MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ftc2ditglem Structured version   Unicode version

Theorem ftc2ditglem 22319
Description: Lemma for ftc2ditg 22320. (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 461 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  <_  B )
21ditgpos 22133 . 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 11615 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
63, 4, 5syl2anc 661 . . . . . 6  |-  ( ph  ->  ( X [,] Y
)  C_  RR )
7 ftc2ditg.a . . . . . 6  |-  ( ph  ->  A  e.  ( X [,] Y ) )
86, 7sseldd 3490 . . . . 5  |-  ( ph  ->  A  e.  RR )
98adantr 465 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR )
10 ftc2ditg.b . . . . . 6  |-  ( ph  ->  B  e.  ( X [,] Y ) )
116, 10sseldd 3490 . . . . 5  |-  ( ph  ->  B  e.  RR )
1211adantr 465 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR )
13 ax-resscn 9552 . . . . . . . 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 21270 . . . . . . . . 9  |-  ( F  e.  ( ( X [,] Y ) -cn-> CC )  ->  F :
( X [,] Y
) --> CC )
1715, 16syl 16 . . . . . . . 8  |-  ( ph  ->  F : ( X [,] Y ) --> CC )
1817adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  F :
( X [,] Y
) --> CC )
196adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( X [,] Y )  C_  RR )
20 iccssre 11615 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
218, 11, 20syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  C_  RR )
2221adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( A [,] B )  C_  RR )
23 eqid 2443 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 21181 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2523, 24dvres 22188 . . . . . . 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 1230 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
27 iccntr 21199 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
288, 11, 27syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2928adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( A (,) B ) )
3029reseq2d 5263 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) ) )
3126, 30eqtrd 2484 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( A (,) B
) ) )
323rexrd 9646 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR* )
33 elicc2 11598 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
343, 4, 33syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
357, 34mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) )
3635simp2d 1010 . . . . . . . . 9  |-  ( ph  ->  X  <_  A )
37 iooss1 11573 . . . . . . . . 9  |-  ( ( X  e.  RR*  /\  X  <_  A )  ->  ( A (,) B )  C_  ( X (,) B ) )
3832, 36, 37syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) B ) )
394rexrd 9646 . . . . . . . . 9  |-  ( ph  ->  Y  e.  RR* )
40 elicc2 11598 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
413, 4, 40syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
4210, 41mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) )
4342simp3d 1011 . . . . . . . . 9  |-  ( ph  ->  B  <_  Y )
44 iooss2 11574 . . . . . . . . 9  |-  ( ( Y  e.  RR*  /\  B  <_  Y )  ->  ( X (,) B )  C_  ( X (,) Y ) )
4539, 43, 44syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( X (,) B
)  C_  ( X (,) Y ) )
4638, 45sstrd 3499 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) Y ) )
4746adantr 465 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  C_  ( X (,) Y ) )
48 ftc2ditg.c . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( X (,) Y )
-cn-> CC ) )
4948adantr 465 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  ( ( X (,) Y
) -cn-> CC ) )
50 rescncf 21274 . . . . . 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 60 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  e.  ( ( A (,) B )
-cn-> CC ) )
5231, 51eqeltrd 2531 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  ( ( A (,) B )
-cn-> CC ) )
53 cncff 21270 . . . . . . . . . . 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 5911 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F ) `
 t ) ) )
5655adantr 465 . . . . . . . 8  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
) )
5756reseq1d 5262 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) ) )
5847resmptd 5315 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( (
t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) ) )
5957, 58eqtrd 2484 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
6031, 59eqtrd 2484 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
61 ioombl 21848 . . . . . . 7  |-  ( A (,) B )  e. 
dom  vol
6261a1i 11 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  e.  dom  vol )
63 fvex 5866 . . . . . . 7  |-  ( ( RR  _D  F ) `
 t )  e. 
_V
6463a1i 11 . . . . . 6  |-  ( ( ( ph  /\  A  <_  B )  /\  t  e.  ( X (,) Y
) )  ->  (
( RR  _D  F
) `  t )  e.  _V )
65 ftc2ditg.i . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
)  e.  L^1 )
6665adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  L^1 )
6756, 66eqeltrrd 2532 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( X (,) Y
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
6847, 62, 64, 67iblss 22084 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
6960, 68eqeltrd 2531 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  L^1 )
70 iccss2 11604 . . . . . . 7  |-  ( ( A  e.  ( X [,] Y )  /\  B  e.  ( X [,] Y ) )  -> 
( A [,] B
)  C_  ( X [,] Y ) )
717, 10, 70syl2anc 661 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  ( X [,] Y ) )
72 rescncf 21274 . . . . . 6  |-  ( ( A [,] B ) 
C_  ( X [,] Y )  ->  ( F  e.  ( ( X [,] Y ) -cn-> CC )  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
7371, 15, 72sylc 60 . . . . 5  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
7473adantr 465 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
759, 12, 1, 52, 69, 74ftc2 22318 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  _d t  =  ( ( ( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) ) )
7631fveq1d 5858 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `  t
)  =  ( ( ( RR  _D  F
)  |`  ( A (,) B ) ) `  t ) )
77 fvres 5870 . . . . 5  |-  ( t  e.  ( A (,) B )  ->  (
( ( RR  _D  F )  |`  ( A (,) B ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
7876, 77sylan9eq 2504 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
7978itgeq2dv 22061 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  _d t  =  S. ( A (,) B ) ( ( RR  _D  F ) `  t
)  _d t )
809rexrd 9646 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR* )
8112rexrd 9646 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR* )
82 ubicc2 11646 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
83 lbicc2 11645 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
84 fvres 5870 . . . . . 6  |-  ( B  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 B )  =  ( F `  B
) )
85 fvres 5870 . . . . . 6  |-  ( A  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 A )  =  ( F `  A
) )
8684, 85oveqan12d 6300 . . . . 5  |-  ( ( B  e.  ( A [,] B )  /\  A  e.  ( A [,] B ) )  -> 
( ( ( F  |`  ( A [,] B
) ) `  B
)  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A )
) )
8782, 83, 86syl2anc 661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  (
( ( F  |`  ( A [,] B ) ) `  B )  -  ( ( F  |`  ( A [,] B
) ) `  A
) )  =  ( ( F `  B
)  -  ( F `
 A ) ) )
8880, 81, 1, 87syl3anc 1229 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  ( (
( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
8975, 79, 883eqtr3d 2492 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  =  (
( F `  B
)  -  ( F `
 A ) ) )
902, 89eqtrd 2484 1  |-  ( (
ph  /\  A  <_  B )  ->  S__ [ A  ->  B ] ( ( RR  _D  F ) `
 t )  _d t  =  ( ( F `  B )  -  ( F `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989   ran crn 4990    |` cres 4991   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   RR*cxr 9630    <_ cle 9632    - cmin 9810   (,)cioo 11538   [,]cicc 11541   TopOpenctopn 14696   topGenctg 14712  ℂfldccnfld 18294   intcnt 19391   -cn->ccncf 21253   volcvol 21748   L^1cibl 21899   S.citg 21900   S__cdit 22123    _D cdv 22140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cc 8818  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-rlim 13291  df-sum 13488  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-cmp 19760  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-ovol 21749  df-vol 21750  df-mbf 21901  df-itg1 21902  df-itg2 21903  df-ibl 21904  df-itg 21905  df-0p 21950  df-ditg 22124  df-limc 22143  df-dv 22144
This theorem is referenced by:  ftc2ditg  22320
  Copyright terms: Public domain W3C validator