Theorem cncfiooicc 40107

Description: A continuous function F on an open interval ( A (,) B ) can be extended to a continuous function G on the corresponding closed interval, if it has a finite right limit R in A and a finite left limit L in B. F can be complex valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
cncfiooicc.x	|- F/ x ph
cncfiooicc.g	|- G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
cncfiooicc.a	|- ( ph -> A e. RR )
cncfiooicc.b	|- ( ph -> B e. RR )
cncfiooicc.f	|- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
cncfiooicc.l	|- ( ph -> L e. ( F lim CC B ) )
cncfiooicc.r	|- ( ph -> R e. ( F lim CC A ) )

Assertion

Ref	Expression
cncfiooicc	|- ( ph -> G e. ( ( A [,] B ) -cn-> CC ) )

Distinct variable groups:   x, A   x, B   x, F   x, L   x, R   ph, x

Allowed substitution hint:   G( x)

Proof of Theorem cncfiooicc

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ x ( ph /\ A < B )
2		cncfiooicc.g	. . 3 |- G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
3		cncfiooicc.a	. . . 4 |- ( ph -> A e. RR )
4	3	adantr 481	. . 3 |- ( ( ph /\ A < B ) -> A e. RR )
5		cncfiooicc.b	. . . 4 |- ( ph -> B e. RR )
6	5	adantr 481	. . 3 |- ( ( ph /\ A < B ) -> B e. RR )
7		simpr 477	. . 3 |- ( ( ph /\ A < B ) -> A < B )
8		cncfiooicc.f	. . . 4 |- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
9	8	adantr 481	. . 3 |- ( ( ph /\ A < B ) -> F e. ( ( A (,) B ) -cn-> CC ) )
10		cncfiooicc.l	. . . 4 |- ( ph -> L e. ( F lim CC B ) )
11	10	adantr 481	. . 3 |- ( ( ph /\ A < B ) -> L e. ( F lim CC B ) )
12		cncfiooicc.r	. . . 4 |- ( ph -> R e. ( F lim CC A ) )
13	12	adantr 481	. . 3 |- ( ( ph /\ A < B ) -> R e. ( F lim CC A ) )
14	1, 2, 4, 6, 7, 9, 11, 13	cncfiooicclem1 40106	. 2 |- ( ( ph /\ A < B ) -> G e. ( ( A [,] B ) -cn-> CC ) )
15		limccl 23639	. . . . . . . . . 10 |- ( F lim CC A ) C_ CC
16	15, 12	sseldi 3601	. . . . . . . . 9 |- ( ph -> R e. CC )
17	16	snssd 4340	. . . . . . . 8 |- ( ph -> { R } C_ CC )
18		ssid 3624	. . . . . . . . 9 |- CC C_ CC
19	18	a1i 11	. . . . . . . 8 |- ( ph -> CC C_ CC )
20		cncfss 22702	. . . . . . . 8 |- ( ( { R } C_ CC /\ CC C_ CC ) -> ( { A } -cn-> { R } ) C_ ( { A } -cn-> CC ) )
21	17, 19, 20	syl2anc 693	. . . . . . 7 |- ( ph -> ( { A } -cn-> { R } ) C_ ( { A } -cn-> CC ) )
22	21	adantr 481	. . . . . 6 |- ( ( ph /\ A = B ) -> ( { A } -cn-> { R } ) C_ ( { A } -cn-> CC ) )
23	3	rexrd 10089	. . . . . . . . . . . 12 |- ( ph -> A e. RR* )
24		iccid 12220	. . . . . . . . . . . 12 |- ( A e. RR* -> ( A [,] A ) = { A } )
25	23, 24	syl 17	. . . . . . . . . . 11 |- ( ph -> ( A [,] A ) = { A } )
26		oveq2 6658	. . . . . . . . . . 11 |- ( A = B -> ( A [,] A ) = ( A [,] B ) )
27	25, 26	sylan9req 2677	. . . . . . . . . 10 |- ( ( ph /\ A = B ) -> { A } = ( A [,] B ) )
28	27	eqcomd 2628	. . . . . . . . 9 |- ( ( ph /\ A = B ) -> ( A [,] B ) = { A } )
29		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ph /\ A = B ) /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
30	28	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ A = B ) /\ x e. ( A [,] B ) ) -> ( A [,] B ) = { A } )
