Theorem limciccioolb 37798

Description: The limit of a function at the lower 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
limciccioolb.1	|- ( ph -> A e. RR )
limciccioolb.2	|- ( ph -> B e. RR )
limciccioolb.3	|- ( ph -> A < B )
limciccioolb.4	|- ( ph -> F : ( A [,] B ) --> CC )

Assertion

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

Proof of Theorem limciccioolb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limciccioolb.4	. 2 |- ( ph -> F : ( A [,] B ) --> CC )
2		ioossicc 11745	. . 3 |- ( A (,) B ) C_ ( A [,] B )
3	2	a1i 11	. 2 |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
4		limciccioolb.1	. . . 4 |- ( ph -> A e. RR )
5		limciccioolb.2	. . . 4 |- ( ph -> B e. RR )
6	4, 5	iccssred 37698	. . 3 |- ( ph -> ( A [,] B ) C_ RR )
7		ax-resscn 9614	. . 3 |- RR C_ CC
8	6, 7	syl6ss 3430	. 2 |- ( ph -> ( A [,] B ) C_ CC )
9		eqid 2471	. 2 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10		eqid 2471	. 2 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) )
11		retop 21860	. . . . . . . . 9 |- ( topGen ` ran (,) ) e. Top
12	11	a1i 11	. . . . . . . 8 |- ( ph -> ( topGen ` ran (,) ) e. Top )
13	5	rexrd 9708	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
14		icossre 11740	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
15	4, 13, 14	syl2anc 673	. . . . . . . . . 10 |- ( ph -> ( A [,) B ) C_ RR )
16		difssd 3550	. . . . . . . . . 10 |- ( ph -> ( RR \ ( A [,] B ) ) C_ RR )
17	15, 16	unssd 3601	. . . . . . . . 9 |- ( ph -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ RR )
18		uniretop 21861	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
19	17, 18	syl6sseq 3464	. . . . . . . 8 |- ( ph -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) )
20		elioore 11691	. . . . . . . . . . . . . . 15 |- ( x e. ( -oo (,) B ) -> x e. RR )
21	20	ad2antlr 741	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x e. RR )
22		simpr 468	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> A <_ x )
23		simpr 468	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x e. ( -oo (,) B ) )
24		mnfxr 11437	. . . . . . . . . . . . . . . . . . 19 |- -oo e. RR*
25	24	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> -oo e. RR* )
26	13	adantr 472	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> B e. RR* )
27		elioo2 11702	. . . . . . . . . . . . . . . . . 18 |- ( ( -oo e. RR* /\ B e. RR* ) -> ( x e. ( -oo (,) B ) <-> ( x e. RR /\ -oo < x /\ x < B ) ) )
28	25, 26, 27	syl2anc 673	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. ( -oo (,) B ) <-> ( x e. RR /\ -oo < x /\ x < B ) ) )
29	23, 28	mpbid 215	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. RR /\ -oo < x /\ x < B ) )
30	29	simp3d 1044	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x < B )
31	30	adantr 472	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x < B )
32	4	ad2antrr 740	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> A e. RR )
33	13	ad2antrr 740	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> B e. RR* )
34		elico2 11723	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR /\ A <_ x /\ x < B ) ) )
35	32, 33, 34	syl2anc 673	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> ( x e. ( A [,) B ) <-> ( x e. RR /\ A <_ x /\ x < B ) ) )
36	21, 22, 31, 35	mpbir3and 1213	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x e. ( A [,) B ) )
37	36	orcd 399	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
38	20	ad2antlr 741	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. RR )
39		simpr 468	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. A <_ x )
40	39	intnanrd 931	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. ( A <_ x /\ x <_ B ) )
41	4	rexrd 9708	. . . . . . . . . . . . . . . . 17 |- ( ph -> A e. RR* )
42	41	ad2antrr 740	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> A e. RR* )
43	13	ad2antrr 740	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> B e. RR* )
44	38	rexrd 9708	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. RR* )
45		elicc4 11726	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( x e. ( A [,] B ) <-> ( A <_ x /\ x <_ B ) ) )
46	42, 43, 44, 45	syl3anc 1292	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> ( x e. ( A [,] B ) <-> ( A <_ x /\ x <_ B ) ) )
47	40, 46	mtbird 308	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. x e. ( A [,] B ) )
48	38, 47	eldifd 3401	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. ( RR \ ( A [,] B ) ) )
49	48	olcd 400	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
50	37, 49	pm2.61dan 808	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
51		elun 3565	. . . . . . . . . . 11 |- ( x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) <-> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
52	50, 51	sylibr 217	. . . . . . . . . 10 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
53	52	ralrimiva 2809	. . . . . . . . 9 |- ( ph -> A. x e. ( -oo (,) B ) x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
