Theorem limcicciooub 39869

Description: The limit of a function at the upper bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
limcicciooub.1	|- ( ph -> A e. RR )
limcicciooub.2	|- ( ph -> B e. RR )
limcicciooub.3	|- ( ph -> A < B )
limcicciooub.4	|- ( ph -> F : ( A [,] B ) --> CC )

Assertion

Ref	Expression
limcicciooub	|- ( ph -> ( ( F |` ( A (,) B ) ) lim CC B ) = ( F lim CC B ) )

Proof of Theorem limcicciooub

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limcicciooub.4	. 2 |- ( ph -> F : ( A [,] B ) --> CC )
2		ioossicc 12259	. . 3 |- ( A (,) B ) C_ ( A [,] B )
3	2	a1i 11	. 2 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
4		limcicciooub.1	. . . 4 |- ( ph -> A e. RR )
5		limcicciooub.2	. . . 4 |- ( ph -> B e. RR )
6	4, 5	iccssred 39727	. . 3 |- ( ph -> ( A [,] B ) C_ RR )
7		ax-resscn 9993	. . 3 |- RR C_ CC
8	6, 7	syl6ss 3615	. 2 |- ( ph -> ( A [,] B ) C_ CC )
9		eqid 2622	. 2 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10		eqid 2622	. 2 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) )
11		retop 22565	. . . . . . . . 9 |- ( topGen ` ran (,) ) e. Top
12	11	a1i 11	. . . . . . . 8 |- ( ph -> ( topGen ` ran (,) ) e. Top )
13	4	rexrd 10089	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
14		iocssre 12253	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )
15	13, 5, 14	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( A (,] B ) C_ RR )
16		difssd 3738	. . . . . . . . . 10 |- ( ph -> ( RR \ ( A [,] B ) ) C_ RR )
17	15, 16	unssd 3789	. . . . . . . . 9 |- ( ph -> ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) C_ RR )
18		uniretop 22566	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
19	17, 18	syl6sseq 3651	. . . . . . . 8 |- ( ph -> ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) )
20		elioore 12205	. . . . . . . . . . . . . . 15 |- ( x e. ( A (,) +oo ) -> x e. RR )
21	20	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. RR )
22		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. ( A (,) +oo ) )
23	13	ad2antrr 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> A e. RR* )
24		pnfxr 10092	. . . . . . . . . . . . . . . . 17 |- +oo e. RR*
25		elioo2 12216	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( x e. ( A (,) +oo ) <-> ( x e. RR /\ A < x /\ x < +oo ) ) )
26	23, 24, 25	sylancl 694	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. ( A (,) +oo ) <-> ( x e. RR /\ A < x /\ x < +oo ) ) )
27	22, 26	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. RR /\ A < x /\ x < +oo ) )
28	27	simp2d 1074	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> A < x )
29		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x <_ B )
30	5	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> B e. RR )
31		elioc2 12236	. . . . . . . . . . . . . . 15 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
32	23, 30, 31	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
33	21, 28, 29, 32	mpbir3and 1245	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. ( A (,] B ) )
34	33	orcd 407	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
35	20	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> x e. RR )
36		3mix3 1232	. . . . . . . . . . . . . . . . 17 |- ( -. x <_ B -> ( -. x e. RR \/ -. A <_ x \/ -. x <_ B ) )
37		3ianor 1055	. . . . . . . . . . . . . . . . 17 |- ( -. ( x e. RR /\ A <_ x /\ x <_ B ) <-> ( -. x e. RR \/ -. A <_ x \/ -. x <_ B ) )
38	36, 37	sylibr 224	. . . . . . . . . . . . . . . 16 |- ( -. x <_ B -> -. ( x e. RR /\ A <_ x /\ x <_ B ) )
39	38	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> -. ( x e. RR /\ A <_ x /\ x <_ B ) )
40	4	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> A e. RR )
41	5	ad2antrr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> B e. RR )
42		elicc2 12238	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
43	40, 41, 42	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
44	39, 43	mtbird 315	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> -. x e. ( A [,] B ) )
45	35, 44	eldifd 3585	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> x e. ( RR \ ( A [,] B ) ) )
46	45	olcd 408	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
47	34, 46	pm2.61dan 832	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A (,) +oo ) ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
48		elun 3753	. . . . . . . . . . 11 |- ( x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) <-> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
49	47, 48	sylibr 224	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) +oo ) ) -> x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
50	49	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. x e. ( A (,) +oo ) x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