31	29, 30	eleqtrd 2703	. . . . . . . . . . 11 |- ( ( ( ph /\ A = B ) /\ x e. ( A [,] B ) ) -> x e. { A } )
32		elsni 4194	. . . . . . . . . . 11 |- ( x e. { A } -> x = A )
33	31, 32	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ A = B ) /\ x e. ( A [,] B ) ) -> x = A )
34	33	iftrued 4094	. . . . . . . . 9 |- ( ( ( ph /\ A = B ) /\ x e. ( A [,] B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
35	28, 34	mpteq12dva 4732	. . . . . . . 8 |- ( ( ph /\ A = B ) -> ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) = ( x e. { A } |-> R ) )
36	2, 35	syl5eq 2668	. . . . . . 7 |- ( ( ph /\ A = B ) -> G = ( x e. { A } |-> R ) )
37	3	recnd 10068	. . . . . . . . 9 |- ( ph -> A e. CC )
38	37	adantr 481	. . . . . . . 8 |- ( ( ph /\ A = B ) -> A e. CC )
39	16	adantr 481	. . . . . . . 8 |- ( ( ph /\ A = B ) -> R e. CC )
40		cncfdmsn 40103	. . . . . . . 8 |- ( ( A e. CC /\ R e. CC ) -> ( x e. { A } |-> R ) e. ( { A } -cn-> { R } ) )
41	38, 39, 40	syl2anc 693	. . . . . . 7 |- ( ( ph /\ A = B ) -> ( x e. { A } |-> R ) e. ( { A } -cn-> { R } ) )
42	36, 41	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ A = B ) -> G e. ( { A } -cn-> { R } ) )
43	22, 42	sseldd 3604	. . . . 5 |- ( ( ph /\ A = B ) -> G e. ( { A } -cn-> CC ) )
44	27	oveq1d 6665	. . . . 5 |- ( ( ph /\ A = B ) -> ( { A } -cn-> CC ) = ( ( A [,] B ) -cn-> CC ) )
45	43, 44	eleqtrd 2703	. . . 4 |- ( ( ph /\ A = B ) -> G e. ( ( A [,] B ) -cn-> CC ) )
46	45	adantlr 751	. . 3 |- ( ( ( ph /\ -. A < B ) /\ A = B ) -> G e. ( ( A [,] B ) -cn-> CC ) )
47		simpll 790	. . . 4 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> ph )
48		eqcom 2629	. . . . . . . . 9 |- ( B = A <-> A = B )
49	48	biimpi 206	. . . . . . . 8 |- ( B = A -> A = B )
50	49	con3i 150	. . . . . . 7 |- ( -. A = B -> -. B = A )
51	50	adantl 482	. . . . . 6 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> -. B = A )
52		simplr 792	. . . . . 6 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> -. A < B )
53		pm4.56 516	. . . . . . 7 |- ( ( -. B = A /\ -. A < B ) <-> -. ( B = A \/ A < B ) )
54	53	biimpi 206	. . . . . 6 |- ( ( -. B = A /\ -. A < B ) -> -. ( B = A \/ A < B ) )
55	51, 52, 54	syl2anc 693	. . . . 5 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> -. ( B = A \/ A < B ) )
56	47, 5	syl 17	. . . . . 6 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> B e. RR )
57	47, 3	syl 17	. . . . . 6 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> A e. RR )
58	56, 57	lttrid 10175	. . . . 5 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> ( B < A <-> -. ( B = A \/ A < B ) ) )
59	55, 58	mpbird 247	. . . 4 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> B < A )
60		0ss 3972	. . . . . . . 8 |- (/) C_ CC
61		eqid 2622	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
62	61	cnfldtop 22587	. . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) e. Top
63		rest0 20973	. . . . . . . . . . 11 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t (/) ) = { (/) } )
64	62, 63	ax-mp 5	. . . . . . . . . 10 |- ( ( TopOpen ` ℂfld ) ↾t (/) ) = { (/) }
65	64	eqcomi 2631	. . . . . . . . 9 |- { (/) } = ( ( TopOpen ` ℂfld ) ↾t (/) )