54		dfss3 3408	. . . . . . . . 9 |- ( ( -oo (,) B ) C_ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) <-> A. x e. ( -oo (,) B ) x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
55	53, 54	sylibr 217	. . . . . . . 8 |- ( ph -> ( -oo (,) B ) C_ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
56		eqid 2471	. . . . . . . . 9 |- U. ( topGen ` ran (,) ) = U. ( topGen ` ran (,) )
57	56	ntrss 20147	. . . . . . . 8 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) /\ ( -oo (,) B ) C_ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) )
58	12, 19, 55, 57	syl3anc 1292	. . . . . . 7 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) C_ ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) )
59	24	a1i 11	. . . . . . . . 9 |- ( ph -> -oo e. RR* )
60	4	mnfltd 11449	. . . . . . . . 9 |- ( ph -> -oo < A )
61		limciccioolb.3	. . . . . . . . 9 |- ( ph -> A < B )
62	59, 13, 4, 60, 61	eliood 37691	. . . . . . . 8 |- ( ph -> A e. ( -oo (,) B ) )
63		iooretop 21864	. . . . . . . . . 10 |- ( -oo (,) B ) e. ( topGen ` ran (,) )
64	63	a1i 11	. . . . . . . . 9 |- ( ph -> ( -oo (,) B ) e. ( topGen ` ran (,) ) )
65		isopn3i 20175	. . . . . . . . 9 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( -oo (,) B ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) = ( -oo (,) B ) )
66	12, 64, 65	syl2anc 673	. . . . . . . 8 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) = ( -oo (,) B ) )
67	62, 66	eleqtrrd 2552	. . . . . . 7 |- ( ph -> A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) )
68	58, 67	sseldd 3419	. . . . . 6 |- ( ph -> A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) )
69	4	leidd 10201	. . . . . . 7 |- ( ph -> A <_ A )
70	4, 5, 61	ltled 9800	. . . . . . 7 |- ( ph -> A <_ B )
71	4, 5, 4, 69, 70	eliccd 37697	. . . . . 6 |- ( ph -> A e. ( A [,] B ) )
72	68, 71	elind 3609	. . . . 5 |- ( ph -> A e. ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
73		icossicc 11746	. . . . . . 7 |- ( A [,) B ) C_ ( A [,] B )
74	73	a1i 11	. . . . . 6 |- ( ph -> ( A [,) B ) C_ ( A [,] B ) )
75		eqid 2471	. . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) )
76	18, 75	restntr 20275	. . . . . 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 ) ) )
77	12, 6, 74, 76	syl3anc 1292	. . . . 5 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
78	72, 77	eleqtrrd 2552	. . . 4 |- ( ph -> A e. ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
79		eqid 2471	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
80	9, 79	rerest 21900	. . . . . . . 8 |- ( ( A [,] B ) C_ RR -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
81	6, 80	syl 17	. . . . . . 7 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
82	81	eqcomd 2477	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
83	82	fveq2d 5883	. . . . 5 |- ( ph -> ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) )
84	83	fveq1d 5881	. . . 4 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
85	78, 84	eleqtrd 2551	. . 3 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
86	71	snssd 4108	. . . . . . . 8 |- ( ph -> { A } C_ ( A [,] B ) )
87		ssequn2 3598	. . . . . . . 8 |- ( { A } C_ ( A [,] B ) <-> ( ( A [,] B ) u. { A } ) = ( A [,] B ) )
88	86, 87	sylib 201	. . . . . . 7 |- ( ph -> ( ( A [,] B ) u. { A } ) = ( A [,] B ) )
89	88	eqcomd 2477	. . . . . 6 |- ( ph -> ( A [,] B ) = ( ( A [,] B ) u. { A } ) )
90	89	oveq2d 6324	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) )
91	90	fveq2d 5883	. . . 4 |- ( ph -> ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) )
92		uncom 3569	. . . . 5 |- ( ( A (,) B ) u. { A } ) = ( { A } u. ( A (,) B ) )
93		snunioo 11784	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
94	41, 13, 61, 93	syl3anc 1292	. . . . 5 |- ( ph -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
95	92, 94	syl5req 2518	. . . 4 |- ( ph -> ( A [,) B ) = ( ( A (,) B ) u. { A } ) )
96	91, 95	fveq12d 5885	. . 3 |- ( ph -> ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) ` ( ( A (,) B ) u. { A } ) ) )
97	85, 96	eleqtrd 2551	. 2 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) ` ( ( A (,) B ) u. { A } ) ) )
98	1, 3, 8, 9, 10, 97	limcres 22920	1 |- ( ph -> ( ( F |` ( A (,) B ) ) lim CC A ) = ( F lim CC A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   \ cdif 3387   u. cun 3388   i^i cin 3389   C_ wss 3390   {csn 3959   U.cuni 4190   class class class wbr 4395   ran crn 4840   |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   -oocmnf 9691   RR*cxr 9692   < clt 9693   <_ cle 9694   (,)cioo 11660   [,)cico 11662   [,]cicc 11663   ↾_t crest 15397   TopOpenctopn 15398   topGenctg 15414  ℂ_fldccnfld 19047   Topctop 19994   intcnt 20109   lim CC climc 22896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-starv 15283  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-rest 15399  df-topn 15400  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-cnp 20321  df-xms 21413  df-ms 21414  df-limc 22900

This theorem is referenced by:  cncfiooicclem1  37868  fourierdlem82  38164  fourierdlem93  38175  fourierdlem111  38193