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Theorem dvfsumrlim2 19477
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 10801 . . . . . . 7  |-  ( T (,)  +oo )  C_  RR
31, 2eqsstri 3284 . . . . . 6  |-  S  C_  RR
4 dvfsumrlim2.1 . . . . . 6  |-  ( ph  ->  X  e.  S )
53, 4sseldi 3254 . . . . 5  |-  ( ph  ->  X  e.  RR )
65rexrd 8968 . . . 4  |-  ( ph  ->  X  e.  RR* )
75renepnfd 8969 . . . 4  |-  ( ph  ->  X  =/=  +oo )
8 icopnfsup 11058 . . . 4  |-  ( ( X  e.  RR*  /\  X  =/=  +oo )  ->  sup ( ( X [,)  +oo ) ,  RR* ,  <  )  =  +oo )
96, 7, 8syl2anc 642 . . 3  |-  ( ph  ->  sup ( ( X [,)  +oo ) ,  RR* ,  <  )  =  +oo )
109adantr 451 . 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 19470 . . . . . . 7  |-  ( ph  ->  G : S --> RR )
2322ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  G : S --> RR )
244ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  X  e.  S )
25 ffvelrn 5743 . . . . . 6  |-  ( ( G : S --> RR  /\  X  e.  S )  ->  ( G `  X
)  e.  RR )
2623, 24, 25syl2anc 642 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  X )  e.  RR )
2726recnd 8948 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  X )  e.  CC )
2815rexrd 8968 . . . . . . . . . 10  |-  ( ph  ->  T  e.  RR* )
294, 1syl6eleq 2448 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  ( T (,)  +oo ) )
30 elioopnf 10826 . . . . . . . . . . . . 13  |-  ( T  e.  RR*  ->  ( X  e.  ( T (,)  +oo )  <->  ( X  e.  RR  /\  T  < 
X ) ) )
3128, 30syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( X  e.  ( T (,)  +oo )  <->  ( X  e.  RR  /\  T  <  X ) ) )
3229, 31mpbid 201 . . . . . . . . . . 11  |-  ( ph  ->  ( X  e.  RR  /\  T  <  X ) )
3332simprd 449 . . . . . . . . . 10  |-  ( ph  ->  T  <  X )
34 df-ioo 10749 . . . . . . . . . . 11  |-  (,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <  w  /\  w  <  v ) } )
35 df-ico 10751 . . . . . . . . . . 11  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
36 xrltletr 10577 . . . . . . . . . . 11  |-  ( ( T  e.  RR*  /\  X  e.  RR*  /\  z  e. 
RR* )  ->  (
( T  <  X  /\  X  <_  z )  ->  T  <  z
) )
3734, 35, 36ixxss1 10763 . . . . . . . . . 10  |-  ( ( T  e.  RR*  /\  T  <  X )  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3828, 33, 37syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3938, 1syl6sseqr 3301 . . . . . . . 8  |-  ( ph  ->  ( X [,)  +oo )  C_  S )
4039adantr 451 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,)  +oo )  C_  S
)
4140sselda 3256 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  y  e.  S )
42 ffvelrn 5743 . . . . . 6  |-  ( ( G : S --> RR  /\  y  e.  S )  ->  ( G `  y
)  e.  RR )
4323, 41, 42syl2anc 642 . . . . 5  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  y )  e.  RR )
4443recnd 8948 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  y )  e.  CC )
4527, 44subcld 9244 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  (
( G `  X
)  -  ( G `
 y ) )  e.  CC )
46 pnfxr 10544 . . . . . . 7  |-  +oo  e.  RR*
47 icossre 10819 . . . . . . 7  |-  ( ( X  e.  RR  /\  +oo 
e.  RR* )  ->  ( X [,)  +oo )  C_  RR )
485, 46, 47sylancl 643 . . . . . 6  |-  ( ph  ->  ( X [,)  +oo )  C_  RR )
4948adantr 451 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,)  +oo )  C_  RR )
50 rlimf 12065 . . . . . . . 8  |-  ( G  ~~> r  L  ->  G : dom  G --> CC )
5150adantl 452 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : dom  G --> CC )
52 ovex 5967 . . . . . . . . 9  |-  ( sum_ k  e.  ( M ... ( |_ `  x
) ) C  -  A )  e.  _V
5352, 21dmmpti 5452 . . . . . . . 8  |-  dom  G  =  S
5453feq2i 5464 . . . . . . 7  |-  ( G : dom  G --> CC  <->  G : S
--> CC )
5551, 54sylib 188 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : S
--> CC )
564adantr 451 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  X  e.  S )
57 ffvelrn 5743 . . . . . 6  |-  ( ( G : S --> CC  /\  X  e.  S )  ->  ( G `  X
)  e.  CC )
