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Theorem dvfsumrlim2 21504
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 11357 . . . . . . 7  |-  ( T (,) +oo )  C_  RR
31, 2eqsstri 3386 . . . . . 6  |-  S  C_  RR
4 dvfsumrlim2.1 . . . . . 6  |-  ( ph  ->  X  e.  S )
53, 4sseldi 3354 . . . . 5  |-  ( ph  ->  X  e.  RR )
65rexrd 9433 . . . 4  |-  ( ph  ->  X  e.  RR* )
75renepnfd 9434 . . . 4  |-  ( ph  ->  X  =/= +oo )
8 icopnfsup 11704 . . . 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 21497 . . . . . . 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 5844 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  X )  e.  RR )
2625recnd 9412 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  X )  e.  CC )
2715rexrd 9433 . . . . . . . . . 10  |-  ( ph  ->  T  e.  RR* )
284, 1syl6eleq 2533 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  ( T (,) +oo ) )
29 elioopnf 11383 . . . . . . . . . . . . 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 11304 . . . . . . . . . . 11  |-  (,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <  w  /\  w  <  v ) } )
34 df-ico 11306 . . . . . . . . . . 11  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
35 xrltletr 11131 . . . . . . . . . . 11  |-  ( ( T  e.  RR*  /\  X  e.  RR*  /\  z  e. 
RR* )  ->  (
( T  <  X  /\  X  <_  z )  ->  T  <  z
) )
3633, 34, 35ixxss1 11318 . . . . . . . . . 10  |-  ( ( T  e.  RR*  /\  T  <  X )  ->  ( X [,) +oo )  C_  ( T (,) +oo )
)
3727, 32, 36syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( X [,) +oo )  C_  ( T (,) +oo ) )
3837, 1syl6sseqr 3403 . . . . . . . 8  |-  ( ph  ->  ( X [,) +oo )  C_  S )
3938adantr 465 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,) +oo )  C_  S
)
4039sselda 3356 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  y  e.  S )
4123, 40ffvelrnd 5844 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  y )  e.  RR )
4241recnd 9412 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( G `  y )  e.  CC )
4326, 42subcld 9719 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  (
( G `  X
)  -  ( G `
 y ) )  e.  CC )
44 pnfxr 11092 . . . . . . 7  |- +oo  e.  RR*
45 icossre 11376 . . . . . . 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 12979 . . . . . . . 8  |-  ( G  ~~> r  L  ->  G : dom  G --> CC )
4948adantl 466 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : dom  G --> CC )
50 ovex 6116 . . . . . . . . 9  |-  ( sum_ k  e.  ( M ... ( |_ `  x
) ) C  -  A )  e.  _V
5150, 21dmmpti 5540 . . . . . . . 8  |-  dom  G  =  S
5251feq2i 5552 . . . . . . 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 5844 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( G `  X )  e.  CC )
56 rlimconst 13022 . . . . 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 5744 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  =  ( y  e.  S  |->  ( G `  y
) ) )
59 simpr 461 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  ~~> r  L
)
6058, 59eqbrtrrd 4314 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  S  |->  ( G `
 y ) )  ~~> r  L )
6139, 60rlimres2 13039 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  ( G `  y ) )  ~~> r  L
)
6226, 42, 57, 61rlimsub 13121 . . 3  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,) +oo )  |->  ( ( G `
 X )  -  ( G `  y ) ) )  ~~> r  ( ( G `  X
)  -  L ) )
6343, 62rlimabs 13086 . 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 21496 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  RR )
6665ralrimiva 2799 . . . . . 6  |-  ( ph  ->  A. x  e.  S  B  e.  RR )
67 nfcsb1v 3304 . . . . . . . 8  |-  F/_ x [_ X  /  x ]_ B
6867nfel1 2589 . . . . . . 7  |-  F/ x [_ X  /  x ]_ B  e.  RR
69 csbeq1a 3297 . . . . . . . 8  |-  ( x  =  X  ->  B  =  [_ X  /  x ]_ B )
7069eleq1d 2509 . . . . . . 7  |-  ( x  =  X  ->  ( B  e.  RR  <->  [_ X  /  x ]_ B  e.  RR ) )
7168, 70rspc 3067 . . . . . 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 9412 . . . 4  |-  ( ph  ->  [_ X  /  x ]_ B  e.  CC )
74 rlimconst 13022 . . . 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 12922 . 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 12939 . . 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 985 . . . . . . 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 1253 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_ +oo ) )  ->  C  <_  B )
92913adant1r 1211 . . . . 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 21502 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x ) )  -> 
0  <_  B )
95943adantr3 1149 . . . . . 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 3356 . . . . 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 11385 . . . . . . 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 9433 . . . . . 6  |-  ( (
ph  /\  y  e.  ( X [,) +oo )
)  ->  y  e.  RR* )
106 pnfge 11110 . . . . . 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 21501 . . . 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 4312 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,) +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  <_  [_ X  /  x ]_ B )
11110, 63, 76, 77, 78, 110rlimle 13125 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   [_csb 3288    C_ wss 3328   class class class wbr 4292    e. cmpt 4350   dom cdm 4840   -->wf 5414   ` cfv 5418  (class class class)co 6091   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285   +oocpnf 9415   RR*cxr 9417    < clt 9418    <_ cle 9419    - cmin 9595   ZZcz 10646   ZZ>=cuz 10861   (,)cioo 11300   [,)cico 11302   ...cfz 11437   |_cfl 11640   abscabs 12723    ~~> r crli 12963   sum_csu 13163    _D cdv 21338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-clim 12966  df-rlim 12967  df-sum 13164  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-cmp 18990  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342
This theorem is referenced by:  dvfsumrlim3  21505
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