Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem69 Structured version   Unicode version

Theorem fourierdlem69 32197
Description: A piecewise continuous function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem69.p  |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  ( 0 ... m ) )  |  ( ( ( p `
 0 )  =  A  /\  ( p `
 m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )
fourierdlem69.m  |-  ( ph  ->  M  e.  NN )
fourierdlem69.q  |-  ( ph  ->  Q  e.  ( P `
 M ) )
fourierdlem69.f  |-  ( ph  ->  F : ( A [,] B ) --> CC )
fourierdlem69.fcn  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
fourierdlem69.r  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
fourierdlem69.l  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
Assertion
Ref Expression
fourierdlem69  |-  ( ph  ->  F  e.  L^1 )
Distinct variable groups:    A, i, m, p    B, i, m, p    i, F    i, M, m, p    Q, i, p    ph, i
Allowed substitution hints:    ph( m, p)    P( i, m, p)    Q( m)    R( i, m, p)    F( m, p)    L( i, m, p)

Proof of Theorem fourierdlem69
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fourierdlem69.f . . . 4  |-  ( ph  ->  F : ( A [,] B ) --> CC )
2 fourierdlem69.q . . . . . . . . . 10  |-  ( ph  ->  Q  e.  ( P `
 M ) )
3 fourierdlem69.m . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN )
4 fourierdlem69.p . . . . . . . . . . . 12  |-  P  =  ( m  e.  NN  |->  { p  e.  ( RR  ^m  ( 0 ... m ) )  |  ( ( ( p `
 0 )  =  A  /\  ( p `
 m )  =  B )  /\  A. i  e.  ( 0..^ m ) ( p `
 i )  < 
( p `  (
i  +  1 ) ) ) } )
54fourierdlem2 32130 . . . . . . . . . . 11  |-  ( M  e.  NN  ->  ( Q  e.  ( P `  M )  <->  ( Q  e.  ( RR  ^m  (
0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
63, 5syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( Q  e.  ( P `  M )  <-> 
( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) ) )
72, 6mpbid 210 . . . . . . . . 9  |-  ( ph  ->  ( Q  e.  ( RR  ^m  ( 0 ... M ) )  /\  ( ( ( Q `  0 )  =  A  /\  ( Q `  M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) ) )
87simprd 461 . . . . . . . 8  |-  ( ph  ->  ( ( ( Q `
 0 )  =  A  /\  ( Q `
 M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
98simpld 457 . . . . . . 7  |-  ( ph  ->  ( ( Q ` 
0 )  =  A  /\  ( Q `  M )  =  B ) )
109simpld 457 . . . . . 6  |-  ( ph  ->  ( Q `  0
)  =  A )
119simprd 461 . . . . . 6  |-  ( ph  ->  ( Q `  M
)  =  B )
1210, 11oveq12d 6288 . . . . 5  |-  ( ph  ->  ( ( Q ` 
0 ) [,] ( Q `  M )
)  =  ( A [,] B ) )
1312feq2d 5700 . . . 4  |-  ( ph  ->  ( F : ( ( Q `  0
) [,] ( Q `
 M ) ) --> CC  <->  F : ( A [,] B ) --> CC ) )
141, 13mpbird 232 . . 3  |-  ( ph  ->  F : ( ( Q `  0 ) [,] ( Q `  M ) ) --> CC )
1514feqmptd 5901 . 2  |-  ( ph  ->  F  =  ( x  e.  ( ( Q `
 0 ) [,] ( Q `  M
) )  |->  ( F `
 x ) ) )
16 nfv 1712 . . 3  |-  F/ x ph
17 0zd 10872 . . 3  |-  ( ph  ->  0  e.  ZZ )
18 nnuz 11117 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
19 1e0p1 11004 . . . . . 6  |-  1  =  ( 0  +  1 )
2019fveq2i 5851 . . . . 5  |-  ( ZZ>= ` 
1 )  =  (
ZZ>= `  ( 0  +  1 ) )
2118, 20eqtri 2483 . . . 4  |-  NN  =  ( ZZ>= `  ( 0  +  1 ) )
223, 21syl6eleq 2552 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= `  ( 0  +  1 ) ) )
237simpld 457 . . . . 5  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
24 elmapi 7433 . . . . 5  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  ->  Q : ( 0 ... M ) --> RR )
2523, 24syl 16 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
2625fnvinran 31629 . . 3  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
278simprd 461 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
2827r19.21bi 2823 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
291adantr 463 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  F : ( A [,] B ) --> CC )
30 simpr 459 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )
3110adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  ( Q `  0 )  =  A )
3211adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  ( Q `  M )  =  B )
3331, 32oveq12d 6288 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  (
( Q `  0
) [,] ( Q `
 M ) )  =  ( A [,] B ) )
3430, 33eleqtrd 2544 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  x  e.  ( A [,] B
) )
3529, 34ffvelrnd 6008 . . 3  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  ( F `  x )  e.  CC )
3625adantr 463 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> RR )
37 elfzofz 11819 . . . . . 6  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
3837adantl 464 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0 ... M ) )
3936, 38ffvelrnd 6008 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  e.  RR )
40 fzofzp1 11890 . . . . . 6  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
4140adantl 464 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  +  1 )  e.  ( 0 ... M ) )
