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Theorem mbfi1fseq 19092
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 5882 . . . . . 6  |-  ( j  =  k  ->  (
2 ^ j )  =  ( 2 ^ k ) )
43oveq2d 5890 . . . . 5  |-  ( j  =  k  ->  (
( F `  z
)  x.  ( 2 ^ j ) )  =  ( ( F `
 z )  x.  ( 2 ^ k
) ) )
54fveq2d 5545 . . . 4  |-  ( j  =  k  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  =  ( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) ) )
65, 3oveq12d 5892 . . 3  |-  ( j  =  k  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) )  =  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
7 fveq2 5541 . . . . . 6  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
87oveq1d 5889 . . . . 5  |-  ( z  =  y  ->  (
( F `  z
)  x.  ( 2 ^ k ) )  =  ( ( F `
 y )  x.  ( 2 ^ k
) ) )
98fveq2d 5545 . . . 4  |-  ( z  =  y  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  =  ( |_ `  (
( F `  y
)  x.  ( 2 ^ k ) ) ) )
109oveq1d 5889 . . 3  |-  ( z  =  y  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) )  /  ( 2 ^ k ) )  =  ( ( |_
`  ( ( F `
 y )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
116, 10cbvmpt2v 5942 . 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 2356 . . . . . 6  |-  ( y  =  x  ->  (
y  e.  ( -u m [,] m )  <->  x  e.  ( -u m [,] m
) ) )
13 oveq2 5882 . . . . . . . 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 4049 . . . . . . 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 2297 . . . . . . 7  |-  ( y  =  x  ->  m  =  m )
1614, 13, 15ifbieq12d 3600 . . . . . 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 2297 . . . . . 6  |-  ( y  =  x  ->  0  =  0 )
1812, 16, 17ifbieq12d 3600 . . . . 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 4127 . . . 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 9060 . . . . . . . 8  |-  ( m  =  k  ->  -u m  =  -u k )
21 id 19 . . . . . . . 8  |-  ( m  =  k  ->  m  =  k )
2220, 21oveq12d 5892 . . . . . . 7  |-  ( m  =  k  ->  ( -u m [,] m )  =  ( -u k [,] k ) )
2322eleq2d 2363 . . . . . 6  |-  ( m  =  k  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u k [,] k
) ) )
24 oveq1 5881 . . . . . . . 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 4052 . . . . . . 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 3600 . . . . . 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 2297 . . . . . 6  |-  ( m  =  k  ->  0  =  0 )
2823, 26, 27ifbieq12d 3600 . . . . 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 4123 . . . 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 2340 . . 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 4127 . 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 19091 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 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   ifcif 3578   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    o Rcofr 6093   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880    <_ cle 8884   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   [,)cico 10674   [,]cicc 10675   |_cfl 10940   ^cexp 11120    ~~> cli 11974  MblFncmbf 18985   S.1citg1 18986   0 pc0p 19040
This theorem is referenced by:  mbfi1flimlem  19093  itg2add  19130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-rest 13343  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cmp 17130  df-ovol 18840  df-vol 18841  df-mbf 18991  df-itg1 18992  df-0p 19041
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