5855, 56, 57syl2anc 642 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( G `  X )  e.  CC )
59 rlimconst 12108 . . . . 5  |-  ( ( ( X [,)  +oo )  C_  RR  /\  ( G `  X )  e.  CC )  ->  (
y  e.  ( X [,)  +oo )  |->  ( G `
 X ) )  ~~> r  ( G `  X ) )
6049, 58, 59syl2anc 642 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( G `  X ) )  ~~> r  ( G `  X ) )
6155feqmptd 5655 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  =  ( y  e.  S  |->  ( G `  y
) ) )
62 simpr 447 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  ~~> r  L
)
6361, 62eqbrtrrd 4124 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  S  |->  ( G `
 y ) )  ~~> r  L )
6440, 63rlimres2 12125 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( G `  y ) )  ~~> r  L
)
6527, 44, 60, 64rlimsub 12207 . . 3  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( ( G `
 X )  -  ( G `  y ) ) )  ~~> r  ( ( G `  X
)  -  L ) )
6645, 65rlimabs 12172 . 2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( abs `  (
( G `  X
)  -  ( G `
 y ) ) ) )  ~~> r  ( abs `  ( ( G `  X )  -  L ) ) )
673a1i 10 . . . . . . . 8  |-  ( ph  ->  S  C_  RR )
6867, 16, 17, 19dvmptrecl 19469 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  RR )
6968ralrimiva 2702 . . . . . 6  |-  ( ph  ->  A. x  e.  S  B  e.  RR )
70 nfcsb1v 3189 . . . . . . . 8  |-  F/_ x [_ X  /  x ]_ B
7170nfel1 2504 . . . . . . 7  |-  F/ x [_ X  /  x ]_ B  e.  RR
72 csbeq1a 3165 . . . . . . . 8  |-  ( x  =  X  ->  B  =  [_ X  /  x ]_ B )
7372eleq1d 2424 . . . . . . 7  |-  ( x  =  X  ->  ( B  e.  RR  <->  [_ X  /  x ]_ B  e.  RR ) )
7471, 73rspc 2954 . . . . . 6  |-  ( X  e.  S  ->  ( A. x  e.  S  B  e.  RR  ->  [_ X  /  x ]_ B  e.  RR )
)
754, 69, 74sylc 56 . . . . 5  |-  ( ph  ->  [_ X  /  x ]_ B  e.  RR )
7675recnd 8948 . . . 4  |-  ( ph  ->  [_ X  /  x ]_ B  e.  CC )
77 rlimconst 12108 . . . 4  |-  ( ( ( X [,)  +oo )  C_  RR  /\  [_ X  /  x ]_ B  e.  CC )  ->  (
y  e.  ( X [,)  +oo )  |->  [_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B )
7848, 76, 77syl2anc 642 . . 3  |-  ( ph  ->  ( y  e.  ( X [,)  +oo )  |-> 
[_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
7978adantr 451 . 2  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  [_ X  /  x ]_ B )  ~~> r  [_ X  /  x ]_ B
)
8045abscld 12008 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  e.  RR )
8175ad2antrr 706 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  [_ X  /  x ]_ B  e.  RR )
8227, 44abssubd 12025 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  =  ( abs `  ( ( G `  y )  -  ( G `  X ) ) ) )
8312adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  M  e.  ZZ )
8413adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  D  e.  RR )
8514adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  M  <_  ( D  +  1 ) )
8615adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  T  e.  RR )
8716adantlr 695 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  S )  ->  A  e.  RR )
8817adantlr 695 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  S )  ->  B  e.  V )
8918adantlr 695 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  x  e.  Z )  ->  B  e.  RR )
9019adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  ( RR  _D  ( x  e.  S  |->  A ) )  =  ( x  e.  S  |->  B ) )
9146a1i 10 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  +oo  e.  RR* )
92 3simpa 952 . . . . . . 7  |-  ( ( D  <_  x  /\  x  <_  k  /\  k  <_  +oo )  ->  ( D  <_  x  /\  x  <_  k ) )
93 dvfsumrlim.l . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k ) )  ->  C  <_  B )
9492, 93syl3an3 1217 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_  +oo ) )  ->  C  <_  B )
95943adant1r 1175 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  (
x  e.  S  /\  k  e.  S )  /\  ( D  <_  x  /\  x  <_  k  /\  k  <_  +oo ) )  ->  C  <_  B )
96 dvfsumrlim.k . . . . . . . 8  |-  ( ph  ->  ( x  e.  S  |->  B )  ~~> r  0 )
971, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 93, 21, 96dvfsumrlimge0 19475 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x ) )  -> 
0  <_  B )
98973adantr3 1116 . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  D  <_  x  /\  x  <_  +oo )
)  ->  0  <_  B )
9998adantlr 695 . . . . 5  |-  ( ( ( ph  /\  y  e.  ( X [,)  +oo ) )  /\  (
x  e.  S  /\  D  <_  x  /\  x  <_  +oo ) )  -> 
0  <_  B )
1004adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  X  e.  S )
10139sselda 3256 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  S )
102 dvfsumrlim2.2 . . . . . 6  |-  ( ph  ->  D  <_  X )
103102adantr 451 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  D  <_  X )
104 elicopnf 10828 . . . . . . 7  |-  ( X  e.  RR  ->  (
y  e.  ( X [,)  +oo )  <->  ( y  e.  RR  /\  X  <_ 
y ) ) )
1055, 104syl 15 . . . . . 6  |-  ( ph  ->  ( y  e.  ( X [,)  +oo )  <->  ( y  e.  RR  /\  X  <_  y ) ) )
106105simplbda 607 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  X  <_  y )
107105simprbda 606 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  RR )
108107rexrd 8968 . . . . . 6  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  RR* )
109 pnfge 10558 . . . . . 6  |-  ( y  e.  RR*  ->  y  <_  +oo )
110108, 109syl 15 . . . . 5  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  <_  +oo )
1111, 11, 83, 84, 85, 86, 87, 88, 89, 90, 20, 91, 95, 21, 99, 100, 101, 103, 106, 110dvfsumlem4 19474 . . . 4  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  ( abs `  ( ( G `  y )  -  ( G `  X )
) )  <_  [_ X  /  x ]_ B )
112111adantlr 695 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 y )  -  ( G `  X ) ) )  <_  [_ X  /  x ]_ B )
11382, 112eqbrtrd 4122 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  <_  [_ X  /  x ]_ B )
11410, 66, 79, 80, 81, 113rlimle 12211 1  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( abs `  ( ( G `  X )  -  L
) )  <_  [_ X  /  x ]_ B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   [_csb 3157    C_ wss 3228   class class class wbr 4102    e. cmpt 4156   dom cdm 4768   -->wf 5330   ` cfv 5334  (class class class)co 5942   supcsup 7280   CCcc 8822   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827    +oocpnf 8951   RR*cxr 8953    < clt 8954    <_ cle 8955    - cmin 9124   ZZcz 10113   ZZ>=cuz 10319   (,)cioo 10745   [,)cico 10747   ...cfz 10871   |_cfl 11013   abscabs 11809    ~~> r crli 12049   sum_csu 12249    _D cdv 19311
This theorem is referenced by:  dvfsumrlim3  19478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-iin 3987  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-ixp 6903  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-fi 7252  df-sup 7281  df-oi 7312  df-card 7659  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-q 10406  df-rp 10444  df-xneg 10541  df-xadd 10542  df-xmul 10543  df-ioo 10749  df-ico 10751  df-icc 10752  df-fz 10872  df-fzo 10960  df-fl 11014  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-rlim 12053  df-sum 12250  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-hom 13323  df-cco 13324  df-rest 13420  df-topn 13421  df-topgen 13437  df-pt 13438  df-prds 13441  df-xrs 13496  df-0g 13497  df-gsum 13498  df-qtop 13503  df-imas 13504  df-xps 13506  df-mre 13581  df-mrc 13582  df-acs 13584  df-mnd 14460  df-submnd 14509  df-mulg 14585  df-cntz 14886  df-cmn 15184  df-xmet 16469  df-met 16470  df-bl 16471  df-mopn 16472  df-fbas 16473  df-fg 16474  df-cnfld 16477  df-top 16736  df-bases 16738  df-topon 16739  df-topsp 16740  df-cld 16856  df-ntr 16857  df-cls 16858  df-nei 16935  df-lp 16968  df-perf 16969  df-cn 17057  df-cnp 17058  df-haus 17143  df-cmp 17214  df-tx 17357  df-hmeo 17546  df-fil 17637  df-fm 17729  df-flim 17730  df-flf 17731  df-xms 17981  df-ms 17982  df-tms 17983  df-cncf 18479  df-limc 19314  df-dv 19315
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