Theorem limcicciooub 31889

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 11635	. . 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 31785	. . 3 |- ( ph -> ( A [,] B ) C_ RR )
7		ax-resscn 9566	. . 3 |- RR C_ CC
8	6, 7	syl6ss 3511	. 2 |- ( ph -> ( A [,] B ) C_ CC )
9		eqid 2457	. 2 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10		eqid 2457	. 2 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) )
11		retop 21485	. . . . . . . . 9 |- ( topGen ` ran (,) ) e. Top
12	11	a1i 11	. . . . . . . 8 |- ( ph -> ( topGen ` ran (,) ) e. Top )
13	4	rexrd 9660	. . . . . . . . . . 11 |- ( ph -> A e. RR* )
14		iocssre 11629	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR ) -> ( A (,] B ) C_ RR )
15	13, 5, 14	syl2anc 661	. . . . . . . . . 10 |- ( ph -> ( A (,] B ) C_ RR )
16		difssd 3628	. . . . . . . . . 10 |- ( ph -> ( RR \ ( A [,] B ) ) C_ RR )
17	15, 16	unssd 3676	. . . . . . . . 9 |- ( ph -> ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) C_ RR )
18		uniretop 21486	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
19	17, 18	syl6sseq 3545	. . . . . . . 8 |- ( ph -> ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) )
20		elioore 11584	. . . . . . . . . . . . . . 15 |- ( x e. ( A (,) +oo ) -> x e. RR )
21	20	ad2antlr 726	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. RR )
22		simplr 755	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. ( A (,) +oo ) )
23	13	ad2antrr 725	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> A e. RR* )
24		pnfxr 11346	. . . . . . . . . . . . . . . . 17 |- +oo e. RR*
25		elioo2 11595	. . . . . . . . . . . . . . . . 17 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( x e. ( A (,) +oo ) <-> ( x e. RR /\ A < x /\ x < +oo ) ) )
26	23, 24, 25	sylancl 662	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. ( A (,) +oo ) <-> ( x e. RR /\ A < x /\ x < +oo ) ) )
27	22, 26	mpbid 210	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. RR /\ A < x /\ x < +oo ) )
28	27	simp2d 1009	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> A < x )
29		simpr 461	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x <_ B )
30	5	ad2antrr 725	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> B e. RR )
31		elioc2 11612	. . . . . . . . . . . . . . 15 |- ( ( A e. RR* /\ B e. RR ) -> ( x e. ( A (,] B ) <-> ( x e. RR /\ A < x /\ x <_ B ) ) )
32	23, 30, 31	syl2anc 661	. . . . . . . . . . . . . 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 1179	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> x e. ( A (,] B ) )
34	33	orcd 392	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ x <_ B ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
35	20	ad2antlr 726	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> x e. RR )
36		3mix3 1167	. . . . . . . . . . . . . . . . 17 |- ( -. x <_ B -> ( -. x e. RR \/ -. A <_ x \/ -. x <_ B ) )
37		3ianor 990	. . . . . . . . . . . . . . . . 17 |- ( -. ( x e. RR /\ A <_ x /\ x <_ B ) <-> ( -. x e. RR \/ -. A <_ x \/ -. x <_ B ) )
38	36, 37	sylibr 212	. . . . . . . . . . . . . . . 16 |- ( -. x <_ B -> -. ( x e. RR /\ A <_ x /\ x <_ B ) )
39	38	adantl 466	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> -. ( x e. RR /\ A <_ x /\ x <_ B ) )
40	4	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> A e. RR )
41	5	ad2antrr 725	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> B e. RR )
42		elicc2 11614	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
43	40, 41, 42	syl2anc 661	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> ( x e. ( A [,] B ) <-> ( x e. RR /\ A <_ x /\ x <_ B ) ) )
44	39, 43	mtbird 301	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> -. x e. ( A [,] B ) )
45	35, 44	eldifd 3482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> x e. ( RR \ ( A [,] B ) ) )
46	45	olcd 393	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( A (,) +oo ) ) /\ -. x <_ B ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
47	34, 46	pm2.61dan 791	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( A (,) +oo ) ) -> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
48		elun 3641	. . . . . . . . . . 11 |- ( x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) <-> ( x e. ( A (,] B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
49	47, 48	sylibr 212	. . . . . . . . . 10 |- ( ( ph /\ x e. ( A (,) +oo ) ) -> x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
50	49	ralrimiva 2871	. . . . . . . . 9 |- ( ph -> A. x e. ( A (,) +oo ) x e. ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
51		dfss3 3489	. . . . . . . . 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 212	. . . . . . . 8 |- ( ph -> ( A (,) +oo ) C_ ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) )
53		eqid 2457	. . . . . . . . 9 |- U. ( topGen ` ran (,) ) = U. ( topGen ` ran (,) )
