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Theorem ftc2ditglem 19796
Description: Lemma for ftc2ditg 19797. (Contributed by Mario Carneiro, 3-Sep-2014.)
Hypotheses
Ref Expression
ftc2ditg.x  |-  ( ph  ->  X  e.  RR )
ftc2ditg.y  |-  ( ph  ->  Y  e.  RR )
ftc2ditg.a  |-  ( ph  ->  A  e.  ( X [,] Y ) )
ftc2ditg.b  |-  ( ph  ->  B  e.  ( X [,] Y ) )
ftc2ditg.c  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( X (,) Y )
-cn-> CC ) )
ftc2ditg.i  |-  ( ph  ->  ( RR  _D  F
)  e.  L ^1 )
ftc2ditg.f  |-  ( ph  ->  F  e.  ( ( X [,] Y )
-cn-> CC ) )
Assertion
Ref Expression
ftc2ditglem  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  B ] ( ( RR  _D  F ) `
 t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
Distinct variable groups:    t, A    t, B    t, F    ph, t    t, X    t, Y

Proof of Theorem ftc2ditglem
StepHypRef Expression
1 simpr 448 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  <_  B )
21ditgpos 19610 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  B ] ( ( RR  _D  F ) `
 t )  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t )
3 ftc2ditg.x . . . . . . 7  |-  ( ph  ->  X  e.  RR )
4 ftc2ditg.y . . . . . . 7  |-  ( ph  ->  Y  e.  RR )
5 iccssre 10924 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
63, 4, 5syl2anc 643 . . . . . 6  |-  ( ph  ->  ( X [,] Y
)  C_  RR )
7 ftc2ditg.a . . . . . 6  |-  ( ph  ->  A  e.  ( X [,] Y ) )
86, 7sseldd 3292 . . . . 5  |-  ( ph  ->  A  e.  RR )
98adantr 452 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
10 ftc2ditg.b . . . . . 6  |-  ( ph  ->  B  e.  ( X [,] Y ) )
116, 10sseldd 3292 . . . . 5  |-  ( ph  ->  B  e.  RR )
1211adantr 452 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR )
13 ax-resscn 8980 . . . . . . . 8  |-  RR  C_  CC
1413a1i 11 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  RR  C_  CC )
15 ftc2ditg.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( X [,] Y )
-cn-> CC ) )
16 cncff 18794 . . . . . . . . 9  |-  ( F  e.  ( ( X [,] Y ) -cn-> CC )  ->  F :
( X [,] Y
) --> CC )
1715, 16syl 16 . . . . . . . 8  |-  ( ph  ->  F : ( X [,] Y ) --> CC )
1817adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  F :
( X [,] Y
) --> CC )
196adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( X [,] Y )  C_  RR )
20 iccssre 10924 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
218, 11, 20syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  C_  RR )
2221adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( A [,] B )  C_  RR )
23 eqid 2387 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 18705 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2523, 24dvres 19665 . . . . . . 7  |-  ( ( ( RR  C_  CC  /\  F : ( X [,] Y ) --> CC )  /\  ( ( X [,] Y ) 
C_  RR  /\  ( A [,] B )  C_  RR ) )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) ) ) )
2614, 18, 19, 22, 25syl22anc 1185 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
27 iccntr 18723 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
288, 11, 27syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2928adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( A (,) B ) )
3029reseq2d 5086 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) ) )
3126, 30eqtrd 2419 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( A (,) B
) ) )
323rexrd 9067 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR* )
33 elicc2 10907 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
343, 4, 33syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
357, 34mpbid 202 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) )
3635simp2d 970 . . . . . . . . 9  |-  ( ph  ->  X  <_  A )
37 iooss1 10883 . . . . . . . . 9  |-  ( ( X  e.  RR*  /\  X  <_  A )  ->  ( A (,) B )  C_  ( X (,) B ) )
3832, 36, 37syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) B ) )
394rexrd 9067 . . . . . . . . 9  |-  ( ph  ->  Y  e.  RR* )
40 elicc2 10907 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
413, 4, 40syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
4210, 41mpbid 202 . . . . . . . . . 10  |-  ( ph  ->  ( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) )
4342simp3d 971 . . . . . . . . 9  |-  ( ph  ->  B  <_  Y )
44 iooss2 10884 . . . . . . . . 9  |-  ( ( Y  e.  RR*  /\  B  <_  Y )  ->  ( X (,) B )  C_  ( X (,) Y ) )
4539, 43, 44syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( X (,) B
)  C_  ( X (,) Y ) )
4638, 45sstrd 3301 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) Y ) )
4746adantr 452 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  C_  ( X (,) Y ) )
48 ftc2ditg.c . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( X (,) Y )
-cn-> CC ) )
4948adantr 452 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  ( ( X (,) Y
) -cn-> CC ) )
50 rescncf 18798 . . . . . 6  |-  ( ( A (,) B ) 
C_  ( X (,) Y )  ->  (
( RR  _D  F
)  e.  ( ( X (,) Y )
-cn-> CC )  ->  (
( RR  _D  F
)  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> CC ) ) )
5147, 49, 50sylc 58 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  e.  ( ( A (,) B )
-cn-> CC ) )
5231, 51eqeltrd 2461 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  ( ( A (,) B )
-cn-> CC ) )
53 cncff 18794 . . . . . . . . . . 11  |-  ( ( RR  _D  F )  e.  ( ( X (,) Y ) -cn-> CC )  ->  ( RR  _D  F ) : ( X (,) Y ) --> CC )
5448, 53syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
) : ( X (,) Y ) --> CC )
5554feqmptd 5718 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F ) `
 t ) ) )
5655adantr 452 . . . . . . . 8  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
) )
5756reseq1d 5085 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) ) )
58 resmpt 5131 . . . . . . . 8  |-  ( ( A (,) B ) 
C_  ( X (,) Y )  ->  (
( t  e.  ( X (,) Y ) 
|->  ( ( RR  _D  F ) `  t
) )  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) ) )
5947, 58syl 16 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( (
t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) ) )
6057, 59eqtrd 2419 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
6131, 60eqtrd 2419 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
62 ioombl 19326 . . . . . . 7  |-  ( A (,) B )  e. 
