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Theorem ftc2ditglem 19408
Description: Lemma for ftc2ditg 19409. (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 19222 . 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 10747 . . . . . . 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 3194 . . . . 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 3194 . . . . 5  |-  ( ph  ->  B  e.  RR )
1211adantr 451 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR )
13 ax-resscn 8810 . . . . . . . 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 18413 . . . . . . . . 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 10747 . . . . . . . . 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 2296 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 18325 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2523, 24dvres 19277 . . . . . . 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 18342 . . . . . . . . 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 4971 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) ) )
3126, 30eqtrd 2328 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( A (,) B
) ) )
323rexrd 8897 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR* )
33 elicc2 10731 . . . . . . . . . . . 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 10707 . . . . . . . . 9  |-  ( ( X  e.  RR*  /\  X  <_  A )  ->  ( A (,) B )  C_  ( X (,) B ) )
3832, 36, 37syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) B ) )
394rexrd 8897 . . . . . . . . 9  |-  ( ph  ->  Y  e.  RR* )
40 elicc2 10731 . . . . . . . . . . . 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 10708 . . . . . . . . 9  |-  ( ( Y  e.  RR*  /\  B  <_  Y )  ->  ( X (,) B )  C_  ( X (,) Y ) )
4539, 43, 44syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( X (,) B
)  C_  ( X (,) Y ) )
4638, 45sstrd 3202 . . . . . . 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 18417 . . . . . 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 2370 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  ( ( A (,) B )
-cn-> CC ) )
53 cncff 18413 . . . . . . . . . . 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 5591 . . . . . . . . 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 4970 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) ) )
58 resmpt 5016 . . . . . . . 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 2328 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
6131, 60eqtrd 2328 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
62 ioombl 18938 . . . . . . 7  |-  ( A (,) B )  e. 
dom  vol
6362a1i 10 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  e.  dom  vol )
64 fvex 5555 . . . . . . 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 2371 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( X (,) Y
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
6947, 63, 65, 68iblss 19175 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L ^1 )
7061, 69eqeltrd 2370 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  L ^1 )
71 iccss2 10736 . . . . . . 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 18417 . . . . . 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 19407 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  _d t  =  ( ( ( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) ) )
7731fveq1d 5543 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `  t
)  =  ( ( ( RR  _D  F
)  |`  ( A (,) B ) ) `  t ) )
78 fvres 5558 . . . . 5  |-  ( t  e.  ( A (,) B )  ->  (
( ( RR  _D  F )  |`  ( A (,) B ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
7977, 78sylan9eq 2348 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
8079itgeq2dv 19152 . . 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 8897 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR* )
8212rexrd 8897 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR* )
83 ubicc2 10769 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
84 lbicc2 10768 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
85 fvres 5558 . . . . . 6  |-  ( B  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 B )  =  ( F `  B
) )
86 fvres 5558 . . . . . 6  |-  ( A  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 A )  =  ( F `  A
) )
8785, 86oveqan12d 5893 . . . . 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 2336 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  =  (
( F `  B
)  -  ( F `
 A ) ) )
912, 90eqtrd 2328 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   RR*cxr 8882    <_ cle 8884    - cmin 9053   (,)cioo 10672   [,]cicc 10675   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   intcnt 16770   -cn->ccncf 18396   volcvol 18839   L ^1cibl 18988   S.citg 18989   S__cdit 18990    _D cdv 19229
This theorem is referenced by:  ftc2ditg  19409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992  df-itg2 18993  df-ibl 18994  df-itg 18995  df-ditg 18996  df-0p 19041  df-limc 19232  df-dv 19233
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