Theorem limcicciooub 37717

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 11720	. . 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 37602	. . 3 |- ( ph -> ( A [,] B ) C_ RR )
7		ax-resscn 9596	. . 3 |- RR C_ CC
8	6, 7	syl6ss 3444	. 2 |- ( ph -> ( A [,] B ) C_ CC )
9		eqid 2451	. 2 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10		eqid 2451	. 2 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) )
11		retop 21782	. . . . . . . . 9 |- ( topGen ` ran (,) ) e. Top
12	11	a1i 11	. . . . . . . 8 |- ( ph -> ( topGen ` ran (,) ) e. Top )
13	4	rexrd 9690	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
14		iocssre 11714	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )
15	13, 5, 14	syl2anc 667	. . . . . . . . . 10 |- ( ph -> ( A (,] B ) C_ RR )
16		difssd 3561	. . . . . . . . . 10 |- ( ph -> ( RR \ ( A [,] B ) ) C_ RR )
17	15, 16	unssd 3610	. . . . . . . . 9 |- ( ph -> ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) C_ RR )
18		uniretop 21783	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
19	17, 18	syl6sseq 3478	. . . . . . . 8 |- ( ph -> ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) )
20		elioore 11666	. . . . . . . . . . . . . . 15 |- ( x e. ( A (,) +oo ) -> x e. RR )
21	20	ad2antlr 733	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. RR )
22		simplr 762	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. ( A (,) +oo ) )
23	13	ad2antrr 732	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> A e. RR* )
24		pnfxr 11412	. . . . . . . . . . . . . . . . 17 |- +oo e. RR*
25		elioo2 11677	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( x e. ( A (,) +oo ) <-> ( x e. RR /\ A < x /\ x < +oo ) ) )
26	23, 24, 25	sylancl 668	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. ( A (,) +oo ) <-> ( x e. RR /\ A < x /\ x < +oo ) ) )
27	22, 26	mpbid 214	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. RR /\ A < x /\ x < +oo ) )
28	27	simp2d 1021	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> A < x )
29		simpr 463	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x <_ B )
30	5	ad2antrr 732	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> B e. RR )
31		elioc2 11697	. . . . . . . . . . . . . . 15 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
32	23, 30, 31	syl2anc 667	. . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. ( A (,] B ) )
34	33	orcd 394	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
35	20	ad2antlr 733	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> x e. RR )
36		3mix3 1179	. . . . . . . . . . . . . . . . 17 |- ( -. x <_ B -> ( -. x e. RR \/ -. A <_ x \/ -. x <_ B ) )
37		3ianor 1002	. . . . . . . . . . . . . . . . 17 |- ( -. ( x e. RR /\ A <_ x /\ x <_ B ) <-> ( -. x e. RR \/ -. A <_ x \/ -. x <_ B ) )
38	36, 37	sylibr 216	. . . . . . . . . . . . . . . 16 |- ( -. x <_ B -> -. ( x e. RR /\ A <_ x /\ x <_ B ) )
39	38	adantl 468	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> -. ( x e. RR /\ A <_ x /\ x <_ B ) )
40	4	ad2antrr 732	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> A e. RR )
41	5	ad2antrr 732	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> B e. RR )
42		elicc2 11699	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
43	40, 41, 42	syl2anc 667	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
44	39, 43	mtbird 303	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> -. x e. ( A [,] B ) )
45	35, 44	eldifd 3415	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> x e. ( RR \ ( A [,] B ) ) )
46	45	olcd 395	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
47	34, 46	pm2.61dan 800	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A (,) +oo ) ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
48		elun 3574	. . . . . . . . . . 11 |- ( x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) <-> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
49	47, 48	sylibr 216	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) +oo ) ) -> x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
50	49	ralrimiva 2802	. . . . . . . . 9 |- ( ph -> A. x e. ( A (,) +oo ) x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
51		dfss3 3422	. . . . . . . . 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 216	. . . . . . . 8 |- ( ph -> ( A (,) +oo ) C_ ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
