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Theorem dvfsumrlim2 19877
Description: Compare a finite sum to an integral (the integral here is given as a function with a known derivative). The statement here says that if  x  e.  S  |->  B is a decreasing function with antiderivative  A converging to zero, then the difference between  sum_ k  e.  ( M ... ( |_ `  x ) ) B ( k ) and  S. u  e.  ( M [,] x
) B ( u )  _d u  =  A ( x ) converges to a constant limit value, with the remainder term bounded by  B
( x ). (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
dvfsum.s  |-  S  =  ( T (,)  +oo )
dvfsum.z  |-  Z  =  ( ZZ>= `  M )
dvfsum.m  |-  ( ph  ->  M  e.  ZZ )
dvfsum.d  |-  ( ph  ->  D  e.  RR )
dvfsum.md  |-  ( ph  ->  M  <_  ( D  +  1 ) )
dvfsum.t  |-  ( ph  ->  T  e.  RR )
dvfsum.a  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
dvfsum.b1  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  V )
dvfsum.b2  |-  ( (
ph  /\  x  e.  Z )  ->  B  e.  RR )
dvfsum.b3  |-  ( ph  ->  ( RR  _D  (
x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )
dvfsum.c  |-  ( x  =  k  ->  B  =  C )
dvfsumrlim.l  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k ) )  ->  C  <_  B )
dvfsumrlim.g  |-  G  =  ( x  e.  S  |->  ( sum_ k  e.  ( M ... ( |_
`  x ) ) C  -  A ) )
dvfsumrlim.k  |-  ( ph  ->  ( x  e.  S  |->  B )  ~~> r  0 )
dvfsumrlim2.1  |-  ( ph  ->  X  e.  S )
dvfsumrlim2.2  |-  ( ph  ->  D  <_  X )
Assertion
Ref Expression
dvfsumrlim2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( abs `  ( ( G `  X )  -  L
) )  <_  [_ X  /  x ]_ B )
Distinct variable groups:    B, k    x, C    x, k, D    ph, k, x    S, k, x    k, M, x   
x, T    x, Z    k, X, x
Allowed substitution hints:    A( x, k)    B( x)    C( k)    T( k)    G( x, k)    L( x, k)    V( x, k)    Z( k)

Proof of Theorem dvfsumrlim2
Dummy variables  y 
z  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvfsum.s . . . . . . 7  |-  S  =  ( T (,)  +oo )
2 ioossre 10936 . . . . . . 7  |-  ( T (,)  +oo )  C_  RR
31, 2eqsstri 3346 . . . . . 6  |-  S  C_  RR
4 dvfsumrlim2.1 . . . . . 6  |-  ( ph  ->  X  e.  S )
53, 4sseldi 3314 . . . . 5  |-  ( ph  ->  X  e.  RR )
65rexrd 9098 . . . 4  |-  ( ph  ->  X  e.  RR* )
75renepnfd 9099 . . . 4  |-  ( ph  ->  X  =/=  +oo )
8 icopnfsup 11209 . . . 4  |-  ( ( X  e.  RR*  /\  X  =/=  +oo )  ->  sup ( ( X [,)  +oo ) ,  RR* ,  <  )  =  +oo )
96, 7, 8syl2anc 643 . . 3  |-  ( ph  ->  sup ( ( X [,)  +oo ) ,  RR* ,  <  )  =  +oo )
109adantr 452 . 2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  sup (
( X [,)  +oo ) ,  RR* ,  <  )  =  +oo )
11 dvfsum.z . . . . . . . 8  |-  Z  =  ( ZZ>= `  M )
12 dvfsum.m . . . . . . . 8  |-  ( ph  ->  M  e.  ZZ )
13 dvfsum.d . . . . . . . 8  |-  ( ph  ->  D  e.  RR )
14 dvfsum.md . . . . . . . 8  |-  ( ph  ->  M  <_  ( D  +  1 ) )
15 dvfsum.t . . . . . . . 8  |-  ( ph  ->  T  e.  RR )
16 dvfsum.a . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
17 dvfsum.b1 . . . . . . . 8  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  V )
18 dvfsum.b2 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  B  e.  RR )
19 dvfsum.b3 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )
20 dvfsum.c . . . . . . . 8  |-  ( x  =  k  ->  B  =  C )
21 dvfsumrlim.g . . . . . . . 8  |-  G  =  ( x  e.  S  |->  ( sum_ k  e.  ( M ... ( |_
`  x ) ) C  -  A ) )
221, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21dvfsumrlimf 19870 . . . . . . 7  |-  ( ph  ->  G : S --> RR )
2322ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  G : S --> RR )
244ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  X  e.  S )
2523, 24ffvelrnd 5838 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  X )  e.  RR )
2625recnd 9078 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  X )  e.  CC )
2715rexrd 9098 . . . . . . . . . 10  |-  ( ph  ->  T  e.  RR* )
284, 1syl6eleq 2502 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  ( T (,)  +oo ) )
29 elioopnf 10962 . . . . . . . . . . . . 13  |-  ( T  e.  RR*  ->  ( X  e.  ( T (,)  +oo )  <->  ( X  e.  RR  /\  T  < 
X ) ) )
3027, 29syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( X  e.  ( T (,)  +oo )  <->  ( X  e.  RR  /\  T  <  X ) ) )
3128, 30mpbid 202 . . . . . . . . . . 11  |-  ( ph  ->  ( X  e.  RR  /\  T  <  X ) )
3231simprd 450 . . . . . . . . . 10  |-  ( ph  ->  T  <  X )
33 df-ioo 10884 . . . . . . . . . . 11  |-  (,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <  w  /\  w  <  v ) } )
34 df-ico 10886 . . . . . . . . . . 11  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
35 xrltletr 10711 . . . . . . . . . . 11  |-  ( ( T  e.  RR*  /\  X  e.  RR*  /\  z  e. 
