Theorem fourierdlem69 40392

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.

Step	Hyp	Ref	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 ) ) ) } )
5	4	fourierdlem2 40326	. . . . . . . . . . 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 ) ) ) ) ) )
6	3, 5	syl 17	. . . . . . . . . 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 ) ) ) ) ) )
7	2, 6	mpbid 222	. . . . . . . . 9 |- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) )
8	7	simprd 479	. . . . . . . 8 |- ( ph -> ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) )
9	8	simpld 475	. . . . . . 7 |- ( ph -> ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) )
10	9	simpld 475	. . . . . 6 |- ( ph -> ( Q ` 0 ) = A )
11	9	simprd 479	. . . . . 6 |- ( ph -> ( Q ` M ) = B )
12	10, 11	oveq12d 6668	. . . . 5 |- ( ph -> ( ( Q ` 0 ) [,] ( Q ` M ) ) = ( A [,] B ) )
13	12	feq2d 6031	. . . 4 |- ( ph -> ( F : ( ( Q ` 0 ) [,] ( Q ` M ) ) --> CC <-> F : ( A [,] B ) --> CC ) )
14	1, 13	mpbird 247	. . 3 |- ( ph -> F : ( ( Q ` 0 ) [,] ( Q ` M ) ) --> CC )
15	14	feqmptd 6249	. 2 |- ( ph -> F = ( x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) |-> ( F ` x ) ) )
16		nfv 1843	. . 3 |- F/ x ph
17		0zd 11389	. . 3 |- ( ph -> 0 e. ZZ )
18		nnuz 11723	. . . . 5 |- NN = ( ZZ>= ` 1 )
19		1e0p1 11552	. . . . . 6 |- 1 = ( 0 + 1 )
20	19	fveq2i 6194	. . . . 5 |- ( ZZ>= ` 1 ) = ( ZZ>= ` ( 0 + 1 ) )
21	18, 20	eqtri 2644	. . . 4 |- NN = ( ZZ>= ` ( 0 + 1 ) )
22	3, 21	syl6eleq 2711	. . 3 |- ( ph -> M e. ( ZZ>= ` ( 0 + 1 ) ) )
23	7	simpld 475	. . . . 5 |- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) )
24		elmapi 7879	. . . . 5 |- ( Q e. ( RR ^m ( 0 ... M ) ) -> Q : ( 0 ... M ) --> RR )
25	23, 24	syl 17	. . . 4 |- ( ph -> Q : ( 0 ... M ) --> RR )
26	25	ffvelrnda 6359	. . 3 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( Q ` i ) e. RR )
27	8	simprd 479	. . . 4 |- ( ph -> A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) )
28	27	r19.21bi 2932	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) )
29	1	adantr 481	. . . 4 |- ( ( ph /\ x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> F : ( A [,] B ) --> CC )
30		simpr 477	. . . . 5 |- ( ( ph /\ x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) )
31	10	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> ( Q ` 0 ) = A )
32	11	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> ( Q ` M ) = B )
33	31, 32	oveq12d 6668	. . . . 5 |- ( ( ph /\ x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> ( ( Q ` 0 ) [,] ( Q ` M ) ) = ( A [,] B ) )
34	30, 33	eleqtrd 2703	. . . 4 |- ( ( ph /\ x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> x e. ( A [,] B ) )
35	29, 34	ffvelrnd 6360	. . 3 |- ( ( ph /\ x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) ) -> ( F ` x ) e. CC )
36	25	adantr 481	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q : ( 0 ... M ) --> RR )
37		elfzofz 12485	. . . . . 6 |- ( i e. ( 0..^ M ) -> i e. ( 0 ... M ) )
38	37	adantl 482	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i e. ( 0 ... M ) )
39	36, 38	ffvelrnd 6360	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) e. RR )
40		fzofzp1 12565	. . . . . 6 |- ( i e. ( 0..^ M ) -> ( i + 1 ) e. ( 0 ... M ) )
41	40	adantl 482	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( i + 1 ) e. ( 0 ... M ) )
42	36, 41	ffvelrnd 6360	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` ( i + 1 ) ) e. RR )
43	1	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> F : ( A [,] B ) --> CC )
44		ioossicc 12259	. . . . . . . 8 |- ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) )
45	4, 3, 2	fourierdlem11 40335	. . . . . . . . . . . 12 |- ( ph -> ( A e. RR /\ B e. RR /\ A < B ) )
46	45	simp1d 1073	. . . . . . . . . . 11 |- ( ph -> A e. RR )
47	46	rexrd 10089	. . . . . . . . . 10 |- ( ph -> A e. RR* )
48	47	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> A e. RR* )
49	45	simp2d 1074	. . . . . . . . . . 11 |- ( ph -> B e. RR )
50	49	rexrd 10089	. . . . . . . . . 10 |- ( ph -> B e. RR* )
51	50	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> B e. RR* )
52	4, 3, 2	fourierdlem15 40339	. . . . . . . . . 10 |- ( ph -> Q : ( 0 ... M ) --> ( A [,] B ) )
53	52	adantr 481	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q : ( 0 ... M ) --> ( A [,] B ) )
54		simpr 477	. . . . . . . . 9 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i e. ( 0..^ M ) )
55	48, 51, 53, 54	fourierdlem8 40332	. . . . . . . 8 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) C_ ( A [,] B ) )
56	44, 55	syl5ss 3614	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( A [,] B ) )
57	43, 56	feqresmpt 6250	. . . . . 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 ) )
59	57, 58	eqeltrrd 2702	. . . . 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 ) ) ) )
61	57	oveq1d 6665	. . . . . 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 ) ) ) )
62	60, 61	eleqtrd 2703	. . . . 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 ) ) )
64	57	oveq1d 6665	. . . . . 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 ) ) )
65	63, 64	eleqtrd 2703	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( x e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` x ) ) lim CC ( Q ` i ) ) )
66	39, 42, 59, 62, 65	iblcncfioo 40194	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( x e. ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) |-> ( F ` x ) ) e. L^1 )
67	43	adantr 481	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ x e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> F : ( A [,] B ) --> CC )
68	55	sselda 3603	. . . . 5 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ x e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> x e. ( A [,] B ) )
69	67, 68	ffvelrnd 6360	. . . 4 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ x e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) ) -> ( F ` x ) e. CC )
70	39, 42, 66, 69	ibliooicc 40187	. . 3 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( x e. ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) |-> ( F ` x ) ) e. L^1 )
71	16, 17, 22, 26, 28, 35, 70	iblspltprt 40189	. 2 |- ( ph -> ( x e. ( ( Q ` 0 ) [,] ( Q ` M ) ) |-> ( F ` x ) ) e. L^1 )
72	15, 71	eqeltrd 2701	1 |- ( ph -> F e. L^1 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   NNcn 11020   ZZ>=cuz 11687   (,)cioo 12175   [,]cicc 12178   ...cfz 12326  ..^cfzo 12465   -cn->ccncf 22679   L^1cibl 23386   lim CC climc 23626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-limc 23630

This theorem is referenced by:  fourierdlem84  40407  fourierdlem88  40411  fourierdlem100  40423  fourierdlem107  40430  fourierdlem111  40434  fourierdlem112  40435