Theorem iblcncfioo 40194

Description: A continuous function F on an open interval ( A (,) B ) with a finite right limit R in A and a finite left limit L in B is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
iblcncfioo.a	|- ( ph -> A e. RR )
iblcncfioo.b	|- ( ph -> B e. RR )
iblcncfioo.f	|- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
iblcncfioo.l	|- ( ph -> L e. ( F lim CC B ) )
iblcncfioo.r	|- ( ph -> R e. ( F lim CC A ) )

Assertion

Ref	Expression
iblcncfioo	|- ( ph -> F e. L^1 )

Proof of Theorem iblcncfioo

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iblcncfioo.f	. . . . 5 |- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
2		cncff 22696	. . . . 5 |- ( F e. ( ( A (,) B ) -cn-> CC ) -> F : ( A (,) B ) --> CC )
3	1, 2	syl 17	. . . 4 |- ( ph -> F : ( A (,) B ) --> CC )
4	3	feqmptd 6249	. . 3 |- ( ph -> F = ( x e. ( A (,) B ) |-> ( F ` x ) ) )
5		iblcncfioo.a	. . . . . . . . . 10 |- ( ph -> A e. RR )
6	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> A e. RR )
7		eliooord 12233	. . . . . . . . . . 11 |- ( x e. ( A (,) B ) -> ( A < x /\ x < B ) )
8	7	simpld 475	. . . . . . . . . 10 |- ( x e. ( A (,) B ) -> A < x )
9	8	adantl 482	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> A < x )
10	6, 9	gtned 10172	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= A )
11	10	neneqd 2799	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = A )
12	11	iffalsed 4097	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
13		elioore 12205	. . . . . . . . . 10 |- ( x e. ( A (,) B ) -> x e. RR )
14	13	adantl 482	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> x e. RR )
15	7	simprd 479	. . . . . . . . . 10 |- ( x e. ( A (,) B ) -> x < B )
16	15	adantl 482	. . . . . . . . 9 |- ( ( ph /\ x e. ( A (,) B ) ) -> x < B )
17	14, 16	ltned 10173	. . . . . . . 8 |- ( ( ph /\ x e. ( A (,) B ) ) -> x =/= B )
18	17	neneqd 2799	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> -. x = B )
19	18	iffalsed 4097	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
20	12, 19	eqtrd 2656	. . . . 5 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` x ) )
21	20	eqcomd 2628	. . . 4 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) = if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
22	21	mpteq2dva 4744	. . 3 |- ( ph -> ( x e. ( A (,) B ) |-> ( F ` x ) ) = ( x e. ( A (,) B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) )
23	4, 22	eqtrd 2656	. 2 |- ( ph -> F = ( x e. ( A (,) B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) )
24		ioossicc 12259	. . . 4 |- ( A (,) B ) C_ ( A [,] B )
25	24	a1i 11	. . 3 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
26		ioombl 23333	. . . 4 |- ( A (,) B ) e. dom vol
27	26	a1i 11	. . 3 |- ( ph -> ( A (,) B ) e. dom vol )
28		iftrue 4092	. . . . . . 7 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
29	28	adantl 482	. . . . . 6 |- ( ( ph /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
30		limccl 23639	. . . . . . . 8 |- ( F lim CC A ) C_ CC
31		iblcncfioo.r	. . . . . . . 8 |- ( ph -> R e. ( F lim CC A ) )
32	30, 31	sseldi 3601	. . . . . . 7 |- ( ph -> R e. CC )
33	32	adantr 481	. . . . . 6 |- ( ( ph /\ x = A ) -> R e. CC )
34	29, 33	eqeltrd 2701	. . . . 5 |- ( ( ph /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
35	34	adantlr 751	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
36		iffalse 4095	. . . . . . . . 9 |- ( -. x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
37	36	ad2antlr 763	. . . . . . . 8 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
38		iftrue 4092	. . . . . . . . 9 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = L )
39	38	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> if ( x = B , L , ( F ` x ) ) = L )
40	37, 39	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
41		limccl 23639	. . . . . . . . 9 |- ( F lim CC B ) C_ CC
42		iblcncfioo.l	. . . . . . . . 9 |- ( ph -> L e. ( F lim CC B ) )
43	41, 42	sseldi 3601	. . . . . . . 8 |- ( ph -> L e. CC )
44	43	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> L e. CC )
45	40, 44	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
46	45	adantllr 755	. . . . 5 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
47		simplll 798	. . . . . . 7 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ph )
48	5	rexrd 10089	. . . . . . . . . 10 |- ( ph -> A e. RR* )
49	47, 48	syl 17	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A e. RR* )
50		iblcncfioo.b	. . . . . . . . . . 11 |- ( ph -> B e. RR )
51	50	rexrd 10089	. . . . . . . . . 10 |- ( ph -> B e. RR* )