RR* )  ->  (
( T  <  X  /\  X  <_  z )  ->  T  <  z
) )
3633, 34, 35ixxss1 10898 . . . . . . . . . 10  |-  ( ( T  e.  RR*  /\  T  <  X )  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3727, 32, 36syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3837, 1syl6sseqr 3363 . . . . . . . 8  |-  ( ph  ->  ( X [,)  +oo )  C_  S )
3938adantr 452 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,)  +oo )  C_  S
)
4039sselda 3316 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  y  e.  S )
4123, 40ffvelrnd 5838 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  y )  e.  RR )
4241recnd 9078 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  y )  e.  CC )
4326, 42subcld 9375 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  (
( G `  X
)  -  ( G `
 y ) )  e.  CC )
44 pnfxr 10677 . . . . . . 7  |-  +oo  e.  RR*
45 icossre 10955 . . . . . . 7  |-  ( ( X  e.  RR  /\  +oo 
e.  RR* )  ->  ( X [,)  +oo )  C_  RR )
465, 44, 45sylancl 644 . . . . . 6  |-  ( ph  ->  ( X [,)  +oo )  C_  RR )
4746adantr 452 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,)  +oo )  C_  RR )
48 rlimf 12258 . . . . . . . 8  |-  ( G  ~~> r  L  ->  G : dom  G --> CC )
4948adantl 453 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : dom  G --> CC )
50 ovex 6073 . . . . . . . . 9  |-  ( sum_ k  e.  ( M ... ( |_ `  x
) ) C  -  A )  e.  _V
5150, 21dmmpti 5541 . . . . . . . 8  |-  dom  G  =  S
5251feq2i 5553 . . . . . . 7  |-  ( G : dom  G --> CC  <->  G : S
--> CC )
5349, 52sylib 189 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : S
--> CC )
544adantr 452 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  X  e.  S )
5553, 54ffvelrnd 5838 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( G `  X )  e.  CC )
56 rlimconst 12301 . . . . 5  |-  ( ( ( X [,)  +oo )  C_  RR  /\  ( G `  X )  e.  CC )  ->  (
y  e.  ( X [,)  +oo )  |->  ( G `
 X ) )  ~~> r  ( G `  X ) )
5747, 55, 56syl2anc 643 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( G `  X ) )  ~~> r  ( G `  X ) )
5853feqmptd 5746 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  =  ( y  e.  S  |->  ( G `  y
) ) )
59 simpr 448 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  ~~> r  L
)
6058, 59eqbrtrrd 4202 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  S  |->  ( G `
 y ) )  ~~> r  L )
6139, 60rlimres2 12318 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( G `  y ) )  ~~> r  L
)
6226, 42, 57, 61rlimsub 12400 . . 3  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( ( G `
 X )  -  ( G `  y ) ) )  ~~> r  ( ( G `  X
)  -  L ) )
6343, 62rlimabs 12365 . 2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( abs `  (
( G `  X
)  -  ( G `
 y ) ) ) )  ~~> r  ( abs `  ( ( G `  X )  -  L ) ) )
643a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  RR )
6564, 16, 17, 19dvmptrecl 19869 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  RR )
6665ralrimiva 2757 . . . . . 6  |-  ( ph  ->  A. x  e.  S  B  e.  RR )
67 nfcsb1v 3251 . . . . . . . 8  |-  F/_ x [_ X  /  x ]_ B
6867nfel1 2558 . . . . . . 7  |-  F/ x [_ X  /  x ]_ B  e.  RR
69 csbeq1a 3227 . . . . . . . 8  |-  ( x  =  X  ->  B  =  [_ X  /  x ]_ B )
7069eleq1d 2478 . . . . . . 7  |-  ( x  =  X  ->  ( B  e.  RR  <->  [_ X  /  x ]_ B  e.  RR ) )
7168, 70rspc 3014 . . . . . 6  |-  ( X  e.  S  ->  ( A. x  e.  S  B  e.  RR  ->  [_ X  /  x ]_ B  e.  RR )
)
724, 66, 71sylc 58 . . . . 5  |-  ( ph  ->  [_ X  /  x ]_ B  e.  RR )
7372recnd 9078 . . . 4  |-  ( ph  ->  [_ X  /  x ]_ B  e.  CC )
74 rlimconst 12301 . . . 4  |-  ( ( ( X [,)  +oo )  C_  RR  /\  [_ X  /  x ]_ B  e.  CC )  ->  (
y  e.  ( X [,)  +oo )  |->  [_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B )
7546, 73, 74syl2anc 643 . . 3  |-  ( ph  ->  ( y  e.  ( X [,)  +oo )  |-> 
[_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
7675adantr 452 . 2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  [_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
7743abscld 12201 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  e.  RR )
7872ad2antrr 707 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  [_ X  /  x ]_ B  e.  RR )
7926, 42abssubd 12218 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  =  ( abs `  ( ( G `  y )  -  ( G `  X ) ) ) )
8012adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  M  e.  ZZ )
8113adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  D  e.  RR )
