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Theorem ftc2ditglem 19392
Description: Lemma for ftc2ditg 19393. (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 447 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  <_  B )
21ditgpos 19206 . 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 10731 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
63, 4, 5syl2anc 642 . . . . . 6  |-  ( ph  ->  ( X [,] Y
)  C_  RR )
7 ftc2ditg.a . . . . . 6  |-  ( ph  ->  A  e.  ( X [,] Y ) )
86, 7sseldd 3181 . . . . 5  |-  ( ph  ->  A  e.  RR )
98adantr 451 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
10 ftc2ditg.b . . . . . 6  |-  ( ph  ->  B  e.  ( X [,] Y ) )
116, 10sseldd 3181 . . . . 5  |-  ( ph  ->  B  e.  RR )
1211adantr 451 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR )
13 ax-resscn 8794 . . . . . . . 8  |-  RR  C_  CC
1413a1i 10 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  RR  C_  CC )
15 ftc2ditg.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( X [,] Y )
-cn-> CC ) )
16 cncff 18397 . . . . . . . . 9  |-  ( F  e.  ( ( X [,] Y ) -cn-> CC )  ->  F :
( X [,] Y
) --> CC )
1715, 16syl 15 . . . . . . . 8  |-  ( ph  ->  F : ( X [,] Y ) --> CC )
1817adantr 451 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  F :
( X [,] Y
) --> CC )
196adantr 451 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( X [,] Y )  C_  RR )
20 iccssre 10731 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
218, 11, 20syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  C_  RR )
2221adantr 451 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( A [,] B )  C_  RR )
23 eqid 2283 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 18309 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2523, 24dvres 19261 . . . . . . 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 1183 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
27 iccntr 18326 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
288, 11, 27syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2928adantr 451 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( A (,) B ) )
3029reseq2d 4955 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) ) )
3126, 30eqtrd 2315 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( A (,) B
) ) )
323rexrd 8881 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR* )
33 elicc2 10715 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
343, 4, 33syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
357, 34mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) )
3635simp2d 968 . . . . . . . . 9  |-  ( ph  ->  X  <_  A )
37 iooss1 10691 . . . . . . . . 9  |-  ( ( X  e.  RR*  /\  X  <_  A )  ->  ( A (,) B )  C_  ( X (,) B ) )
3832, 36, 37syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) B ) )
394rexrd 8881 . . . . . . . . 9  |-  ( ph  ->  Y  e.  RR* )
40 elicc2 10715 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
413, 4, 40syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
4210, 41mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) )
4342simp3d 969 . . . . . . . . 9  |-  ( ph  ->  B  <_  Y )
44 iooss2 10692 . . . . . . . . 9  |-  ( ( Y  e.  RR*  /\  B  <_  Y )  ->  ( X (,) B )  C_  ( X (,) Y ) )
4539, 43, 44syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( X (,) B
)  C_  ( X (,) Y ) )
4638, 45sstrd 3189 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) Y ) )
4746adantr 451 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  C_  ( X (,) Y ) )
48 ftc2ditg.c . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( X (,) Y )
-cn-> CC ) )
4948adantr 451 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  ( ( X (,) Y
) -cn-> CC ) )
50 rescncf 18401 . . . . . 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 56 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  e.  ( ( A (,) B )
-cn-> CC ) )
5231, 51eqeltrd 2357 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  ( ( A (,) B )
-cn-> CC ) )
53 cncff 18397 . . . . . . . . . . 11  |-  ( ( RR  _D  F )  e.  ( ( X (,) Y ) -cn-> CC )  ->  ( RR  _D  F ) : ( X (,) Y ) --> CC )
5448, 53syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
) : ( X (,) Y ) --> CC )
5554feqmptd 5575 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F ) `
 t ) ) )
5655adantr 451 . . . . . . . 8  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
) )
5756reseq1d 4954 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) ) )
58 resmpt 5000 . . . . . . . 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 15 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( (
t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) ) )
6057, 59eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
6131, 60eqtrd 2315 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
62 ioombl 18922 . . . . . . 7  |-  ( A (,) B )  e. 
dom  vol
6362a1i 10 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  e.  dom  vol )
64 fvex 5539 . . . . . . 7  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
6564a1i 10 . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  L ^1 )
6856, 67eqeltrrd 2358 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( X (,) Y
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
6947, 63, 65, 68iblss 19159 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
7061, 69eqeltrd 2357 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  L ^1 )
71 iccss2 10720 . . . . . . 7  |-  ( ( A  e.  ( X [,] Y )  /\  B  e.  ( X [,] Y ) )  -> 
( A [,] B
)  C_  ( X [,] Y ) )
727, 10, 71syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  ( X [,] Y ) )
73 rescncf 18401 . . . . . 6  |-  ( ( A [,] B ) 
C_  ( X [,] Y )  ->  ( F  e.  ( ( X [,] Y ) -cn-> CC )  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
7472, 15, 73sylc 56 . . . . 5  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
7574adantr 451 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
769, 12, 1, 52, 70, 75ftc2 19391 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  _d t  =  ( ( ( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) ) )
7731fveq1d 5527 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `  t
)  =  ( ( ( RR  _D  F
)  |`  ( A (,) B ) ) `  t ) )
78 fvres 5542 . . . . 5  |-  ( t  e.  ( A (,) B )  ->  (
( ( RR  _D  F )  |`  ( A (,) B ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
7977, 78sylan9eq 2335 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
8079itgeq2dv 19136 . . 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 8881 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR* )
8212rexrd 8881 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR* )
83 ubicc2 10753 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
84 lbicc2 10752 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
85 fvres 5542 . . . . . 6  |-  ( B  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 B )  =  ( F `  B
) )
86 fvres 5542 . . . . . 6  |-  ( A  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 A )  =  ( F `  A
) )
8785, 86oveqan12d 5877 . . . . 5  |-  ( ( B  e.  ( A [,] B )  /\  A  e.  ( A [,] B ) )  -> 
( ( ( F  |`  ( A [,] B
) ) `  B
)  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A )
) )
8883, 84, 87syl2anc 642 . . . 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 1182 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  ( (
( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
9076, 80, 893eqtr3d 2323 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  =  (
( F `  B
)  -  ( F `
 A ) ) )
912, 90eqtrd 2315 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   RR*cxr 8866    <_ cle 8868    - cmin 9037   (,)cioo 10656   [,]cicc 10659   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377   intcnt 16754   -cn->ccncf 18380   volcvol 18823   L ^1cibl 18972   S.citg 18973   S__cdit 18974    _D cdv 19213
This theorem is referenced by:  ftc2ditg  19393
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-cc 8061  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-disj 3994  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-ofr 6079  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-omul 6484  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-acn 7575  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-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  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-ovol 18824  df-vol 18825  df-mbf 18975  df-itg1 18976  df-itg2 18977  df-ibl 18978  df-itg 18979  df-ditg 18980  df-0p 19025  df-limc 19216  df-dv 19217
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