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Theorem dvfsumrlim2 22184
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 11585 . . . . . . 7  |-  ( T (,) +oo )  C_  RR
31, 2eqsstri 3534 . . . . . 6  |-  S  C_  RR
4 dvfsumrlim2.1 . . . . . 6  |-  ( ph  ->  X  e.  S )
53, 4sseldi 3502 . . . . 5  |-  ( ph  ->  X  e.  RR )
65rexrd 9642 . . . 4  |-  ( ph  ->  X  e.  RR* )
75renepnfd 9643 . . . 4  |-  ( ph  ->  X  =/= +oo )
8 icopnfsup 11959 . . . 4  |-  ( ( X  e.  RR*  /\  X  =/= +oo )  ->  sup ( ( X [,) +oo ) ,  RR* ,  <  )  = +oo )
96, 7, 8syl2anc 661 . . 3  |-  ( ph  ->  sup ( ( X [,) +oo ) , 
RR* ,  <  )  = +oo )
109adantr 465 . 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 22177 . . . . . . 7  |-  ( ph  ->  G : S --> RR )
2322ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  G : S --> RR )
244ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  X  e.  S )
2523, 24ffvelrnd 6021 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  X )  e.  RR )
2625recnd 9621 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  X )  e.  CC )
2715rexrd 9642 . . . . . . . . . 10  |-  ( ph  ->  T  e.  RR* )
284, 1syl6eleq 2565 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  ( T (,) +oo ) )
29 elioopnf 11617 . . . . . . . . . . . . 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 210 . . . . . . . . . . 11  |-  ( ph  ->  ( X  e.  RR  /\  T  <  X ) )
3231simprd 463 . . . . . . . . . 10  |-  ( ph  ->  T  <  X )
33 df-ioo 11532 . . . . . . . . . . 11  |-  (,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <  w  /\  w  <  v ) } )
34 df-ico 11534 . . . . . . . . . . 11  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
35 xrltletr 11359 . . . . . . . . . . 11  |-  ( ( T  e.  RR*  /\  X  e.  RR*  /\  z  e. 
RR* )  ->  (
( T  <  X  /\  X  <_  z )  ->  T  <  z
) )
3633, 34, 35ixxss1 11546 . . . . . . . . . 10  |-  ( ( T  e.  RR*  /\  T  <  X )  ->  ( X [,) +oo )  C_  ( T (,) +oo )
)
3727, 32, 36syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( X [,) +oo )  C_  ( T (,) +oo ) )
3837, 1syl6sseqr 3551 . . . . . . . 8  |-  ( ph  ->  ( X [,) +oo )  C_  S )
3938adantr 465 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,) +oo )  C_  S
)
4039sselda 3504 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  y  e.  S )
4123, 40ffvelrnd 6021 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  y )  e.  RR )
4241recnd 9621 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  y )  e.  CC )
4326, 42subcld 9929 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  (
( G `  X
)  -  ( G `
 y ) )  e.  CC )
44 pnfxr 11320 . . . . . . 7  |- +oo  e.  RR*
45 icossre 11604 . . . . . . 7  |-  ( ( X  e.  RR  /\ +oo  e.  RR* )  ->  ( X [,) +oo )  C_  RR )
465, 44, 45sylancl 662 . . . . . 6  |-  ( ph  ->  ( X [,) +oo )  C_  RR )
4746adantr 465 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,) +oo )  C_  RR )
48 rlimf 13286 . . . . . . . 8  |-  ( G  ~~> r  L  ->  G : dom  G --> CC )
4948adantl 466 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : dom  G --> CC )
50 ovex 6308 . . . . . . . . 9  |-  ( sum_ k  e.  ( M ... ( |_ `  x
) ) C  -  A )  e.  _V
5150, 21dmmpti 5709 . . . . . . . 8  |-  dom  G  =  S
5251feq2i 5723 . . . . . . 7  |-  ( G : dom  G --> CC  <->  G : S
--> CC )
5349, 52sylib 196 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : S
--> CC )
544adantr 465 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  X  e.  S )
5553, 54ffvelrnd 6021 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( G `  X )  e.  CC )
56 rlimconst 13329 . . . . 5  |-  ( ( ( X [,) +oo )  C_  RR  /\  ( G `  X )  e.  CC )  ->  (
y  e.  ( X [,) +oo )  |->  ( G `  X ) )  ~~> r  ( G `
 X ) )
5747, 55, 56syl2anc 661 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  ( G `  X ) )  ~~> r  ( G `  X ) )
5853feqmptd 5919 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  =  ( y  e.  S  |->  ( G `  y
) ) )
59 simpr 461 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  ~~> r  L
)
6058, 59eqbrtrrd 4469 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  S  |->  ( G `
 y ) )  ~~> r  L )
6139, 60rlimres2 13346 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  ( G `  y ) )  ~~> r  L
)
6226, 42, 57, 61rlimsub 13428 . . 3  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  ( ( G `
 X )  -  ( G `  y ) ) )  ~~> r  ( ( G `  X
)  -  L ) )
6343, 62rlimabs 13393 . 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 22176 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  RR )
6665ralrimiva 2878 . . . . . 6  |-  ( ph  ->  A. x  e.  S  B  e.  RR )
67 nfcsb1v 3451 . . . . . . . 8  |-  F/_ x [_ X  /  x ]_ B
6867nfel1 2645 . . . . . . 7  |-  F/ x [_ X  /  x ]_ B  e.  RR
69 csbeq1a 3444 . . . . . . . 8  |-  ( x  =  X  ->  B  =  [_ X  /  x ]_ B )
7069eleq1d 2536 . . . . . . 7  |-  ( x  =  X  ->  ( B  e.  RR  <->  [_ X  /  x ]_ B  e.  RR ) )
7168, 70rspc 3208 . . . . . 6  |-  ( X  e.  S  ->  ( A. x  e.  S  B  e.  RR  ->  [_ X  /  x ]_ B  e.  RR )
)
724, 66, 71sylc 60 . . . . 5  |-  ( ph  ->  [_ X  /  x ]_ B  e.  RR )
7372recnd 9621 . . . 4  |-  ( ph  ->  [_ X  /  x ]_ B  e.  CC )
