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Theorem dvfsumrlim2 22411
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 11597 . . . . . . 7  |-  ( T (,) +oo )  C_  RR
31, 2eqsstri 3519 . . . . . 6  |-  S  C_  RR
4 dvfsumrlim2.1 . . . . . 6  |-  ( ph  ->  X  e.  S )
53, 4sseldi 3487 . . . . 5  |-  ( ph  ->  X  e.  RR )
65rexrd 9646 . . . 4  |-  ( ph  ->  X  e.  RR* )
75renepnfd 9647 . . . 4  |-  ( ph  ->  X  =/= +oo )
8 icopnfsup 11974 . . . 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 22404 . . . . . . 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 6017 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  X )  e.  RR )
2625recnd 9625 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  X )  e.  CC )
2715rexrd 9646 . . . . . . . . . 10  |-  ( ph  ->  T  e.  RR* )
284, 1syl6eleq 2541 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  ( T (,) +oo ) )
29 elioopnf 11629 . . . . . . . . . . . . 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 11544 . . . . . . . . . . 11  |-  (,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <  w  /\  w  <  v ) } )
34 df-ico 11546 . . . . . . . . . . 11  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
35 xrltletr 11371 . . . . . . . . . . 11  |-  ( ( T  e.  RR*  /\  X  e.  RR*  /\  z  e. 
RR* )  ->  (
( T  <  X  /\  X  <_  z )  ->  T  <  z
) )
3633, 34, 35ixxss1 11558 . . . . . . . . . 10  |-  ( ( T  e.  RR*  /\  T  <  X )  ->  ( X [,) +oo )  C_  ( T (,) +oo )
)
3727, 32, 36syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( X [,) +oo )  C_  ( T (,) +oo ) )
3837, 1syl6sseqr 3536 . . . . . . . 8  |-  ( ph  ->  ( X [,) +oo )  C_  S )
3938adantr 465 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,) +oo )  C_  S
)
4039sselda 3489 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  y  e.  S )
4123, 40ffvelrnd 6017 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  y )  e.  RR )
4241recnd 9625 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  y )  e.  CC )
4326, 42subcld 9936 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  (
( G `  X
)  -  ( G `
 y ) )  e.  CC )
44 pnfxr 11332 . . . . . . 7  |- +oo  e.  RR*
45 icossre 11616 . . . . . . 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 13306 . . . . . . . 8  |-  ( G  ~~> r  L  ->  G : dom  G --> CC )
4948adantl 466 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : dom  G --> CC )
50 ovex 6309 . . . . . . . . 9  |-  ( sum_ k  e.  ( M ... ( |_ `  x
) ) C  -  A )  e.  _V
5150, 21dmmpti 5700 . . . . . . . 8  |-  dom  G  =  S
5251feq2i 5714 . . . . . . 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 6017 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( G `  X )  e.  CC )
56 rlimconst 13349 . . . . 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 5911 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  =  ( y  e.  S  |->  ( G `  y
) ) )
59 simpr 461 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  ~~> r  L
)
6058, 59eqbrtrrd 4459 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  S  |->  ( G `
 y ) )  ~~> r  L )
6139, 60rlimres2 13366 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  ( G `  y ) )  ~~> r  L
)
6226, 42, 57, 61rlimsub 13448 . . 3  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  ( ( G `
 X )  -  ( G `  y ) ) )  ~~> r  ( ( G `  X
)  -  L ) )
6343, 62rlimabs 13413 . 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 22403 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  RR )
6665ralrimiva 2857 . . . . . 6  |-  ( ph  ->  A. x  e.  S  B  e.  RR )
67 nfcsb1v 3436 . . . . . . . 8  |-  F/_ x [_ X  /  x ]_ B
6867nfel1 2621 . . . . . . 7  |-  F/ x [_ X  /  x ]_ B  e.  RR
69 csbeq1a 3429 . . . . . . . 8  |-  ( x  =  X  ->  B  =  [_ X  /  x ]_ B )
7069eleq1d 2512 . . . . . . 7  |-  ( x  =  X  ->  ( B  e.  RR  <->  [_ X  /  x ]_ B  e.  RR ) )
7168, 70rspc 3190 . . . . . 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 9625 . . . 4  |-  ( ph  ->  [_ X  /  x ]_ B  e.  CC )
74 rlimconst 13349 . . . 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 13249 . 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 13266 . . 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 994 . . . . . . 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 1264 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_ +oo ) )  ->  C  <_  B )
92913adant1r 1222 . . . . 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 22409 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x ) )  -> 
0  <_  B )
95943adantr3 1158 . . . . . 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 3489 . . . . 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 11631 . . . . . . 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 9646 . . . . . 6  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  y  e.  RR* )
106 pnfge 11350 . . . . . 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 22408 . . . 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 4457 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  <_  [_ X  /  x ]_ B )
11110, 63, 76, 77, 78, 110rlimle 13452 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   [_csb 3420    C_ wss 3461   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989   -->wf 5574   ` cfv 5578  (class class class)co 6281   supcsup 7902   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498   +oocpnf 9628   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810   ZZcz 10871   ZZ>=cuz 11092   (,)cioo 11540   [,)cico 11542   ...cfz 11683   |_cfl 11909   abscabs 13049    ~~> r crli 13290   sum_csu 13490    _D cdv 22245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-rlim 13294  df-sum 13491  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-cmp 19865  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-limc 22248  df-dv 22249
This theorem is referenced by:  dvfsumrlim3  22412
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