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Theorem dvfsumrlim2 19947
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 11003 . . . . . . 7  |-  ( T (,)  +oo )  C_  RR
31, 2eqsstri 3364 . . . . . 6  |-  S  C_  RR
4 dvfsumrlim2.1 . . . . . 6  |-  ( ph  ->  X  e.  S )
53, 4sseldi 3332 . . . . 5  |-  ( ph  ->  X  e.  RR )
65rexrd 9165 . . . 4  |-  ( ph  ->  X  e.  RR* )
75renepnfd 9166 . . . 4  |-  ( ph  ->  X  =/=  +oo )
8 icopnfsup 11277 . . . 4  |-  ( ( X  e.  RR*  /\  X  =/=  +oo )  ->  sup ( ( X [,)  +oo ) ,  RR* ,  <  )  =  +oo )
96, 7, 8syl2anc 644 . . 3  |-  ( ph  ->  sup ( ( X [,)  +oo ) ,  RR* ,  <  )  =  +oo )
109adantr 453 . 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 19940 . . . . . . 7  |-  ( ph  ->  G : S --> RR )
2322ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  G : S --> RR )
244ad2antrr 708 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  X  e.  S )
2523, 24ffvelrnd 5900 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  X )  e.  RR )
2625recnd 9145 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  X )  e.  CC )
2715rexrd 9165 . . . . . . . . . 10  |-  ( ph  ->  T  e.  RR* )
284, 1syl6eleq 2532 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  ( T (,)  +oo ) )
29 elioopnf 11029 . . . . . . . . . . . . 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 203 . . . . . . . . . . 11  |-  ( ph  ->  ( X  e.  RR  /\  T  <  X ) )
3231simprd 451 . . . . . . . . . 10  |-  ( ph  ->  T  <  X )
33 df-ioo 10951 . . . . . . . . . . 11  |-  (,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <  w  /\  w  <  v ) } )
34 df-ico 10953 . . . . . . . . . . 11  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
35 xrltletr 10778 . . . . . . . . . . 11  |-  ( ( T  e.  RR*  /\  X  e.  RR*  /\  z  e. 
RR* )  ->  (
( T  <  X  /\  X  <_  z )  ->  T  <  z
) )
3633, 34, 35ixxss1 10965 . . . . . . . . . 10  |-  ( ( T  e.  RR*  /\  T  <  X )  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3727, 32, 36syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3837, 1syl6sseqr 3381 . . . . . . . 8  |-  ( ph  ->  ( X [,)  +oo )  C_  S )
3938adantr 453 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,)  +oo )  C_  S
)
4039sselda 3334 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  y  e.  S )
4123, 40ffvelrnd 5900 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  y )  e.  RR )
4241recnd 9145 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  y )  e.  CC )
4326, 42subcld 9442 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  (
( G `  X
)  -  ( G `
 y ) )  e.  CC )
44 pnfxr 10744 . . . . . . 7  |-  +oo  e.  RR*
45 icossre 11022 . . . . . . 7  |-  ( ( X  e.  RR  /\  +oo 
e.  RR* )  ->  ( X [,)  +oo )  C_  RR )
465, 44, 45sylancl 645 . . . . . 6  |-  ( ph  ->  ( X [,)  +oo )  C_  RR )
4746adantr 453 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,)  +oo )  C_  RR )
48 rlimf 12326 . . . . . . . 8  |-  ( G  ~~> r  L  ->  G : dom  G --> CC )
4948adantl 454 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : dom  G --> CC )
50 ovex 6135 . . . . . . . . 9  |-  ( sum_ k  e.  ( M ... ( |_ `  x
) ) C  -  A )  e.  _V
5150, 21dmmpti 5603 . . . . . . . 8  |-  dom  G  =  S
5251feq2i 5615 . . . . . . 7  |-  ( G : dom  G --> CC  <->  G : S
--> CC )
5349, 52sylib 190 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : S
--> CC )
544adantr 453 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  X  e.  S )
5553, 54ffvelrnd 5900 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( G `  X )  e.  CC )
56 rlimconst 12369 . . . . 5  |-  ( ( ( X [,)  +oo )  C_  RR  /\  ( G `  X )  e.  CC )  ->  (
y  e.  ( X [,)  +oo )  |->  ( G `
 X ) )  ~~> r  ( G `  X ) )
5747, 55, 56syl2anc 644 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( G `  X ) )  ~~> r  ( G `  X ) )
5853feqmptd 5808 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  =  ( y  e.  S  |->  ( G `  y
) ) )
59 simpr 449 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  ~~> r  L
)
6058, 59eqbrtrrd 4259 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  S  |->  ( G `
 y ) )  ~~> r  L )
6139, 60rlimres2 12386 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( G `  y ) )  ~~> r  L
)
6226, 42, 57, 61rlimsub 12468 . . 3  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( ( G `
 X )  -  ( G `  y ) ) )  ~~> r  ( ( G `  X
)  -  L ) )
6343, 62rlimabs 12433 . 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 19939 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  RR )
6665ralrimiva 2795 . . . . . 6  |-  ( ph  ->  A. x  e.  S  B  e.  RR )
67 nfcsb1v 3282 . . . . . . . 8  |-  F/_ x [_ X  /  x ]_ B
6867nfel1 2588 . . . . . . 7  |-  F/ x [_ X  /  x ]_ B  e.  RR
69 csbeq1a 3275 . . . . . . . 8  |-  ( x  =  X  ->  B  =  [_ X  /  x ]_ B )
7069eleq1d 2508 . . . . . . 7  |-  ( x  =  X  ->  ( B  e.  RR  <->  [_ X  /  x ]_ B  e.  RR ) )
7168, 70rspc 3052 . . . . . 6  |-  ( X  e.  S  ->  ( A. x  e.  S  B  e.  RR  ->  [_ X  /  x ]_ B  e.  RR )
)
724, 66, 71sylc 59 . . . . 5  |-  ( ph  ->  [_ X  /  x ]_ B  e.  RR )
7372recnd 9145 . . . 4  |-  ( ph  ->  [_ X  /  x ]_ B  e.  CC )
