Theorem cncfiooiccre 40108

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 is assumed to be real-valued. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   G( x)

Proof of Theorem cncfiooiccre

Step	Hyp	Ref	Expression
1		iftrue 4092	. . . . . . 7 |- ( x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
2	1	adantl 482	. . . . . 6 |- ( ( ph /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = R )
3		cncfiooiccre.f	. . . . . . . . 9 |- ( ph -> F e. ( ( A (,) B ) -cn-> RR ) )
4		cncff 22696	. . . . . . . . 9 |- ( F e. ( ( A (,) B ) -cn-> RR ) -> F : ( A (,) B ) --> RR )
5	3, 4	syl 17	. . . . . . . 8 |- ( ph -> F : ( A (,) B ) --> RR )
6		ioosscn 39716	. . . . . . . . 9 |- ( A (,) B ) C_ CC
7	6	a1i 11	. . . . . . . 8 |- ( ph -> ( A (,) B ) C_ CC )
8		eqid 2622	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
9		cncfiooiccre.b	. . . . . . . . . 10 |- ( ph -> B e. RR )
10	9	rexrd 10089	. . . . . . . . 9 |- ( ph -> B e. RR* )
11		cncfiooiccre.a	. . . . . . . . 9 |- ( ph -> A e. RR )
12		cncfiooiccre.altb	. . . . . . . . 9 |- ( ph -> A < B )
13	8, 10, 11, 12	lptioo1cn 39878	. . . . . . . 8 |- ( ph -> A e. ( ( limPt ` ( TopOpen ` ℂfld ) ) ` ( A (,) B ) ) )
14		cncfiooiccre.r	. . . . . . . 8 |- ( ph -> R e. ( F lim CC A ) )
15	5, 7, 13, 14	limcrecl 39861	. . . . . . 7 |- ( ph -> R e. RR )
16	15	adantr 481	. . . . . 6 |- ( ( ph /\ x = A ) -> R e. RR )
17	2, 16	eqeltrd 2701	. . . . 5 |- ( ( ph /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. RR )
18	17	adantlr 751	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. RR )
19		iffalse 4095	. . . . . . . . 9 |- ( -. x = A -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = if ( x = B , L , ( F ` x ) ) )
20		iftrue 4092	. . . . . . . . 9 |- ( x = B -> if ( x = B , L , ( F ` x ) ) = L )
21	19, 20	sylan9eq 2676	. . . . . . . 8 |- ( ( -. x = A /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
22	21	adantll 750	. . . . . . 7 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = L )
23	11	rexrd 10089	. . . . . . . . . 10 |- ( ph -> A e. RR* )
24	8, 23, 9, 12	lptioo2cn 39877	. . . . . . . . 9 |- ( ph -> B e. ( ( limPt ` ( TopOpen ` ℂfld ) ) ` ( A (,) B ) ) )
25		cncfiooiccre.l	. . . . . . . . 9 |- ( ph -> L e. ( F lim CC B ) )
26	5, 7, 24, 25	limcrecl 39861	. . . . . . . 8 |- ( ph -> L e. RR )
27	26	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> L e. RR )
28	22, 27	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. RR )
29	28	adantllr 755	. . . . 5 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. RR )
30		iffalse 4095	. . . . . . . 8 |- ( -. x = B -> if ( x = B , L , ( F ` x ) ) = ( F ` x ) )
31	19, 30	sylan9eq 2676	. . . . . . 7 |- ( ( -. x = A /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` x ) )
32	31	adantll 750	. . . . . 6 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) = ( F ` x ) )
33	5	ad3antrrr 766	. . . . . . 7 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> F : ( A (,) B ) --> RR )
34	23	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A e. RR* )
35	10	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> B e. RR* )
36	11	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR )
37	9	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
38		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
39		eliccre 39728	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. RR /\ x e. ( A [,] B ) ) -> x e. RR )
40	36, 37, 38, 39	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
41	40	ad2antrr 762	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. RR )
42	11	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A e. RR )
