Theorem limciccioolb 37695

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 11717	. . 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 37596	. . 3 |- ( ph -> ( A [,] B ) C_ RR )
7		ax-resscn 9593	. . 3 |- RR C_ CC
8	6, 7	syl6ss 3443	. 2 |- ( ph -> ( A [,] B ) C_ CC )
9		eqid 2450	. 2 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
10		eqid 2450	. 2 |- ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) )
11		retop 21775	. . . . . . . . 9 |- ( topGen ` ran (,) ) e. Top
12	11	a1i 11	. . . . . . . 8 |- ( ph -> ( topGen ` ran (,) ) e. Top )
13	5	rexrd 9687	. . . . . . . . . . 11 |- ( ph -> B e. RR* )
14		icossre 11712	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR )
15	4, 13, 14	syl2anc 666	. . . . . . . . . 10 |- ( ph -> ( A [,) B ) C_ RR )
16		difssd 3560	. . . . . . . . . 10 |- ( ph -> ( RR \ ( A [,] B ) ) C_ RR )
17	15, 16	unssd 3609	. . . . . . . . 9 |- ( ph -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ RR )
18		uniretop 21776	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
19	17, 18	syl6sseq 3477	. . . . . . . 8 |- ( ph -> ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) C_ U. ( topGen ` ran (,) ) )
20		elioore 11663	. . . . . . . . . . . . . . 15 |- ( x e. ( -oo (,) B ) -> x e. RR )
21	20	ad2antlr 732	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x e. RR )
22		simpr 463	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> A <_ x )
23		simpr 463	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x e. ( -oo (,) B ) )
24		mnfxr 11411	. . . . . . . . . . . . . . . . . . 19 |- -oo e. RR*
25	24	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> -oo e. RR* )
26	13	adantr 467	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> B e. RR* )
27		elioo2 11674	. . . . . . . . . . . . . . . . . 18 |- ( ( -oo e. RR* /\ B e. RR* ) -> ( x e. ( -oo (,) B ) <-> ( x e. RR /\ -oo < x /\ x < B ) ) )
28	25, 26, 27	syl2anc 666	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. ( -oo (,) B ) <-> ( x e. RR /\ -oo < x /\ x < B ) ) )
29	23, 28	mpbid 214	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. RR /\ -oo < x /\ x < B ) )
30	29	simp3d 1021	. . . . . . . . . . . . . . 15 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x < B )
31	30	adantr 467	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x < B )
32	4	ad2antrr 731	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> A e. RR )
33	13	ad2antrr 731	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> B e. RR* )
34		elico2 11695	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ B e. RR* ) -> ( x e. ( A [,) B ) <-> ( x e. RR /\ A <_ x /\ x < B ) ) )
35	32, 33, 34	syl2anc 666	. . . . . . . . . . . . . 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 1190	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> x e. ( A [,) B ) )
37	36	orcd 394	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ A <_ x ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
38	20	ad2antlr 732	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. RR )
39		simpr 463	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. A <_ x )
40	39	intnanrd 927	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. ( A <_ x /\ x <_ B ) )
41	4	rexrd 9687	. . . . . . . . . . . . . . . . 17 |- ( ph -> A e. RR* )
42	41	ad2antrr 731	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> A e. RR* )
43	13	ad2antrr 731	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> B e. RR* )
44	38	rexrd 9687	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. RR* )
45		elicc4 11698	. . . . . . . . . . . . . . . 16 |- ( ( A e. RR* /\ B e. RR* /\ x e. RR* ) -> ( x e. ( A [,] B ) <-> ( A <_ x /\ x <_ B ) ) )
46	42, 43, 44, 45	syl3anc 1267	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> ( x e. ( A [,] B ) <-> ( A <_ x /\ x <_ B ) ) )
47	40, 46	mtbird 303	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> -. x e. ( A [,] B ) )
48	38, 47	eldifd 3414	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> x e. ( RR \ ( A [,] B ) ) )
49	48	olcd 395	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( -oo (,) B ) ) /\ -. A <_ x ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
50	37, 49	pm2.61dan 799	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
51		elun 3573	. . . . . . . . . . 11 |- ( x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) <-> ( x e. ( A [,) B ) \/ x e. ( RR \ ( A [,] B ) ) ) )
52	50, 51	sylibr 216	. . . . . . . . . 10 |- ( ( ph /\ x e. ( -oo (,) B ) ) -> x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
53	52	ralrimiva 2801	. . . . . . . . 9 |- ( ph -> A. x e. ( -oo (,) B ) x e. ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
54		dfss3 3421	. . . . . . . . 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 216	. . . . . . . 8 |- ( ph -> ( -oo (,) B ) C_ ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) )
