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Theorem mbfi1fseq 19614
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  ( 0 p  o R  <_  ( g `  n )  /\  (
g `  n )  o R  <_  ( 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 6090 . . . . . 6  |-  ( j  =  k  ->  (
2 ^ j )  =  ( 2 ^ k ) )
43oveq2d 6098 . . . . 5  |-  ( j  =  k  ->  (
( F `  z
)  x.  ( 2 ^ j ) )  =  ( ( F `
 z )  x.  ( 2 ^ k
) ) )
54fveq2d 5733 . . . 4  |-  ( j  =  k  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  =  ( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) ) )
65, 3oveq12d 6100 . . 3  |-  ( j  =  k  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) )  =  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
7 fveq2 5729 . . . . . 6  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
87oveq1d 6097 . . . . 5  |-  ( z  =  y  ->  (
( F `  z
)  x.  ( 2 ^ k ) )  =  ( ( F `
 y )  x.  ( 2 ^ k
) ) )
98fveq2d 5733 . . . 4  |-  ( z  =  y  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  =  ( |_ `  (
( F `  y
)  x.  ( 2 ^ k ) ) ) )
109oveq1d 6097 . . 3  |-  ( z  =  y  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) )  /  ( 2 ^ k ) )  =  ( ( |_
`  ( ( F `
 y )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
116, 10cbvmpt2v 6153 . 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 2497 . . . . . 6  |-  ( y  =  x  ->  (
y  e.  ( -u m [,] m )  <->  x  e.  ( -u m [,] m
) ) )
13 oveq2 6090 . . . . . . . 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 4223 . . . . . . 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 )
)
15 eqidd 2438 . . . . . . 7  |-  ( y  =  x  ->  m  =  m )
1614, 13, 15ifbieq12d 3762 . . . . . 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 ) )
17 eqidd 2438 . . . . . 6  |-  ( y  =  x  ->  0  =  0 )
1812, 16, 17ifbieq12d 3762 . . . . 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 ) )
1918cbvmptv 4301 . . . 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 ) )
20 negeq 9299 . . . . . . . 8  |-  ( m  =  k  ->  -u m  =  -u k )
21 id 21 . . . . . . . 8  |-  ( m  =  k  ->  m  =  k )
2220, 21oveq12d 6100 . . . . . . 7  |-  ( m  =  k  ->  ( -u m [,] m )  =  ( -u k [,] k ) )
2322eleq2d 2504 . . . . . 6  |-  ( m  =  k  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u k [,] k
) ) )
24 oveq1 6089 . . . . . . . 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 ) )
2524, 21breq12d 4226 . . . . . . 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 )
)
2625, 24, 21ifbieq12d 3762 . . . . . 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 ) )
27 eqidd 2438 . . . . . 6  |-  ( m  =  k  ->  0  =  0 )
2823, 26, 27ifbieq12d 3762 . . . . 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 ) )
2928mpteq2dv 4297 . . . 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 ) ) )
3019, 29syl5eq 2481 . . 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 ) ) )
3130cbvmptv 4301 . 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 ) ) )
321, 2, 11, 31mbfi1fseqlem6 19613 1  |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
A. n  e.  NN  ( 0 p  o R  <_  ( g `  n )  /\  (
g `  n )  o R  <_  ( g `
 ( n  + 
1 ) ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `
 n ) `  x ) )  ~~>  ( F `
 x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    e. wcel 1726   A.wral 2706   ifcif 3740   class class class wbr 4213    e. cmpt 4267   dom cdm 4879   -->wf 5451   ` cfv 5455  (class class class)co 6082    e. cmpt2 6084    o Rcofr 6305   RRcr 8990   0cc0 8991   1c1 8992    + caddc 8994    x. cmul 8996    +oocpnf 9118    <_ cle 9122   -ucneg 9293    / cdiv 9678   NNcn 10001   2c2 10050   [,)cico 10919   [,]cicc 10920   |_cfl 11202   ^cexp 11383    ~~> cli 12279  MblFncmbf 19507   S.1citg1 19508   0 pc0p 19562
This theorem is referenced by:  mbfi1flimlem  19615  itg2add  19652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-ofr 6307  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-n0 10223  df-z 10284  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-fl 11203  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-rlim 12284  df-sum 12481  df-rest 13651  df-topgen 13668  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-top 16964  df-bases 16966  df-topon 16967  df-cmp 17451  df-ovol 19362  df-vol 19363  df-mbf 19513  df-itg1 19514  df-0p 19563
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