66	61, 65, 65	cncfcn 22712	. . . . . . . 8 |- ( ( (/) C_ CC /\ (/) C_ CC ) -> ( (/) -cn-> (/) ) = ( { (/) } Cn { (/) } ) )
67	60, 60, 66	mp2an 708	. . . . . . 7 |- ( (/) -cn-> (/) ) = ( { (/) } Cn { (/) } )
68		cncfss 22702	. . . . . . . 8 |- ( ( (/) C_ CC /\ CC C_ CC ) -> ( (/) -cn-> (/) ) C_ ( (/) -cn-> CC ) )
69	60, 18, 68	mp2an 708	. . . . . . 7 |- ( (/) -cn-> (/) ) C_ ( (/) -cn-> CC )
70	67, 69	eqsstr3i 3636	. . . . . 6 |- ( { (/) } Cn { (/) } ) C_ ( (/) -cn-> CC )
71	2	a1i 11	. . . . . . . 8 |- ( ( ph /\ B < A ) -> G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) )
72		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ B < A ) -> B < A )
73	23	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ B < A ) -> A e. RR* )
74	5	rexrd 10089	. . . . . . . . . . . 12 |- ( ph -> B e. RR* )
75	74	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ B < A ) -> B e. RR* )
76		icc0 12223	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( A [,] B ) = (/) <-> B < A ) )
77	73, 75, 76	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ B < A ) -> ( ( A [,] B ) = (/) <-> B < A ) )
78	72, 77	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ B < A ) -> ( A [,] B ) = (/) )
79	78	mpteq1d 4738	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) = ( x e. (/) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) )
80		mpt0 6021	. . . . . . . . 9 |- ( x e. (/) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) = (/)
81	80	a1i 11	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( x e. (/) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) ) = (/) )
82	71, 79, 81	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ B < A ) -> G = (/) )
83		0cnf 40090	. . . . . . 7 |- (/) e. ( { (/) } Cn { (/) } )
84	82, 83	syl6eqel 2709	. . . . . 6 |- ( ( ph /\ B < A ) -> G e. ( { (/) } Cn { (/) } ) )
85	70, 84	sseldi 3601	. . . . 5 |- ( ( ph /\ B < A ) -> G e. ( (/) -cn-> CC ) )
86	78	eqcomd 2628	. . . . . 6 |- ( ( ph /\ B < A ) -> (/) = ( A [,] B ) )
87	86	oveq1d 6665	. . . . 5 |- ( ( ph /\ B < A ) -> ( (/) -cn-> CC ) = ( ( A [,] B ) -cn-> CC ) )
88	85, 87	eleqtrd 2703	. . . 4 |- ( ( ph /\ B < A ) -> G e. ( ( A [,] B ) -cn-> CC ) )
89	47, 59, 88	syl2anc 693	. . 3 |- ( ( ( ph /\ -. A < B ) /\ -. A = B ) -> G e. ( ( A [,] B ) -cn-> CC ) )
90	46, 89	pm2.61dan 832	. 2 |- ( ( ph /\ -. A < B ) -> G e. ( ( A [,] B ) -cn-> CC ) )
91	14, 90	pm2.61dan 832	1 |- ( ph -> G e. ( ( A [,] B ) -cn-> CC ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   (,)cioo 12175   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   Topctop 20698   Cn ccn 21028   -cn->ccncf 22679   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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  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-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  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-xms 22125  df-ms 22126  df-cncf 22681  df-limc 23630

This theorem is referenced by:  cncfiooiccre  40108  cncfioobd  40110  itgioocnicc  40193  iblcncfioo  40194  fourierdlem73  40396  fourierdlem81  40404  fourierdlem82  40405