53		eqid 2451	. . . . . . . . 9 |- U. ( topGen ` ran (,) ) = U. ( topGen ` ran (,) )
54	53	ntrss 20070	. . . . . . . 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 1268	. . . . . . 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 11423	. . . . . . . . 9 |- ( ph -> B < +oo )
59	13, 56, 5, 57, 58	eliood 37595	. . . . . . . 8 |- ( ph -> B e. ( A (,) +oo ) )
60		iooretop 21786	. . . . . . . . 9 |- ( A (,) +oo ) e. ( topGen ` ran (,) )
61		isopn3i 20098	. . . . . . . . 9 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A (,) +oo ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) = ( A (,) +oo ) )
62	12, 60, 61	sylancl 668	. . . . . . . 8 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) = ( A (,) +oo ) )
63	59, 62	eleqtrrd 2532	. . . . . . 7 |- ( ph -> B e. ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) )
64	55, 63	sseldd 3433	. . . . . 6 |- ( ph -> B e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) )
65	5	rexrd 9690	. . . . . . 7 |- ( ph -> B e. RR* )
66	4, 5, 57	ltled 9783	. . . . . . 7 |- ( ph -> A <_ B )
67		ubicc2 11749	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
68	13, 65, 66, 67	syl3anc 1268	. . . . . 6 |- ( ph -> B e. ( A [,] B ) )
69	64, 68	elind 3618	. . . . 5 |- ( ph -> B e. ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
70		iocssicc 11722	. . . . . . 7 |- ( A (,] B ) C_ ( A [,] B )
71	70	a1i 11	. . . . . 6 |- ( ph -> ( A (,] B ) C_ ( A [,] B ) )
72		eqid 2451	. . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) )
73	18, 72	restntr 20198	. . . . . 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 1268	. . . . 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 2532	. . . 4 |- ( ph -> B e. ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
76		eqid 2451	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
77	9, 76	rerest 21822	. . . . . . . 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 2457	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
80	79	fveq2d 5869	. . . . 5 |- ( ph -> ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) )
81	80	fveq1d 5867	. . . 4 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
82	75, 81	eleqtrd 2531	. . 3 |- ( ph -> B e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
83	68	snssd 4117	. . . . . . . 8 |- ( ph -> { B } C_ ( A [,] B ) )
84		ssequn2 3607	. . . . . . . 8 |- ( { B } C_ ( A [,] B ) <-> ( ( A [,] B ) u. { B } ) = ( A [,] B ) )
85	83, 84	sylib 200	. . . . . . 7 |- ( ph -> ( ( A [,] B ) u. { B } ) = ( A [,] B ) )
86	85	eqcomd 2457	. . . . . 6 |- ( ph -> ( A [,] B ) = ( ( A [,] B ) u. { B } ) )
87	86	oveq2d 6306	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) )
88	87	fveq2d 5869	. . . 4 |- ( ph -> ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) ) )
89		snunioo2 37606	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
90	13, 65, 57, 89	syl3anc 1268	. . . . 5 |- ( ph -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
91	90	eqcomd 2457	. . . 4 |- ( ph -> ( A (,] B ) = ( ( A (,) B ) u. { B } ) )
92	88, 91	fveq12d 5871	. . 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 2531	. 2 |- ( ph -> B e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) ) ` ( ( A (,) B ) u. { B } ) ) )
94	1, 3, 8, 9, 10, 93	limcres 22841	1 |- ( ph -> ( ( F |` ( A (,) B ) ) lim CC B ) = ( F lim CC B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   \/ w3o 984   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   \ cdif 3401   u. cun 3402   i^i cin 3403   C_ wss 3404   {csn 3968   U.cuni 4198   class class class wbr 4402   ran crn 4835   |` cres 4836   -->wf 5578   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   +oocpnf 9672   RR*cxr 9674   < clt 9675   <_ cle 9676   (,)cioo 11635   (,]cioc 11636   [,]cicc 11638   ↾_t crest 15319   TopOpenctopn 15320   topGenctg 15336  ℂ_fldccnfld 18970   Topctop 19917   intcnt 20032   lim CC climc 22817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7925  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-icc 11642  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-rest 15321  df-topn 15322  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-cnp 20244  df-xms 21335  df-ms 21336  df-limc 22821

This theorem is referenced by:  cncfiooicclem1  37771  fourierdlem82  38052  fourierdlem93  38063  fourierdlem111  38081