52	47, 51	syl 17	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> B e. RR* )
53		eliccxr 39737	. . . . . . . . . 10 |- ( x e. ( A [,] B ) -> x e. RR* )
54	53	ad3antlr 767	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. RR* )
55	49, 52, 54	3jca 1242	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( A e. RR* /\ B e. RR* /\ x e. RR* ) )
56	5	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A e. RR )
57	5	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR )
58	50	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
59		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
60		eliccre 39728	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. RR /\ x e. ( A [,] B ) ) -> x e. RR )
61	57, 58, 59, 60	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
62	61	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x e. RR )
63	5, 50	jca 554	. . . . . . . . . . . . . . . 16 |- ( ph -> ( A e. RR /\ B e. RR ) )
64	63	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( A e. RR /\ B e. RR ) )
65		elicc2 12238	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
66	64, 65	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
67	59, 66	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x e. RR /\ A <_ x /\ x <_ B ) )
68	67	simp2d 1074	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A [,] B ) ) -> A <_ x )
69	68	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A <_ x )
70		df-ne 2795	. . . . . . . . . . . . 13 |- ( x =/= A <-> -. x = A )
71	70	biimpri 218	. . . . . . . . . . . 12 |- ( -. x = A -> x =/= A )
72	71	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x =/= A )
73	56, 62, 69, 72	leneltd 10191	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A < x )
74	73	adantr 481	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A < x )
75		nesym 2850	. . . . . . . . . . . . 13 |- ( B =/= x <-> -. x = B )
76	75	biimpri 218	. . . . . . . . . . . 12 |- ( -. x = B -> B =/= x )
77	76	adantl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B =/= x )
78	67	simp3d 1075	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> x <_ B )
79	61, 58, 78	3jca 1242	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x e. RR /\ B e. RR /\ x <_ B ) )
80	79	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> ( x e. RR /\ B e. RR /\ x <_ B ) )
81		leltne 10127	. . . . . . . . . . . 12 |- ( ( x e. RR /\ B e. RR /\ x <_ B ) -> ( x < B <-> B =/= x ) )
82	80, 81	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> ( x < B <-> B =/= x ) )
83	77, 82	mpbird 247	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x < B )
84	83	adantlr 751	. . . . . . . . 9 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x < B )
85	74, 84	jca 554	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( A < x /\ x < B ) )
86		elioo3g 12204	. . . . . . . 8 |- ( x e. ( A (,) B ) <-> ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) /\ ( A < x /\ x < B ) ) )
87	55, 85, 86	sylanbrc 698	. . . . . . 7 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. ( A (,) B ) )
88	47, 87	jca 554	. . . . . 6 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( ph /\ x e. ( A (,) B ) ) )
89	3	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( F ` x ) e. CC )
90	20, 89	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ x e. ( A (,) B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
91	88, 90	syl 17	. . . . 5 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
92	46, 91	pm2.61dan 832	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
93	35, 92	pm2.61dan 832	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. CC )
94		nfv 1843	. . . . 5 |- F/ x ph
95		eqid 2622	. . . . 5 |- ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
96	94, 95, 5, 50, 1, 42, 31	cncfiooicc 40107	. . . 4 |- ( ph -> ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
97		cniccibl 23607	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) e. L^1 )
98	5, 50, 96, 97	syl3anc 1326	. . 3 |- ( ph -> ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) e. L^1 )
99	25, 27, 93, 98	iblss 23571	. 2 |- ( ph -> ( x e. ( A (,) B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) e. L^1 )
100	23, 99	eqeltrd 2701	1 |- ( ph -> F e. L^1 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,]cicc 12178   -cn->ccncf 22679   volcvol 23232   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-0p 23437  df-limc 23630

This theorem is referenced by:  fourierdlem69  40392  fourierdlem73  40396  fourierdlem81  40404  fourierdlem93  40416