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Theorem mbfi1fseq 21333
Description: A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function  G and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1  |-  ( ph  ->  F  e. MblFn )
mbfi1fseq.2  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
Assertion
Ref Expression
mbfi1fseq  |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
A. n  e.  NN  ( 0p  oR  <_  ( g `  n )  /\  (
g `  n )  oR  <_  ( g `
 ( n  + 
1 ) ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `
 n ) `  x ) )  ~~>  ( F `
 x ) ) )
Distinct variable groups:    g, n, x, F    ph, n, x
Allowed substitution hint:    ph( g)

Proof of Theorem mbfi1fseq
Dummy variables  j 
k  m  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfi1fseq.1 . 2  |-  ( ph  ->  F  e. MblFn )
2 mbfi1fseq.2 . 2  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
3 oveq2 6209 . . . . . 6  |-  ( j  =  k  ->  (
2 ^ j )  =  ( 2 ^ k ) )
43oveq2d 6217 . . . . 5  |-  ( j  =  k  ->  (
( F `  z
)  x.  ( 2 ^ j ) )  =  ( ( F `
 z )  x.  ( 2 ^ k
) ) )
54fveq2d 5804 . . . 4  |-  ( j  =  k  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  =  ( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) ) )
65, 3oveq12d 6219 . . 3  |-  ( j  =  k  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) )  =  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
7 fveq2 5800 . . . . . 6  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
87oveq1d 6216 . . . . 5  |-  ( z  =  y  ->  (
( F `  z
)  x.  ( 2 ^ k ) )  =  ( ( F `
 y )  x.  ( 2 ^ k
) ) )
98fveq2d 5804 . . . 4  |-  ( z  =  y  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  =  ( |_ `  (
( F `  y
)  x.  ( 2 ^ k ) ) ) )
109oveq1d 6216 . . 3  |-  ( z  =  y  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) )  /  ( 2 ^ k ) )  =  ( ( |_
`  ( ( F `
 y )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
116, 10cbvmpt2v 6276 . 2  |-  ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) )  =  ( k  e.  NN ,  y  e.  RR  |->  ( ( |_
`  ( ( F `
 y )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
12 eleq1 2526 . . . . . 6  |-  ( y  =  x  ->  (
y  e.  ( -u m [,] m )  <->  x  e.  ( -u m [,] m
) ) )
13 oveq2 6209 . . . . . . . 8  |-  ( y  =  x  ->  (
m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) y )  =  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) )
1413breq1d 4411 . . . . . . 7  |-  ( y  =  x  ->  (
( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m  <->  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  <_  m )
)
1514, 13ifbieq1d 3921 . . . . . 6  |-  ( y  =  x  ->  if ( ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m ,  ( m
( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) y ) ,  m )  =  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  m ) )
1612, 15ifbieq1d 3921 . . . . 5  |-  ( y  =  x  ->  if ( y  e.  (
-u m [,] m
) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y ) ,  m ) ,  0 )  =  if ( x  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  m ) ,  0 ) )
1716cbvmptv 4492 . . . 4  |-  ( y  e.  RR  |->  if ( y  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) y ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m ,  ( m
( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) ,  m ) ,  0 ) )
18 negeq 9714 . . . . . . . 8  |-  ( m  =  k  ->  -u m  =  -u k )
19 id 22 . . . . . . . 8  |-  ( m  =  k  ->  m  =  k )
2018, 19oveq12d 6219 . . . . . . 7  |-  ( m  =  k  ->  ( -u m [,] m )  =  ( -u k [,] k ) )
2120eleq2d 2524 . . . . . 6  |-  ( m  =  k  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u k [,] k
) ) )
22 oveq1 6208 . . . . . . . 8  |-  ( m  =  k  ->  (
m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  =  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) )
2322, 19breq12d 4414 . . . . . . 7  |-  ( m  =  k  ->  (
( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m  <->  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  <_  k )
)
2423, 22, 19ifbieq12d 3925 . . . . . 6  |-  ( m  =  k  ->  if ( ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m ,  ( m
( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) ,  m )  =  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  k ) )
2521, 24ifbieq1d 3921 . . . . 5  |-  ( m  =  k  ->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  <_  m , 
( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  m
) ,  0 )  =  if ( x  e.  ( -u k [,] k ) ,  if ( ( k ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_ 
k ,  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) ,  k ) ,  0 ) )
2625mpteq2dv 4488 . . . 4  |-  ( m  =  k  ->  (
x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m ,  ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  (
-u k [,] k
) ,  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  k ) ,  0 ) ) )
2717, 26syl5eq 2507 . . 3  |-  ( m  =  k  ->  (
y  e.  RR  |->  if ( y  e.  (
-u m [,] m
) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  (
-u k [,] k
) ,  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  k ) ,  0 ) ) )
2827cbvmptv 4492 . 2  |-  ( m  e.  NN  |->  ( y  e.  RR  |->  if ( y  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) y ) ,  m ) ,  0 ) ) )  =  ( k  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u k [,] k ) ,  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  k ) ,  0 ) ) )
291, 2, 11, 28mbfi1fseqlem6 21332 1  |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
A. n  e.  NN  ( 0p  oR  <_  ( g `  n )  /\  (
g `  n )  oR  <_  ( g `
 ( n  + 
1 ) ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `
 n ) `  x ) )  ~~>  ( F `
 x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   E.wex 1587    e. wcel 1758   A.wral 2799   ifcif 3900   class class class wbr 4401    |-> cmpt 4459   dom cdm 4949   -->wf 5523   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203    oRcofr 6430   RRcr 9393   0cc0 9394   1c1 9395    + caddc 9397    x. cmul 9399   +oocpnf 9527    <_ cle 9531   -ucneg 9708    / cdiv 10105   NNcn 10434   2c2 10483   [,)cico 11414   [,]cicc 11415   |_cfl 11758   ^cexp 11983    ~~> cli 13081  MblFncmbf 21228   S.1citg1 21229   0pc0p 21281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-ofr 6432  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fi 7773  df-sup 7803  df-oi 7836  df-card 8221  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-q 11066  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-ioo 11416  df-ico 11418  df-icc 11419  df-fz 11556  df-fzo 11667  df-fl 11760  df-seq 11925  df-exp 11984  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-rlim 13086  df-sum 13283  df-rest 14481  df-topgen 14502  df-psmet 17935  df-xmet 17936  df-met 17937  df-bl 17938  df-mopn 17939  df-top 18636  df-bases 18638  df-topon 18639  df-cmp 19123  df-ovol 21081  df-vol 21082  df-mbf 21233  df-itg1 21234  df-0p 21282
This theorem is referenced by:  mbfi1flimlem  21334  itg2add  21371
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