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Theorem i1fres 21847
Description: The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside  A.) (Contributed by Mario Carneiro, 29-Jun-2014.)
Hypothesis
Ref Expression
i1fres.1  |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )
Assertion
Ref Expression
i1fres  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    G( x)

Proof of Theorem i1fres
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 i1ff 21818 . . . . . . . 8  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
21adantr 465 . . . . . . 7  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F : RR --> RR )
3 ffn 5729 . . . . . . 7  |-  ( F : RR --> RR  ->  F  Fn  RR )
42, 3syl 16 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  F  Fn  RR )
5 fnfvelrn 6016 . . . . . 6  |-  ( ( F  Fn  RR  /\  x  e.  RR )  ->  ( F `  x
)  e.  ran  F
)
64, 5sylan 471 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  ( F `  x )  e.  ran  F )
7 i1f0rn 21824 . . . . . 6  |-  ( F  e.  dom  S.1  ->  0  e.  ran  F )
87ad2antrr 725 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  0  e.  ran  F )
9 ifcl 3981 . . . . 5  |-  ( ( ( F `  x
)  e.  ran  F  /\  0  e.  ran  F )  ->  if (
x  e.  A , 
( F `  x
) ,  0 )  e.  ran  F )
106, 8, 9syl2anc 661 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  x  e.  RR )  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  e.  ran  F
)
11 i1fres.1 . . . 4  |-  G  =  ( x  e.  RR  |->  if ( x  e.  A ,  ( F `  x ) ,  0 ) )
1210, 11fmptd 6043 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> ran  F )
13 frn 5735 . . . 4  |-  ( F : RR --> RR  ->  ran 
F  C_  RR )
142, 13syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  C_  RR )
15 fss 5737 . . 3  |-  ( ( G : RR --> ran  F  /\  ran  F  C_  RR )  ->  G : RR --> RR )
1612, 14, 15syl2anc 661 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G : RR --> RR )
17 i1frn 21819 . . . 4  |-  ( F  e.  dom  S.1  ->  ran 
F  e.  Fin )
1817adantr 465 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  F  e.  Fin )
19 frn 5735 . . . 4  |-  ( G : RR --> ran  F  ->  ran  G  C_  ran  F )
2012, 19syl 16 . . 3  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  C_  ran  F )
21 ssfi 7737 . . 3  |-  ( ( ran  F  e.  Fin  /\ 
ran  G  C_  ran  F
)  ->  ran  G  e. 
Fin )
2218, 20, 21syl2anc 661 . 2  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  ran  G  e.  Fin )
23 eleq1 2539 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  (
x  e.  A  <->  z  e.  A ) )
24 fveq2 5864 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
2523, 24ifbieq1d 3962 . . . . . . . . . . . . 13  |-  ( x  =  z  ->  if ( x  e.  A ,  ( F `  x ) ,  0 )  =  if ( z  e.  A , 
( F `  z
) ,  0 ) )
26 fvex 5874 . . . . . . . . . . . . . 14  |-  ( F `
 z )  e. 
