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Theorem dvfsumrlim2 19379
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 10712 . . . . . . 7  |-  ( T (,)  +oo )  C_  RR
31, 2eqsstri 3208 . . . . . 6  |-  S  C_  RR
4 dvfsumrlim2.1 . . . . . 6  |-  ( ph  ->  X  e.  S )
53, 4sseldi 3178 . . . . 5  |-  ( ph  ->  X  e.  RR )
65rexrd 8881 . . . 4  |-  ( ph  ->  X  e.  RR* )
75renepnfd 8882 . . . 4  |-  ( ph  ->  X  =/=  +oo )
8 icopnfsup 10969 . . . 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 19372 . . . . . . 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 5663 . . . . . 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 8861 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  X )  e.  CC )
2815rexrd 8881 . . . . . . . . . 10  |-  ( ph  ->  T  e.  RR* )
294, 1syl6eleq 2373 . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  ( T (,)  +oo ) )
30 elioopnf 10737 . . . . . . . . . . . . 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 10660 . . . . . . . . . . 11  |-  (,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <  w  /\  w  <  v ) } )
35 df-ico 10662 . . . . . . . . . . 11  |-  [,)  =  ( u  e.  RR* ,  v  e.  RR*  |->  { w  e.  RR*  |  ( u  <_  w  /\  w  <  v ) } )
36 xrltletr 10488 . . . . . . . . . . 11  |-  ( ( T  e.  RR*  /\  X  e.  RR*  /\  z  e. 
RR* )  ->  (
( T  <  X  /\  X  <_  z )  ->  T  <  z
) )
3734, 35, 36ixxss1 10674 . . . . . . . . . 10  |-  ( ( T  e.  RR*  /\  T  <  X )  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3828, 33, 37syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( X [,)  +oo )  C_  ( T (,)  +oo ) )
3938, 1syl6sseqr 3225 . . . . . . . 8  |-  ( ph  ->  ( X [,)  +oo )  C_  S )
4039adantr 451 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( X [,)  +oo )  C_  S
)
4140sselda 3180 . . . . . 6  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  y  e.  S )
42 ffvelrn 5663 . . . . . 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 8861 . . . 4  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( G `  y )  e.  CC )
4527, 44subcld 9157 . . 3  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  (
( G `  X
)  -  ( G `
 y ) )  e.  CC )
46 pnfxr 10455 . . . . . . 7  |-  +oo  e.  RR*
47 icossre 10730 . . . . . . 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 11975 . . . . . . . 8  |-  ( G  ~~> r  L  ->  G : dom  G --> CC )
5150adantl 452 . . . . . . 7  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G : dom  G --> CC )
52 ovex 5883 . . . . . . . . 9  |-  ( sum_ k  e.  ( M ... ( |_ `  x
) ) C  -  A )  e.  _V
5352, 21dmmpti 5373 . . . . . . . 8  |-  dom  G  =  S
5453feq2i 5384 . . . . . . 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 5663 . . . . . 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 12018 . . . . 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 5575 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  =  ( y  e.  S  |->  ( G `  y
) ) )
62 simpr 447 . . . . . 6  |-  ( (
ph  /\  G  ~~> r  L
)  ->  G  ~~> r  L
)
6361, 62eqbrtrrd 4045 . . . . 5  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  S  |->  ( G `
 y ) )  ~~> r  L )
6440, 63rlimres2 12035 . . . 4  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( G `  y ) )  ~~> r  L
)
6527, 44, 60, 64rlimsub 12117 . . 3  |-  ( (
ph  /\  G  ~~> r  L
)  ->  ( y  e.  ( X [,)  +oo )  |->  ( ( G `
 X )  -  ( G `  y ) ) )  ~~> r  ( ( G `  X
)  -  L ) )
6645, 65rlimabs 12082 . 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 19371 . . . . . . 7  |-  ( (
ph  /\  x  e.  S )  ->  B  e.  RR )
6968ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. x  e.  S  B  e.  RR )
70 nfcsb1v 3113 . . . . . . . 8  |-  F/_ x [_ X  /  x ]_ B
7170nfel1 2429 . . . . . . 7  |-  F/ x [_ X  /  x ]_ B  e.  RR
72 csbeq1a 3089 . . . . . . . 8  |-  ( x  =  X  ->  B  =  [_ X  /  x ]_ B )
7372eleq1d 2349 . . . . . . 7  |-  ( x  =  X  ->  ( B  e.  RR  <->  [_ X  /  x ]_ B  e.  RR ) )
7471, 73rspc 2878 . . . . . 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 8861 . . . 4  |-  ( ph  ->  [_ X  /  x ]_ B  e.  CC )
77 rlimconst 12018 . . . 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 11918 . 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 11935 . . 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 19377 . . . . . . 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 3180 . . . . 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 10739 . . . . . . 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 8881 . . . . . 6  |-  ( (
ph  /\  y  e.  ( X [,)  +oo )
)  ->  y  e.  RR* )
109 pnfge 10469 . . . . . 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 19376 . . . 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 4043 . 2  |-  ( ( ( ph  /\  G  ~~> r  L )  /\  y  e.  ( X [,)  +oo ) )  ->  ( abs `  ( ( G `
 X )  -  ( G `  y ) ) )  <_  [_ X  /  x ]_ B )
11410, 66, 79, 80, 81, 113rlimle 12121 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   [_csb 3081    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   ZZcz 10024   ZZ>=cuz 10230   (,)cioo 10656   [,)cico 10658   ...cfz 10782   |_cfl 10924   abscabs 11719    ~~> r crli 11959   sum_csu 12158    _D cdv 19213
This theorem is referenced by:  dvfsumrlim3  19380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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