8214adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  M  <_  ( D  +  1 ) )
8315adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  T  e.  RR )
8416adantlr 696 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  S )  ->  A  e.  RR )
8517adantlr 696 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  S )  ->  B  e.  V )
8618adantlr 696 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  Z )  ->  B  e.  RR )
8719adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )
8844a1i 11 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  +oo  e.  RR* )
89 3simpa 954 . . . . . . 7  |-  ( ( D  <_  x  /\  x  <_  k  /\  k  <_  +oo )  ->  ( D  <_  x  /\  x  <_  k ) )
90 dvfsumrlim.l . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k ) )  ->  C  <_  B )
9189, 90syl3an3 1219 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_  +oo ) )  ->  C  <_  B )
92913adant1r 1177 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  (
x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_  +oo ) )  ->  C  <_  B )
93 dvfsumrlim.k . . . . . . . 8  |-  ( ph  ->  ( x  e.  S  |->  B )  ~~> r  0 )
941, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 90, 21, 93dvfsumrlimge0 19875 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x ) )  -> 
0  <_  B )
95943adantr3 1118 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x  /\  x  <_  +oo )
)  ->  0  <_  B )
9695adantlr 696 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  (
x  e.  S  /\  D  <_  x  /\  x  <_  +oo ) )  -> 
0  <_  B )
974adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  X  e.  S )
9838sselda 3316 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  S )
99 dvfsumrlim2.2 . . . . . 6  |-  ( ph  ->  D  <_  X )
10099adantr 452 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  D  <_  X )
101 elicopnf 10964 . . . . . . 7  |-  ( X  e.  RR  ->  (
y  e.  ( X [,)  +oo )  <->  ( y  e.  RR  /\  X  <_ 
y ) ) )
1025, 101syl 16 . . . . . 6  |-  ( ph  ->  ( y  e.  ( X [,)  +oo )  <->  ( y  e.  RR  /\  X  <_  y ) ) )
103102simplbda 608 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  X  <_  y )
104102simprbda 607 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  RR )
105104rexrd 9098 . . . . . 6  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  RR* )
106 pnfge 10691 . . . . . 6  |-  ( y  e.  RR*  ->  y  <_  +oo )
107105, 106syl 16 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  <_  +oo )
1081, 11, 80, 81, 82, 83, 84, 85, 86, 87, 20, 88, 92, 21, 96, 97, 98, 100, 103, 107dvfsumlem4 19874 . . . 4  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  ( abs `  ( ( G `  y )  -  ( G `  X )
) )  <_  [_ X  /  x ]_ B )
109108adantlr 696 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 y )  -  ( G `  X ) ) )  <_  [_ X  /  x ]_ B )
11079, 109eqbrtrd 4200 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  <_  [_ X  /  x ]_ B )
11110, 63, 76, 77, 78, 110rlimle 12404 1  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( abs `  ( ( G `  X )  -  L
) )  <_  [_ X  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   [_csb 3219    C_ wss 3288   class class class wbr 4180    e. cmpt 4234   dom cdm 4845   -->wf 5417   ` cfv 5421  (class class class)co 6048   supcsup 7411   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955    + caddc 8957    +oocpnf 9081   RR*cxr 9083    < clt 9084    <_ cle 9085    - cmin 9255   ZZcz 10246   ZZ>=cuz 10452   (,)cioo 10880   [,)cico 10882   ...cfz 11007   |_cfl 11164   abscabs 12002    ~~> r crli 12242   sum_csu 12442    _D cdv 19711
This theorem is referenced by:  dvfsumrlim3  19878
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-inf2 7560  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  ax-addf 9033  ax-mulf 9034
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-se 4510  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-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  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-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-fbas 16662  df-fg 16663  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-ntr 17047  df-cls 17048  df-nei 17125  df-lp 17163  df-perf 17164  df-cn 17253  df-cnp 17254  df-haus 17341  df-cmp 17412  df-tx 17555  df-hmeo 17748  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933  df-xms 18311  df-ms 18312  df-tms 18313  df-cncf 18869  df-limc 19714  df-dv 19715
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