74 rlimconst 13329 . . . 4  |-  ( ( ( X [,) +oo )  C_  RR  /\  [_ X  /  x ]_ B  e.  CC )  ->  (
y  e.  ( X [,) +oo )  |->  [_ X  /  x ]_ B
)  ~~> r  [_ X  /  x ]_ B )
7546, 73, 74syl2anc 661 . . 3  |-  ( ph  ->  ( y  e.  ( X [,) +oo )  |-> 
[_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
7675adantr 465 . 2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  [_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
7743abscld 13229 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  e.  RR )
7872ad2antrr 725 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  [_ X  /  x ]_ B  e.  RR )
7926, 42abssubd 13246 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  =  ( abs `  ( ( G `  y )  -  ( G `  X ) ) ) )
8012adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  M  e.  ZZ )
8113adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  D  e.  RR )
8214adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  M  <_  ( D  +  1 ) )
8315adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  T  e.  RR )
8416adantlr 714 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,) +oo ) )  /\  x  e.  S )  ->  A  e.  RR )
8517adantlr 714 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,) +oo ) )  /\  x  e.  S )  ->  B  e.  V )
8618adantlr 714 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,) +oo ) )  /\  x  e.  Z )  ->  B  e.  RR )
8719adantr 465 . . . . 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 993 . . . . . . 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 1263 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_ +oo ) )  ->  C  <_  B )
92913adant1r 1221 . . . . 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 22182 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x ) )  -> 
0  <_  B )
95943adantr3 1157 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x  /\  x  <_ +oo )
)  ->  0  <_  B )
9695adantlr 714 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,) +oo ) )  /\  (
x  e.  S  /\  D  <_  x  /\  x  <_ +oo ) )  -> 
0  <_  B )
974adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  X  e.  S )
9838sselda 3504 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  y  e.  S )
99 dvfsumrlim2.2 . . . . . 6  |-  ( ph  ->  D  <_  X )
10099adantr 465 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  D  <_  X )
101 elicopnf 11619 . . . . . . 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 624 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  X  <_  y )
104102simprbda 623 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  y  e.  RR )
105104rexrd 9642 . . . . . 6  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  y  e.  RR* )
106 pnfge 11338 . . . . . 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 22181 . . . 4  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  ( abs `  ( ( G `  y )  -  ( G `  X )
) )  <_  [_ X  /  x ]_ B )
109108adantlr 714 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( abs `  ( ( G `
 y )  -  ( G `  X ) ) )  <_  [_ X  /  x ]_ B )
11079, 109eqbrtrd 4467 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  <_  [_ X  /  x ]_ B )
11110, 63, 76, 77, 78, 110rlimle 13432 1  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( abs `  ( ( G `  X )  -  L
) )  <_  [_ X  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   [_csb 3435    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   -->wf 5583   ` cfv 5587  (class class class)co 6283   supcsup 7899   CCcc 9489   RRcr 9490   0cc0 9491   1c1 9492    + caddc 9494   +oocpnf 9624   RR*cxr 9626    < clt 9627    <_ cle 9628    - cmin 9804   ZZcz 10863   ZZ>=cuz 11081   (,)cioo 11528   [,)cico 11530   ...cfz 11671   |_cfl 11894   abscabs 13029    ~~> r crli 13270   sum_csu 13470    _D cdv 22018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570  ax-mulf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7934  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-q 11182  df-rp 11220  df-xneg 11317  df-xadd 11318  df-xmul 11319  df-ioo 11532  df-ico 11534  df-icc 11535  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-starv 14569  df-sca 14570  df-vsca 14571  df-ip 14572  df-tset 14573  df-ple 14574  df-ds 14576  df-unif 14577  df-hom 14578  df-cco 14579  df-rest 14677  df-topn 14678  df-0g 14696  df-gsum 14697  df-topgen 14698  df-pt 14699  df-prds 14702  df-xrs 14756  df-qtop 14761  df-imas 14762  df-xps 14764  df-mre 14840  df-mrc 14841  df-acs 14843  df-mnd 15731  df-submnd 15784  df-mulg 15867  df-cntz 16157  df-cmn 16603  df-psmet 18198  df-xmet 18199  df-met 18200  df-bl 18201  df-mopn 18202  df-fbas 18203  df-fg 18204  df-cnfld 18208  df-top 19182  df-bases 19184  df-topon 19185  df-topsp 19186  df-cld 19302  df-ntr 19303  df-cls 19304  df-nei 19381  df-lp 19419  df-perf 19420  df-cn 19510  df-cnp 19511  df-haus 19598  df-cmp 19669  df-tx 19814  df-hmeo 20007  df-fil 20098  df-fm 20190  df-flim 20191  df-flf 20192  df-xms 20574  df-ms 20575  df-tms 20576  df-cncf 21133  df-limc 22021  df-dv 22022
This theorem is referenced by:  dvfsumrlim3  22185
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