  RR* )
75 blssm 20649 . . . 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 1223 . . 3  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( 1st `  ( F `  A )
) ( ball `  D
) ( 2nd `  ( F `  A )
) )  C_  X
)
7769, 76eqsstrd 3531 . 2  |-  ( (
ph  /\  ( 1st  o.  F ) ( ~~> t `  J ) P  /\  A  e.  NN )  ->  ( ( ball `  D
) `  ( F `  A ) )  C_  X )
781, 6, 8, 9, 62, 77lmcls 19562 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   <.cop 4026   class class class wbr 4440    X. cxp 4990    o. ccom 4996   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   1c1 9482    + caddc 9484   RR*cxr 9616   NNcn 10525   ZZcz 10853   ZZ>=cuz 11071   RR+crp 11209   *Metcxmt 18167   ballcbl 18169   MetOpencmopn 18172  TopOnctopon 19155   clsccl 19278   ~~> tclm 19486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-topgen 14688  df-psmet 18175  df-xmet 18176  df-bl 18178  df-mopn 18179  df-top 19159  df-bases 19161  df-topon 19162  df-cld 19279  df-ntr 19280  df-cls 19281  df-lm 19489
This theorem is referenced by:  bcthlem3  21493  heiborlem8  29904
  Copyright terms: Public domain W3C validator