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Theorem ftc2ditglem 22612
Description: Lemma for ftc2ditg 22613. (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 459 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  A  <_  B )
21ditgpos 22426 . 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 11609 . . . . . . 7  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
63, 4, 5syl2anc 659 . . . . . 6  |-  ( ph  ->  ( X [,] Y
)  C_  RR )
7 ftc2ditg.a . . . . . 6  |-  ( ph  ->  A  e.  ( X [,] Y ) )
86, 7sseldd 3490 . . . . 5  |-  ( ph  ->  A  e.  RR )
98adantr 463 . . . 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 463 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR )
13 ax-resscn 9538 . . . . . . . 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 21563 . . . . . . . . 9  |-  ( F  e.  ( ( X [,] Y ) -cn-> CC )  ->  F :
( X [,] Y
) --> CC )
1715, 16syl 16 . . . . . . . 8  |-  ( ph  ->  F : ( X [,] Y ) --> CC )
1817adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  F :
( X [,] Y
) --> CC )
196adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( X [,] Y )  C_  RR )
20 iccssre 11609 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
218, 11, 20syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( A [,] B
)  C_  RR )
2221adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( A [,] B )  C_  RR )
23 eqid 2454 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
2423tgioo2 21474 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
2523, 24dvres 22481 . . . . . . 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 1227 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
27 iccntr 21492 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
288, 11, 27syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
2928adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) )  =  ( A (,) B ) )
3029reseq2d 5262 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) ) )
3126, 30eqtrd 2495 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( ( RR  _D  F )  |`  ( A (,) B
) ) )
323rexrd 9632 . . . . . . . . 9  |-  ( ph  ->  X  e.  RR* )
33 elicc2 11592 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( A  e.  ( X [,] Y )  <-> 
( A  e.  RR  /\  X  <_  A  /\  A  <_  Y ) ) )
343, 4, 33syl2anc 659 . . . . . . . . . . 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 1007 . . . . . . . . 9  |-  ( ph  ->  X  <_  A )
37 iooss1 11567 . . . . . . . . 9  |-  ( ( X  e.  RR*  /\  X  <_  A )  ->  ( A (,) B )  C_  ( X (,) B ) )
3832, 36, 37syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) B ) )
394rexrd 9632 . . . . . . . . 9  |-  ( ph  ->  Y  e.  RR* )
40 elicc2 11592 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( B  e.  ( X [,] Y )  <-> 
( B  e.  RR  /\  X  <_  B  /\  B  <_  Y ) ) )
413, 4, 40syl2anc 659 . . . . . . . . . . 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 1008 . . . . . . . . 9  |-  ( ph  ->  B  <_  Y )
44 iooss2 11568 . . . . . . . . 9  |-  ( ( Y  e.  RR*  /\  B  <_  Y )  ->  ( X (,) B )  C_  ( X (,) Y ) )
4539, 43, 44syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( X (,) B
)  C_  ( X (,) Y ) )
4638, 45sstrd 3499 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  ( X (,) Y ) )
4746adantr 463 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  C_  ( X (,) Y ) )
48 ftc2ditg.c . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  e.  ( ( X (,) Y )
-cn-> CC ) )
4948adantr 463 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  ( ( X (,) Y
) -cn-> CC ) )
50 rescncf 21567 . . . . . 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 2542 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  ( ( A (,) B )
-cn-> CC ) )
53 cncff 21563 . . . . . . . . . . 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 5901 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F ) `
 t ) ) )
5655adantr 463 . . . . . . . 8  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  =  ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
) )
5756reseq1d 5261 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( ( t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) ) )
5847resmptd 5313 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( (
t  e.  ( X (,) Y )  |->  ( ( RR  _D  F
) `  t )
)  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B ) 
|->  ( ( RR  _D  F ) `  t
) ) )
5957, 58eqtrd 2495 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
6031, 59eqtrd 2495 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  =  ( t  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 t ) ) )
61 ioombl 22141 . . . . . . 7  |-  ( A (,) B )  e. 