_V
27 c0ex 9586 . . . . . . . . . . . . . 14  |-  0  e.  _V
2826, 27ifex 4008 . . . . . . . . . . . . 13  |-  if ( z  e.  A , 
( F `  z
) ,  0 )  e.  _V
2925, 11, 28fvmpt 5948 . . . . . . . . . . . 12  |-  ( z  e.  RR  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3029adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( G `  z )  =  if ( z  e.  A ,  ( F `
 z ) ,  0 ) )
3130eqeq1d 2469 . . . . . . . . . 10  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  if (
z  e.  A , 
( F `  z
) ,  0 )  =  y ) )
32 eldifsni 4153 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  ->  y  =/=  0
)
3332ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  y  =/=  0 )
3433necomd 2738 . . . . . . . . . . . . 13  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  0  =/=  y )
35 iffalse 3948 . . . . . . . . . . . . . 14  |-  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  0 )
3635neeq1d 2744 . . . . . . . . . . . . 13  |-  ( -.  z  e.  A  -> 
( if ( z  e.  A ,  ( F `  z ) ,  0 )  =/=  y  <->  0  =/=  y
) )
3734, 36syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( -.  z  e.  A  ->  if ( z  e.  A ,  ( F `
 z ) ,  0 )  =/=  y
) )
3837necon4bd 2689 . . . . . . . . . . 11  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  -> 
z  e.  A ) )
3938pm4.71rd 635 . . . . . . . . . 10  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( z  e.  A  /\  if ( z  e.  A , 
( F `  z
) ,  0 )  =  y ) ) )
4031, 39bitrd 253 . . . . . . . . 9  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  ( z  e.  A  /\  if ( z  e.  A , 
( F `  z
) ,  0 )  =  y ) ) )
41 iftrue 3945 . . . . . . . . . . 11  |-  ( z  e.  A  ->  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  ( F `
 z ) )
4241eqeq1d 2469 . . . . . . . . . 10  |-  ( z  e.  A  ->  ( if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y  <->  ( F `  z )  =  y ) )
4342pm5.32i 637 . . . . . . . . 9  |-  ( ( z  e.  A  /\  if ( z  e.  A ,  ( F `  z ) ,  0 )  =  y )  <-> 
( z  e.  A  /\  ( F `  z
)  =  y ) )
4440, 43syl6bb 261 . . . . . . . 8  |-  ( ( ( ( F  e. 
dom  S.1  /\  A  e. 
dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( G `  z
)  =  y  <->  ( z  e.  A  /\  ( F `  z )  =  y ) ) )
4544pm5.32da 641 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  RR  /\  ( G `  z )  =  y )  <->  ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z )  =  y ) ) ) )
46 an12 795 . . . . . . 7  |-  ( ( z  e.  RR  /\  ( z  e.  A  /\  ( F `  z
)  =  y ) )  <->  ( z  e.  A  /\  ( z  e.  RR  /\  ( F `  z )  =  y ) ) )
4745, 46syl6bb 261 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  RR  /\  ( G `  z )  =  y )  <->  ( z  e.  A  /\  (
z  e.  RR  /\  ( F `  z )  =  y ) ) ) )
48 ffn 5729 . . . . . . . . 9  |-  ( G : RR --> ran  F  ->  G  Fn  RR )
4912, 48syl 16 . . . . . . . 8  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  Fn  RR )
5049adantr 465 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  G  Fn  RR )
51 fniniseg 6000 . . . . . . 7  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
5250, 51syl 16 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
534adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  F  Fn  RR )
54 fniniseg 6000 . . . . . . . 8  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { y } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  y ) ) )
5553, 54syl 16 . . . . . . 7  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' F " { y } )  <-> 
( z  e.  RR  /\  ( F `  z
)  =  y ) ) )
5655anbi2d 703 . . . . . 6  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( (
z  e.  A  /\  z  e.  ( `' F " { y } ) )  <->  ( z  e.  A  /\  (
z  e.  RR  /\  ( F `  z )  =  y ) ) ) )
5747, 52, 563bitr4d 285 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
( z  e.  A  /\  z  e.  ( `' F " { y } ) ) ) )
58 elin 3687 . . . . 5  |-  ( z  e.  ( A  i^i  ( `' F " { y } ) )  <->  ( z  e.  A  /\  z  e.  ( `' F " { y } ) ) )
5957, 58syl6bbr 263 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( z  e.  ( `' G " { y } )  <-> 
z  e.  ( A  i^i  ( `' F " { y } ) ) ) )
6059eqrdv 2464 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  =  ( A  i^i  ( `' F " { y } ) ) )
61 simplr 754 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  A  e.  dom  vol )
62 i1fima 21820 . . . . 5  |-  ( F  e.  dom  S.1  ->  ( `' F " { y } )  e.  dom  vol )
6362ad2antrr 725 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  e.  dom  vol )
64 inmbl 21687 . . . 4  |-  ( ( A  e.  dom  vol  /\  ( `' F " { y } )  e.  dom  vol )  ->  ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol )
6561, 63, 64syl2anc 661 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( A  i^i  ( `' F " { y } ) )  e.  dom  vol )
6660, 65eqeltrd 2555 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' G " { y } )  e.  dom  vol )
6760fveq2d 5868 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  =  ( vol `  ( A  i^i  ( `' F " { y } ) ) ) )
68 mblvol 21676 . . . . 5  |-  ( ( A  i^i  ( `' F " { y } ) )  e. 