51		dfss3 3592	. . . . . . . . 9 |- ( ( A (,) +oo ) C_ ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) <-> A. x e. ( A (,) +oo ) x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
52	50, 51	sylibr 224	. . . . . . . 8 |- ( ph -> ( A (,) +oo ) C_ ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
53		eqid 2622	. . . . . . . . 9 |- U. ( topGen ` ran (,) ) = U. ( topGen ` ran (,) )
54	53	ntrss 20859	. . . . . . . 8 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) /\ ( A (,) +oo ) C_ ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) )
55	12, 19, 52, 54	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) )
56	24	a1i 11	. . . . . . . . 9 |- ( ph -> +oo e. RR* )
57		limcicciooub.3	. . . . . . . . 9 |- ( ph -> A < B )
58	5	ltpnfd 11955	. . . . . . . . 9 |- ( ph -> B < +oo )
59	13, 56, 5, 57, 58	eliood 39720	. . . . . . . 8 |- ( ph -> B e. ( A (,) +oo ) )
60		iooretop 22569	. . . . . . . . 9 |- ( A (,) +oo ) e. ( topGen ` ran (,) )
61		isopn3i 20886	. . . . . . . . 9 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A (,) +oo ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) = ( A (,) +oo ) )
62	12, 60, 61	sylancl 694	. . . . . . . 8 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) = ( A (,) +oo ) )
63	59, 62	eleqtrrd 2704	. . . . . . 7 |- ( ph -> B e. ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) )
64	55, 63	sseldd 3604	. . . . . 6 |- ( ph -> B e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) )
65	5	rexrd 10089	. . . . . . 7 |- ( ph -> B e. RR* )
66	4, 5, 57	ltled 10185	. . . . . . 7 |- ( ph -> A <_ B )
67		ubicc2 12289	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
68	13, 65, 66, 67	syl3anc 1326	. . . . . 6 |- ( ph -> B e. ( A [,] B ) )
69	64, 68	elind 3798	. . . . 5 |- ( ph -> B e. ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
70		iocssicc 12261	. . . . . . 7 |- ( A (,] B ) C_ ( A [,] B )
71	70	a1i 11	. . . . . 6 |- ( ph -> ( A (,] B ) C_ ( A [,] B ) )
72		eqid 2622	. . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) )
73	18, 72	restntr 20986	. . . . . 6 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A [,] B ) C_ RR /\ ( A (,] B ) C_ ( A [,] B ) ) -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) = ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
74	12, 6, 71, 73	syl3anc 1326	. . . . 5 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) = ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
75	69, 74	eleqtrrd 2704	. . . 4 |- ( ph -> B e. ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
76		eqid 2622	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
77	9, 76	rerest 22607	. . . . . . . 8 |- ( ( A [,] B ) C_ RR -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
78	6, 77	syl 17	. . . . . . 7 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
79	78	eqcomd 2628	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
80	79	fveq2d 6195	. . . . 5 |- ( ph -> ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) )
81	80	fveq1d 6193	. . . 4 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
82	75, 81	eleqtrd 2703	. . 3 |- ( ph -> B e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
83	68	snssd 4340	. . . . . . . 8 |- ( ph -> { B } C_ ( A [,] B ) )
84		ssequn2 3786	. . . . . . . 8 |- ( { B } C_ ( A [,] B ) <-> ( ( A [,] B ) u. { B } ) = ( A [,] B ) )
85	83, 84	sylib 208	. . . . . . 7 |- ( ph -> ( ( A [,] B ) u. { B } ) = ( A [,] B ) )
86	85	eqcomd 2628	. . . . . 6 |- ( ph -> ( A [,] B ) = ( ( A [,] B ) u. { B } ) )
87	86	oveq2d 6666	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) )
88	87	fveq2d 6195	. . . 4 |- ( ph -> ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) ) )
89		snunioo2 39731	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
90	13, 65, 57, 89	syl3anc 1326	. . . . 5 |- ( ph -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
91	90	eqcomd 2628	. . . 4 |- ( ph -> ( A (,] B ) = ( ( A (,) B ) u. { B } ) )
92	88, 91	fveq12d 6197	. . 3 |- ( ph -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) ) ` ( ( A (,) B ) u. { B } ) ) )
93	82, 92	eleqtrd 2703	. 2 |- ( ph -> B e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) ) ` ( ( A (,) B ) u. { B } ) ) )
94	1, 3, 8, 9, 10, 93	limcres 23650	1 |- ( ph -> ( ( F |` ( A (,) B ) ) lim CC B ) = ( F lim CC B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   U.cuni 4436   class class class wbr 4653   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   (,]cioc 12176   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698   intcnt 20821   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-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-cnp 21032  df-xms 22125  df-ms 22126  df-limc 23630

This theorem is referenced by:  cncfiooicclem1  40106  fourierdlem82  40405  fourierdlem93  40416  fourierdlem111  40434