56		eqid 2450	. . . . . . . . 9 |- U. ( topGen ` ran (,) ) = U. ( topGen ` ran (,) )
57	56	ntrss 20063	. . . . . . . 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 1267	. . . . . . 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 11423	. . . . . . . . 9 |- ( ph -> -oo < A )
61		limciccioolb.3	. . . . . . . . 9 |- ( ph -> A < B )
62	59, 13, 4, 60, 61	eliood 37589	. . . . . . . 8 |- ( ph -> A e. ( -oo (,) B ) )
63		iooretop 21779	. . . . . . . . . 10 |- ( -oo (,) B ) e. ( topGen ` ran (,) )
64	63	a1i 11	. . . . . . . . 9 |- ( ph -> ( -oo (,) B ) e. ( topGen ` ran (,) ) )
65		isopn3i 20091	. . . . . . . . 9 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( -oo (,) B ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) = ( -oo (,) B ) )
66	12, 64, 65	syl2anc 666	. . . . . . . 8 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) = ( -oo (,) B ) )
67	62, 66	eleqtrrd 2531	. . . . . . 7 |- ( ph -> A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( -oo (,) B ) ) )
68	58, 67	sseldd 3432	. . . . . 6 |- ( ph -> A e. ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) )
69	4	leidd 10177	. . . . . . 7 |- ( ph -> A <_ A )
70	4, 5, 61	ltled 9780	. . . . . . 7 |- ( ph -> A <_ B )
71	4, 5, 4, 69, 70	eliccd 37595	. . . . . 6 |- ( ph -> A e. ( A [,] B ) )
72	68, 71	elind 3617	. . . . 5 |- ( ph -> A e. ( ( ( int ` ( topGen ` ran (,) ) ) ` ( ( A [,) B ) u. ( RR \ ( A [,] B ) ) ) ) i^i ( A [,] B ) ) )
73		icossicc 11718	. . . . . . 7 |- ( A [,) B ) C_ ( A [,] B )
74	73	a1i 11	. . . . . 6 |- ( ph -> ( A [,) B ) C_ ( A [,] B ) )
75		eqid 2450	. . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) )
76	18, 75	restntr 20191	. . . . . 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 1267	. . . . 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 2531	. . . 4 |- ( ph -> A e. ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
79		eqid 2450	. . . . . . . . 9 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
80	9, 79	rerest 21815	. . . . . . . 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 2456	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
83	82	fveq2d 5867	. . . . 5 |- ( ph -> ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) )
84	83	fveq1d 5865	. . . 4 |- ( ph -> ( ( int ` ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) = ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
85	78, 84	eleqtrd 2530	. . 3 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) ` ( A [,) B ) ) )
86	71	snssd 4116	. . . . . . . 8 |- ( ph -> { A } C_ ( A [,] B ) )
87		ssequn2 3606	. . . . . . . 8 |- ( { A } C_ ( A [,] B ) <-> ( ( A [,] B ) u. { A } ) = ( A [,] B ) )
88	86, 87	sylib 200	. . . . . . 7 |- ( ph -> ( ( A [,] B ) u. { A } ) = ( A [,] B ) )
89	88	eqcomd 2456	. . . . . 6 |- ( ph -> ( A [,] B ) = ( ( A [,] B ) u. { A } ) )
90	89	oveq2d 6304	. . . . 5 |- ( ph -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) )
91	90	fveq2d 5867	. . . 4 |- ( ph -> ( int ` ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) ) = ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) )
92		uncom 3577	. . . . 5 |- ( ( A (,) B ) u. { A } ) = ( { A } u. ( A (,) B ) )
93		snunioo 11755	. . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
94	41, 13, 61, 93	syl3anc 1267	. . . . 5 |- ( ph -> ( { A } u. ( A (,) B ) ) = ( A [,) B ) )
95	92, 94	syl5req 2497	. . . 4 |- ( ph -> ( A [,) B ) = ( ( A (,) B ) u. { A } ) )
96	91, 95	fveq12d 5869	. . 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 2530	. 2 |- ( ph -> A e. ( ( int ` ( ( TopOpen ` ℂfld ) ↾t ( ( A [,] B ) u. { A } ) ) ) ` ( ( A (,) B ) u. { A } ) ) )
98	1, 3, 8, 9, 10, 97	limcres 22834	1 |- ( ph -> ( ( F |` ( A (,) B ) ) lim CC A ) = ( F lim CC A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   \ cdif 3400   u. cun 3401   i^i cin 3402   C_ wss 3403   {csn 3967   U.cuni 4197   class class class wbr 4401   ran crn 4834   |` cres 4835   -->wf 5577   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   -oocmnf 9670   RR*cxr 9671   < clt 9672   <_ cle 9673   (,)cioo 11632   [,)cico 11634   [,]cicc 11635   ↾_t crest 15312   TopOpenctopn 15313   topGenctg 15329  ℂ_fldccnfld 18963   Topctop 19910   intcnt 20025   lim CC climc 22810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fi 7922  df-sup 7953  df-inf 7954  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ico 11638  df-icc 11639  df-fz 11782  df-seq 12211  df-exp 12270  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-plusg 15196  df-mulr 15197  df-starv 15198  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-rest 15314  df-topn 15315  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-cnp 20237  df-xms 21328  df-ms 21329  df-limc 22814

This theorem is referenced by:  cncfiooicclem1  37765  fourierdlem82  38046  fourierdlem93  38057  fourierdlem111  38075