43	40	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x e. RR )
44	23	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A e. RR* )
45	10	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> B e. RR* )
46	38	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x e. ( A [,] B ) )
47		iccgelb 12230	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> A <_ x )
48	44, 45, 46, 47	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A <_ x )
49		neqne 2802	. . . . . . . . . . 11 |- ( -. x = A -> x =/= A )
50	49	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> x =/= A )
51	42, 43, 48, 50	leneltd 10191	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> A < x )
52	51	adantr 481	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> A < x )
53	40	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x e. RR )
54	9	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B e. RR )
55	23	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> A e. RR* )
56	10	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B e. RR* )
57	38	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x e. ( A [,] B ) )
58		iccleub 12229	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ x e. ( A [,] B ) ) -> x <_ B )
59	55, 56, 57, 58	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x <_ B )
60		neqne 2802	. . . . . . . . . . . 12 |- ( -. x = B -> x =/= B )
61	60	necomd 2849	. . . . . . . . . . 11 |- ( -. x = B -> B =/= x )
62	61	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> B =/= x )
63	53, 54, 59, 62	leneltd 10191	. . . . . . . . 9 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = B ) -> x < B )
64	63	adantlr 751	. . . . . . . 8 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x < B )
65	34, 35, 41, 52, 64	eliood 39720	. . . . . . 7 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> x e. ( A (,) B ) )
66	33, 65	ffvelrnd 6360	. . . . . 6 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> ( F ` x ) e. RR )
67	32, 66	eqeltrd 2701	. . . . 5 |- ( ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) /\ -. x = B ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. RR )
68	29, 67	pm2.61dan 832	. . . 4 |- ( ( ( ph /\ x e. ( A [,] B ) ) /\ -. x = A ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. RR )
69	18, 68	pm2.61dan 832	. . 3 |- ( ( ph /\ x e. ( A [,] B ) ) -> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) e. RR )
70		cncfiooiccre.g	. . 3 |- G = ( x e. ( A [,] B ) |-> if ( x = A , R , if ( x = B , L , ( F ` x ) ) ) )
71	69, 70	fmptd 6385	. 2 |- ( ph -> G : ( A [,] B ) --> RR )
72		ax-resscn 9993	. . 3 |- RR C_ CC
73		cncfiooiccre.x	. . . 4 |- F/ x ph
74		ssid 3624	. . . . . 6 |- CC C_ CC
75		cncfss 22702	. . . . . 6 |- ( ( RR C_ CC /\ CC C_ CC ) -> ( ( A (,) B ) -cn-> RR ) C_ ( ( A (,) B ) -cn-> CC ) )
76	72, 74, 75	mp2an 708	. . . . 5 |- ( ( A (,) B ) -cn-> RR ) C_ ( ( A (,) B ) -cn-> CC )
77	76, 3	sseldi 3601	. . . 4 |- ( ph -> F e. ( ( A (,) B ) -cn-> CC ) )
78	73, 70, 11, 9, 77, 25, 14	cncfiooicc 40107	. . 3 |- ( ph -> G e. ( ( A [,] B ) -cn-> CC ) )
79		cncffvrn 22701	. . 3 |- ( ( RR C_ CC /\ G e. ( ( A [,] B ) -cn-> CC ) ) -> ( G e. ( ( A [,] B ) -cn-> RR ) <-> G : ( A [,] B ) --> RR ) )
80	72, 78, 79	sylancr 695	. 2 |- ( ph -> ( G e. ( ( A [,] B ) -cn-> RR ) <-> G : ( A [,] B ) --> RR ) )
81	71, 80	mpbird 247	1 |- ( ph -> G e. ( ( A [,] B ) -cn-> RR ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082  ℂ_fldccnfld 19746   -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-nei 20902  df-lp 20940  df-cn 21031  df-cnp 21032  df-xms 22125  df-ms 22126  df-cncf 22681  df-limc 23630

This theorem is referenced by: (None)