4236, 41ffvelrnd 6008 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  ( i  +  1 ) )  e.  RR )
431adantr 463 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  F : ( A [,] B ) --> CC )
44 ioossicc 11613 . . . . . . . 8  |-  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  C_  ( ( Q `  i ) [,] ( Q `  ( i  +  1 ) ) )
454, 3, 2fourierdlem11 32139 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) )
4645simp1d 1006 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
4746rexrd 9632 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR* )
4847adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  A  e.  RR* )
4945simp2d 1007 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
5049rexrd 9632 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
5150adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  B  e.  RR* )
524, 3, 2fourierdlem15 32143 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> ( A [,] B ) )
5352adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( A [,] B
) )
54 simpr 459 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
5548, 51, 53, 54fourierdlem8 32136 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) [,] ( Q `  (
i  +  1 ) ) )  C_  ( A [,] B ) )
5644, 55syl5ss 3500 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) )  C_  ( A [,] B ) )
5743, 56feqresmpt 5902 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  =  ( x  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  |->  ( F `  x ) ) )
58 fourierdlem69.fcn . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) )  e.  ( ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) -cn-> CC ) )
5957, 58eqeltrrd 2543 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( x  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  |->  ( F `  x ) )  e.  ( ( ( Q `
 i ) (,) ( Q `  (
i  +  1 ) ) ) -cn-> CC ) )
60 fourierdlem69.l . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
6157oveq1d 6285 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  ( i  +  1 ) ) )  =  ( ( x  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  |->  ( F `  x ) ) lim CC  ( Q `
 ( i  +  1 ) ) ) )
6260, 61eleqtrd 2544 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  L  e.  ( ( x  e.  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) 
|->  ( F `  x
) ) lim CC  ( Q `  ( i  +  1 ) ) ) )
63 fourierdlem69.r . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( F  |`  (
( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) ) lim CC  ( Q `
 i ) ) )
6457oveq1d 6285 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( ( F  |`  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) ) ) lim CC  ( Q `  i )
)  =  ( ( x  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  |->  ( F `  x ) ) lim CC  ( Q `
 i ) ) )
6563, 64eleqtrd 2544 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  R  e.  ( ( x  e.  ( ( Q `  i
) (,) ( Q `
 ( i  +  1 ) ) ) 
|->  ( F `  x
) ) lim CC  ( Q `  i )
) )
6639, 42, 59, 62, 65iblcncfioo 32016 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( x  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  |->  ( F `  x ) )  e.  L^1 )
6743adantr 463 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( Q `  i ) [,] ( Q `  ( i  +  1 ) ) ) )  ->  F : ( A [,] B ) --> CC )
6855sselda 3489 . . . . 5  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( Q `  i ) [,] ( Q `  ( i  +  1 ) ) ) )  ->  x  e.  ( A [,] B
) )
6967, 68ffvelrnd 6008 . . . 4  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( Q `  i ) [,] ( Q `  ( i  +  1 ) ) ) )  ->  ( F `  x )  e.  CC )
7039, 42, 66, 69ibliooicc 32009 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( x  e.  ( ( Q `  i ) [,] ( Q `  ( i  +  1 ) ) )  |->  ( F `  x ) )  e.  L^1 )
7116, 17, 22, 26, 28, 35, 70iblspltprt 32011 . 2  |-  ( ph  ->  ( x  e.  ( ( Q `  0
) [,] ( Q `
 M ) ) 
|->  ( F `  x
) )  e.  L^1 )
7215, 71eqeltrd 2542 1  |-  ( ph  ->  F  e.  L^1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   class class class wbr 4439    |-> cmpt 4497    |` cres 4990   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484   RR*cxr 9616    < clt 9617   NNcn 10531   ZZ>=cuz 11082   (,)cioo 11532   [,]cicc 11535   ...cfz 11675  ..^cfzo 11799   -cn->ccncf 21546   L^1cibl 22192   lim CC climc 22432
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-limsup 13376  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-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-cn 19895  df-cnp 19896  df-cmp 20054  df-tx 20229  df-hmeo 20422  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-limc 22436
This theorem is referenced by:  fourierdlem84  32212  fourierdlem88  32216  fourierdlem100  32228  fourierdlem107  32235  fourierdlem111  32239  fourierdlem112  32240
  Copyright terms: Public domain W3C validator