dom  vol  ->  ( vol `  ( A  i^i  ( `' F " { y } ) ) )  =  ( vol* `  ( A  i^i  ( `' F " { y } ) ) ) )
6965, 68syl 16 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( A  i^i  ( `' F " { y } ) ) )  =  ( vol* `  ( A  i^i  ( `' F " { y } ) ) ) )
7067, 69eqtrd 2508 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  =  ( vol* `  ( A  i^i  ( `' F " { y } ) ) ) )
71 inss2 3719 . . . . 5  |-  ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } )
7271a1i 11 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } ) )
73 mblss 21677 . . . . 5  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( `' F " { y } ) 
C_  RR )
7463, 73syl 16 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( `' F " { y } )  C_  RR )
75 mblvol 21676 . . . . . 6  |-  ( ( `' F " { y } )  e.  dom  vol 
->  ( vol `  ( `' F " { y } ) )  =  ( vol* `  ( `' F " { y } ) ) )
7663, 75syl 16 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  =  ( vol* `  ( `' F " { y } ) ) )
77 i1fima2sn 21822 . . . . . 6  |-  ( ( F  e.  dom  S.1  /\  y  e.  ( ran 
G  \  { 0 } ) )  -> 
( vol `  ( `' F " { y } ) )  e.  RR )
7877adantlr 714 . . . . 5  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' F " { y } ) )  e.  RR )
7976, 78eqeltrrd 2556 . . . 4  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol* `  ( `' F " { y } ) )  e.  RR )
80 ovolsscl 21632 . . . 4  |-  ( ( ( A  i^i  ( `' F " { y } ) )  C_  ( `' F " { y } )  /\  ( `' F " { y } )  C_  RR  /\  ( vol* `  ( `' F " { y } ) )  e.  RR )  ->  ( vol* `  ( A  i^i  ( `' F " { y } ) ) )  e.  RR )
8172, 74, 79, 80syl3anc 1228 . . 3  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol* `  ( A  i^i  ( `' F " { y } ) ) )  e.  RR )
8270, 81eqeltrd 2555 . 2  |-  ( ( ( F  e.  dom  S.1 
/\  A  e.  dom  vol )  /\  y  e.  ( ran  G  \  { 0 } ) )  ->  ( vol `  ( `' G " { y } ) )  e.  RR )
8316, 22, 66, 82i1fd 21823 1  |-  ( ( F  e.  dom  S.1  /\  A  e.  dom  vol )  ->  G  e.  dom  S.1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    i^i cin 3475    C_ wss 3476   ifcif 3939   {csn 4027    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    Fn wfn 5581   -->wf 5582   ` cfv 5586   Fincfn 7513   RRcr 9487   0cc0 9488   vol*covol 21609   volcvol 21610   S.1citg1 21759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-sum 13468  df-rest 14674  df-topgen 14695  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-top 19166  df-bases 19168  df-topon 19169  df-cmp 19653  df-ovol 21611  df-vol 21612  df-mbf 21763  df-itg1 21764
This theorem is referenced by:  i1fpos  21848  itg1climres  21856  itg2uba  21885  itg2splitlem  21890  itg2monolem1  21892  ftc1anclem5  29671  ftc1anclem7  29673
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