54	53	ntrss 19774	. . . . . . . 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 1228	. . . . . . 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 31726	. . . . . . . . 9 |- ( ph -> B < +oo )
59	13, 56, 5, 57, 58	eliood 31777	. . . . . . . 8 |- ( ph -> B e. ( A (,) +oo ) )
60		iooretop 21490	. . . . . . . . 9 |- ( A (,) +oo ) e. ( topGen ` ran (,) )
61		isopn3i 19801	. . . . . . . . 9 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( A (,) +oo ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) = ( A (,) +oo ) )
62	12, 60, 61	sylancl 662	. . . . . . . 8 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) = ( A (,) +oo ) )
63	59, 62	eleqtrrd 2548	. . . . . . 7 |- ( ph -> B e. ( ( int ` ( topGen ` ran (,) ) ) ` ( A (,) +oo ) ) )
64	55, 63	sseldd 3500	. . . . . 6 |- ( ph -> B e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) )
65	5	rexrd 9660	. . . . . . 7 |- ( ph -> B e. RR* )
66	4, 5, 57	ltled 9750	. . . . . . 7 |- ( ph -> A <_ B )
67		ubicc2 11662	. . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
68	13, 65, 66, 67	syl3anc 1228	. . . . . 6 |- ( ph -> B e. ( A [,] B ) )
69	64, 68	elind 3684	. . . . 5 |- ( ph -> B e. ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A (,] B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
70		iocssicc 11637	. . . . . . 7 |- ( A (,] B ) C_ ( A [,] B )
71	70	a1i 11	. . . . . 6 |- ( ph -> ( A (,] B ) C_ ( A [,] B ) )
72		eqid 2457	. . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) )
73	18, 72	restntr 19901	. . . . . 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 1228	. . . . 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 2548	. . . 4 |- ( ph -> B e. ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
76		eqid 2457	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
77	9, 76	rerest 21526	. . . . . . . 8 |- ( ( A [,] B ) C_ RR -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
78	6, 77	syl 16	. . . . . . 7 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
79	78	eqcomd 2465	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
80	79	fveq2d 5876	. . . . 5 |- ( ph -> ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) )
81	80	fveq1d 5874	. . . 4 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
82	75, 81	eleqtrd 2547	. . 3 |- ( ph -> B e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A (,] B ) ) )
83	68	snssd 4177	. . . . . . . 8 |- ( ph -> { B } C_ ( A [,] B ) )
84		ssequn2 3673	. . . . . . . 8 |- ( { B } C_ ( A [,] B ) <-> ( ( A [,] B ) u. { B } ) = ( A [,] B ) )
85	83, 84	sylib 196	. . . . . . 7 |- ( ph -> ( ( A [,] B ) u. { B } ) = ( A [,] B ) )
86	85	eqcomd 2465	. . . . . 6 |- ( ph -> ( A [,] B ) = ( ( A [,] B ) u. { B } ) )
87	86	oveq2d 6312	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) )
88	87	fveq2d 5876	. . . 4 |- ( ph -> ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) ) )
89		snunioo2 31790	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
90	13, 65, 57, 89	syl3anc 1228	. . . . 5 |- ( ph -> ( ( A (,) B ) u. { B } ) = ( A (,] B ) )
91	90	eqcomd 2465	. . . 4 |- ( ph -> ( A (,] B ) = ( ( A (,) B ) u. { B } ) )
92	88, 91	fveq12d 5878	. . 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 2547	. 2 |- ( ph -> B e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { B } ) ) ) ` ( ( A (,) B ) u. { B } ) ) )
94	1, 3, 8, 9, 10, 93	limcres 22507	1 |- ( ph -> ( ( F |` ( A (,) B ) ) lim CC B ) = ( F lim CC B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/ w3o 972   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   \ cdif 3468   u. cun 3469   i^i cin 3470   C_ wss 3471   {csn 4032   U.cuni 4251   class class class wbr 4456   ran crn 5009   |` cres 5010   -->wf 5590   ` cfv 5594  (class class class)co 6296   CCcc 9507   RRcr 9508   +oocpnf 9642   RR*cxr 9644   < clt 9645   <_ cle 9646   (,)cioo 11554   (,]cioc 11555   [,]cicc 11557   ↾_t crest 14929   TopOpenctopn 14930   topGenctg 14946  ℂ_fldccnfld 18638   Topctop 19612   intcnt 19736   lim CC climc 22483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-icc 11561  df-fz 11698  df-seq 12111  df-exp 12170  df-cj 13035  df-re 13036  df-im 13037  df-sqrt 13171  df-abs 13172  df-struct 14737  df-ndx 14738  df-slot 14739  df-base 14740  df-plusg 14816  df-mulr 14817  df-starv 14818  df-tset 14822  df-ple 14823  df-ds 14825  df-unif 14826  df-rest 14931  df-topn 14932  df-topgen 14952  df-psmet 18629  df-xmet 18630  df-met 18631  df-bl 18632  df-mopn 18633  df-cnfld 18639  df-top 19617  df-bases 19619  df-topon 19620  df-topsp 19621  df-cld 19738  df-ntr 19739  df-cls 19740  df-cnp 19947  df-xms 21040  df-ms 21041  df-limc 22487

This theorem is referenced by:  cncfiooicclem1  31942  fourierdlem82  32217  fourierdlem93  32228  fourierdlem111  32246