dom  vol
6362a1i 11 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  e.  dom  vol )
64 fvex 5682 . . . . . . 7  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
6564a1i 11 . . . . . 6  |-  ( ( ( ph  /\  A  <_  B )  /\  t  e.  ( X (,) Y
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
66 ftc2ditg.i . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
)  e.  L ^1 )
6766adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  L ^1 )
6856, 67eqeltrrd 2462 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( X (,) Y
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
6947, 63, 65, 68iblss 19563 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
7061, 69eqeltrd 2461 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  L ^1 )
71 iccss2 10913 . . . . . . 7  |-  ( ( A  e.  ( X [,] Y )  /\  B  e.  ( X [,] Y ) )  -> 
( A [,] B
)  C_  ( X [,] Y ) )
727, 10, 71syl2anc 643 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  ( X [,] Y ) )
73 rescncf 18798 . . . . . 6  |-  ( ( A [,] B ) 
C_  ( X [,] Y )  ->  ( F  e.  ( ( X [,] Y ) -cn-> CC )  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
7472, 15, 73sylc 58 . . . . 5  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
7574adantr 452 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
769, 12, 1, 52, 70, 75ftc2 19795 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  _d t  =  ( ( ( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) ) )
7731fveq1d 5670 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `  t
)  =  ( ( ( RR  _D  F
)  |`  ( A (,) B ) ) `  t ) )
78 fvres 5685 . . . . 5  |-  ( t  e.  ( A (,) B )  ->  (
( ( RR  _D  F )  |`  ( A (,) B ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
7977, 78sylan9eq 2439 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
8079itgeq2dv 19540 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t )
819rexrd 9067 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR* )
8212rexrd 9067 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR* )
83 ubicc2 10946 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
84 lbicc2 10945 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
85 fvres 5685 . . . . . 6  |-  ( B  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 B )  =  ( F `  B
) )
86 fvres 5685 . . . . . 6  |-  ( A  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 A )  =  ( F `  A
) )
8785, 86oveqan12d 6039 . . . . 5  |-  ( ( B  e.  ( A [,] B )  /\  A  e.  ( A [,] B ) )  -> 
( ( ( F  |`  ( A [,] B
) ) `  B
)  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A )
) )
8883, 84, 87syl2anc 643 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( ( F  |`  ( A [,] B ) ) `  B )  -  ( ( F  |`  ( A [,] B
) ) `  A
) )  =  ( ( F `  B
)  -  ( F `
 A ) ) )
8981, 82, 1, 88syl3anc 1184 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  ( (
( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
9076, 80, 893eqtr3d 2427 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  =  (
( F `  B
)  -  ( F `
 A ) ) )
912, 90eqtrd 2419 1  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  B ] ( ( RR  _D  F ) `
 t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   class class class wbr 4153    e. cmpt 4207   dom cdm 4818   ran crn 4819    |` cres 4820   -->wf 5390   ` cfv 5394  (class class class)co 6020   CCcc 8921   RRcr 8922   RR*cxr 9052    <_ cle 9054    - cmin 9223   (,)cioo 10848   [,]cicc 10851   TopOpenctopn 13576   topGenctg 13592  ℂfldccnfld 16626   intcnt 17004   -cn->ccncf 18777   volcvol 19227   L ^1cibl 19376   S.citg 19377   S__cdit 19378    _D cdv 19617
This theorem is referenced by:  ftc2ditg  19797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cc 8248  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-disj 4124  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-ofr 6245  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-acn 7762  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-cmp 17372  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-ovol 19228  df-vol 19229  df-mbf 19379  df-itg1 19380  df-itg2 19381  df-ibl 19382  df-itg 19383  df-ditg 19384  df-0p 19429  df-limc 19620  df-dv 19621
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