74 rlimconst 12369 . . . 4  |-  ( ( ( X [,)  +oo )  C_  RR  /\  [_ X  /  x ]_ B  e.  CC )  ->  (
y  e.  ( X [,)  +oo )  |->  [_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B )
7546, 73, 74syl2anc 644 . . 3  |-  ( ph  ->  ( y  e.  ( X [,)  +oo )  |-> 
[_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
7675adantr 453 . 2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  [_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
7743abscld 12269 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  e.  RR )
7872ad2antrr 708 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  [_ X  /  x ]_ B  e.  RR )
7926, 42abssubd 12286 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  =  ( abs `  ( ( G `  y )  -  ( G `  X ) ) ) )
8012adantr 453 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  M  e.  ZZ )
8113adantr 453 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  D  e.  RR )
8214adantr 453 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  M  <_  ( D  +  1 ) )
8315adantr 453 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  T  e.  RR )
8416adantlr 697 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  S )  ->  A  e.  RR )
8517adantlr 697 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  S )  ->  B  e.  V )
8618adantlr 697 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  Z )  ->  B  e.  RR )
8719adantr 453 . . . . 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 955 . . . . . . 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 1220 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_  +oo ) )  ->  C  <_  B )
92913adant1r 1178 . . . . 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 19945 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x ) )  -> 
0  <_  B )
95943adantr3 1119 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x  /\  x  <_  +oo )
)  ->  0  <_  B )
9695adantlr 697 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  (
x  e.  S  /\  D  <_  x  /\  x  <_  +oo ) )  -> 
0  <_  B )
974adantr 453 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  X  e.  S )
9838sselda 3334 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  S )
99 dvfsumrlim2.2 . . . . . 6  |-  ( ph  ->  D  <_  X )
10099adantr 453 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  D  <_  X )
101 elicopnf 11031 . . . . . . 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 609 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  X  <_  y )
104102simprbda 608 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  RR )
105104rexrd 9165 . . . . . 6  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  RR* )
106 pnfge 10758 . . . . . 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 19944 . . . 4  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  ( abs `  ( ( G `  y )  -  ( G `  X )
) )  <_  [_ X  /  x ]_ B )
109108adantlr 697 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 y )  -  ( G `  X ) ) )  <_  [_ X  /  x ]_ B )
11079, 109eqbrtrd 4257 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  <_  [_ X  /  x ]_ B )
11110, 63, 76, 77, 78, 110rlimle 12472 1  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( abs `  ( ( G `  X )  -  L
) )  <_  [_ X  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   [_csb 3267    C_ wss 3306   class class class wbr 4237    e. cmpt 4291   dom cdm 4907   -->wf 5479   ` cfv 5483  (class class class)co 6110   supcsup 7474   CCcc 9019   RRcr 9020   0cc0 9021   1c1 9022    + caddc 9024    +oocpnf 9148   RR*cxr 9150    < clt 9151    <_ cle 9152    - cmin 9322   ZZcz 10313   ZZ>=cuz 10519   (,)cioo 10947   [,)cico 10949   ...cfz 11074   |_cfl 11232   abscabs 12070    ~~> r crli 12310   sum_csu 12510    _D cdv 19781
This theorem is referenced by:  dvfsumrlim3  19948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-sup 7475  df-oi 7508  df-card 7857  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-q 10606  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-ioo 10951  df-ico 10953  df-icc 10954  df-fz 11075  df-fzo 11167  df-fl 11233  df-seq 11355  df-exp 11414  df-hash 11650  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-clim 12313  df-rlim 12314  df-sum 12511  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-starv 13575  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-unif 13583  df-hom 13584  df-cco 13585  df-rest 13681  df-topn 13682  df-topgen 13698  df-pt 13699  df-prds 13702  df-xrs 13757  df-0g 13758  df-gsum 13759  df-qtop 13764  df-imas 13765  df-xps 13767  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-submnd 14770  df-mulg 14846  df-cntz 15147  df-cmn 15445  df-psmet 16725  df-xmet 16726  df-met 16727  df-bl 16728  df-mopn 16729  df-fbas 16730  df-fg 16731  df-cnfld 16735  df-top 16994  df-bases 16996  df-topon 16997  df-topsp 16998  df-cld 17114  df-ntr 17115  df-cls 17116  df-nei 17193  df-lp 17231  df-perf 17232  df-cn 17322  df-cnp 17323  df-haus 17410  df-cmp 17481  df-tx 17625  df-hmeo 17818  df-fil 17909  df-fm 18001  df-flim 18002  df-flf 18003  df-xms 18381  df-ms 18382  df-tms 18383  df-cncf 18939  df-limc 19784  df-dv 19785
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