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Theorem fourierdlem69 31847
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 31780 . . . . . . . . . . 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 463 . . . . . . . 8  |-  ( ph  ->  ( ( ( Q `
 0 )  =  A  /\  ( Q `
 M )  =  B )  /\  A. i  e.  ( 0..^ M ) ( Q `
 i )  < 
( Q `  (
i  +  1 ) ) ) )
98simpld 459 . . . . . . 7  |-  ( ph  ->  ( ( Q ` 
0 )  =  A  /\  ( Q `  M )  =  B ) )
109simpld 459 . . . . . 6  |-  ( ph  ->  ( Q `  0
)  =  A )
119simprd 463 . . . . . 6  |-  ( ph  ->  ( Q `  M
)  =  B )
1210, 11oveq12d 6299 . . . . 5  |-  ( ph  ->  ( ( Q ` 
0 ) [,] ( Q `  M )
)  =  ( A [,] B ) )
1312feq2d 5708 . . . 4  |-  ( ph  ->  ( F : ( ( Q `  0
) [,] ( Q `
 M ) ) --> CC  <->  F : ( A [,] B ) --> CC ) )
141, 13mpbird 232 . . 3  |-  ( ph  ->  F : ( ( Q `  0 ) [,] ( Q `  M ) ) --> CC )
1514feqmptd 5911 . 2  |-  ( ph  ->  F  =  ( x  e.  ( ( Q `
 0 ) [,] ( Q `  M
) )  |->  ( F `
 x ) ) )
16 nfv 1694 . . 3  |-  F/ x ph
17 0zd 10882 . . 3  |-  ( ph  ->  0  e.  ZZ )
18 nnuz 11125 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
19 1e0p1 11012 . . . . . 6  |-  1  =  ( 0  +  1 )
2019fveq2i 5859 . . . . 5  |-  ( ZZ>= ` 
1 )  =  (
ZZ>= `  ( 0  +  1 ) )
2118, 20eqtri 2472 . . . 4  |-  NN  =  ( ZZ>= `  ( 0  +  1 ) )
223, 21syl6eleq 2541 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= `  ( 0  +  1 ) ) )
237simpld 459 . . . . 5  |-  ( ph  ->  Q  e.  ( RR 
^m  ( 0 ... M ) ) )
24 elmapi 7442 . . . . 5  |-  ( Q  e.  ( RR  ^m  ( 0 ... M
) )  ->  Q : ( 0 ... M ) --> RR )
2523, 24syl 16 . . . 4  |-  ( ph  ->  Q : ( 0 ... M ) --> RR )
2625fnvinran 31343 . . 3  |-  ( (
ph  /\  i  e.  ( 0 ... M
) )  ->  ( Q `  i )  e.  RR )
278simprd 463 . . . 4  |-  ( ph  ->  A. i  e.  ( 0..^ M ) ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
2827r19.21bi 2812 . . 3  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  <  ( Q `  ( i  +  1 ) ) )
291adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  F : ( A [,] B ) --> CC )
30 simpr 461 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )
3110adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  ( Q `  0 )  =  A )
3211adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  ( Q `  M )  =  B )
3331, 32oveq12d 6299 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  (
( Q `  0
) [,] ( Q `
 M ) )  =  ( A [,] B ) )
3430, 33eleqtrd 2533 . . . 4  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  x  e.  ( A [,] B
) )
3529, 34ffvelrnd 6017 . . 3  |-  ( (
ph  /\  x  e.  ( ( Q ` 
0 ) [,] ( Q `  M )
) )  ->  ( F `  x )  e.  CC )
3625adantr 465 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> RR )
37 elfzofz 11822 . . . . . 6  |-  ( i  e.  ( 0..^ M )  ->  i  e.  ( 0 ... M
) )
3837adantl 466 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0 ... M ) )
3936, 38ffvelrnd 6017 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  i )  e.  RR )
40 fzofzp1 11888 . . . . . 6  |-  ( i  e.  ( 0..^ M )  ->  ( i  +  1 )  e.  ( 0 ... M
) )
4140adantl 466 . . . . 5  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( i  +  1 )  e.  ( 0 ... M ) )
4236, 41ffvelrnd 6017 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( Q `  ( i  +  1 ) )  e.  RR )
431adantr 465 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  F : ( A [,] B ) --> CC )
44 ioossicc 11619 . . . . . . . 8  |-  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  C_  ( ( Q `  i ) [,] ( Q `  ( i  +  1 ) ) )
454, 3, 2fourierdlem11 31789 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) )
4645simp1d 1009 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
4746rexrd 9646 . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR* )
4847adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  A  e.  RR* )
4945simp2d 1010 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
5049rexrd 9646 . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR* )
5150adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  B  e.  RR* )
524, 3, 2fourierdlem15 31793 . . . . . . . . . 10  |-  ( ph  ->  Q : ( 0 ... M ) --> ( A [,] B ) )
5352adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  Q : ( 0 ... M ) --> ( A [,] B
) )
54 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  i  e.  ( 0..^ M ) )
5548, 51, 53, 54fourierdlem8 31786 . . . . . . . 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 5912 . . . . . 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 2532 . . . . 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 6296 . . . . . 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 2533 . . . . 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 6296 . . . . . 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 2533 . . . . 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 31667 . . . 4  |-  ( (
ph  /\  i  e.  ( 0..^ M ) )  ->  ( x  e.  ( ( Q `  i ) (,) ( Q `  ( i  +  1 ) ) )  |->  ( F `  x ) )  e.  L^1 )
6743adantr 465 . . . . 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 6017 . . . 4  |-  ( ( ( ph  /\  i  e.  ( 0..^ M ) )  /\  x  e.  ( ( Q `  i ) [,] ( Q `  ( i  +  1 ) ) ) )  ->  ( F `  x )  e.  CC )
7039, 42, 66, 69ibliooicc 31660 . . 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 31662 . 2  |-  ( ph  ->  ( x  e.  ( ( Q `  0
) [,] ( Q `
 M ) ) 
|->  ( F `  x
) )  e.  L^1 )
7215, 71eqeltrd 2531 1  |-  ( ph  ->  F  e.  L^1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   {crab 2797   class class class wbr 4437    |-> cmpt 4495    |` cres 4991   -->wf 5574   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498   RR*cxr 9630    < clt 9631   NNcn 10542   ZZ>=cuz 11090   (,)cioo 11538   [,]cicc 11541   ...cfz 11681  ..^cfzo 11803   -cn->ccncf 21253   L^1cibl 21899   lim CC climc 22139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cc 8818  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-mulg 15934  df-cntz 16229  df-cmn 16674  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-cnfld 18295  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-cn 19601  df-cnp 19602  df-cmp 19760  df-tx 19936  df-hmeo 20129  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-ovol 21749  df-vol 21750  df-mbf 21901  df-itg1 21902  df-itg2 21903  df-ibl 21904  df-itg 21905  df-0p 21950  df-limc 22143
This theorem is referenced by:  fourierdlem84  31862  fourierdlem88  31866  fourierdlem100  31878  fourierdlem107  31885  fourierdlem111  31889  fourierdlem112  31890
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