dom  vol
6261a1i 11 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( A (,) B )  e.  dom  vol )
63 fvex 5858 . . . . . . 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 463 . . . . . . 7  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  F )  e.  L^1 )
6756, 66eqeltrrd 2543 . . . . . 6  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( X (,) Y
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
6847, 62, 64, 67iblss 22377 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( t  e.  ( A (,) B
)  |->  ( ( RR 
_D  F ) `  t ) )  e.  L^1 )
6960, 68eqeltrd 2542 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( RR  _D  ( F  |`  ( A [,] B ) ) )  e.  L^1 )
70 iccss2 11598 . . . . . . 7  |-  ( ( A  e.  ( X [,] Y )  /\  B  e.  ( X [,] Y ) )  -> 
( A [,] B
)  C_  ( X [,] Y ) )
717, 10, 70syl2anc 659 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  ( X [,] Y ) )
72 rescncf 21567 . . . . . 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 463 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) )
759, 12, 1, 52, 69, 74ftc2 22611 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  _d t  =  ( ( ( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) ) )
7631fveq1d 5850 . . . . 5  |-  ( (
ph  /\  A  <_  B )  ->  ( ( RR  _D  ( F  |`  ( A [,] B ) ) ) `  t
)  =  ( ( ( RR  _D  F
)  |`  ( A (,) B ) ) `  t ) )
77 fvres 5862 . . . . 5  |-  ( t  e.  ( A (,) B )  ->  (
( ( RR  _D  F )  |`  ( A (,) B ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
7876, 77sylan9eq 2515 . . . 4  |-  ( ( ( ph  /\  A  <_  B )  /\  t  e.  ( A (,) B
) )  ->  (
( RR  _D  ( F  |`  ( A [,] B ) ) ) `
 t )  =  ( ( RR  _D  F ) `  t
) )
7978itgeq2dv 22354 . . 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 9632 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  A  e.  RR* )
8112rexrd 9632 . . . 4  |-  ( (
ph  /\  A  <_  B )  ->  B  e.  RR* )
82 ubicc2 11640 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
83 lbicc2 11639 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
84 fvres 5862 . . . . . 6  |-  ( B  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 B )  =  ( F `  B
) )
85 fvres 5862 . . . . . 6  |-  ( A  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 A )  =  ( F `  A
) )
8684, 85oveqan12d 6289 . . . . 5  |-  ( ( B  e.  ( A [,] B )  /\  A  e.  ( A [,] B ) )  -> 
( ( ( F  |`  ( A [,] B
) ) `  B
)  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A )
) )
8782, 83, 86syl2anc 659 . . . 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 1226 . . 3  |-  ( (
ph  /\  A  <_  B )  ->  ( (
( F  |`  ( A [,] B ) ) `
 B )  -  ( ( F  |`  ( A [,] B ) ) `  A ) )  =  ( ( F `  B )  -  ( F `  A ) ) )
8975, 79, 883eqtr3d 2503 . 2  |-  ( (
ph  /\  A  <_  B )  ->  S. ( A (,) B ) ( ( RR  _D  F
) `  t )  _d t  =  (
( F `  B
)  -  ( F `
 A ) ) )
902, 89eqtrd 2495 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497   dom cdm 4988   ran crn 4989    |` cres 4990   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   RR*cxr 9616    <_ cle 9618    - cmin 9796   (,)cioo 11532   [,]cicc 11535   TopOpenctopn 14911   topGenctg 14927  ℂfldccnfld 18615   intcnt 19685   -cn->ccncf 21546   volcvol 22041   L^1cibl 22192   S.citg 22193   S__cdit 22416    _D cdv 22433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cc 8806  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-ofr 6514  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-cmp 20054  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-ovol 22042  df-vol 22043  df-mbf 22194  df-itg1 22195  df-itg2 22196  df-ibl 22197  df-itg 22198  df-0p 22243  df-ditg 22417  df-limc 22436  df-dv 22437
This theorem is referenced by:  ftc2ditg  22613
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