Theorem dirkercncf 40324

Description: For any natural number N, the Dirichlet Kernel ( D ` N ) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
dirkercncf.d	|- D = ( n e. NN |-> ( y e. RR |-> if ( ( y mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) )

Assertion

Ref	Expression
dirkercncf	|- ( N e. NN -> ( D ` N ) e. ( RR -cn-> RR ) )

Distinct variable groups:   y, D   y, N   y, n

Allowed substitution hints:   D( n)   N( n)

Proof of Theorem dirkercncf

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dirkercncf.d	. . . 4 |- D = ( n e. NN |-> ( y e. RR |-> if ( ( y mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) )
2	1	dirkerf 40314	. . 3 |- ( N e. NN -> ( D ` N ) : RR --> RR )
3		ax-resscn 9993	. . . . . . . . . . 11 |- RR C_ CC
4	3	a1i 11	. . . . . . . . . 10 |- ( N e. NN -> RR C_ CC )
5	2, 4	fssd 6057	. . . . . . . . 9 |- ( N e. NN -> ( D ` N ) : RR --> CC )
6	5	ad2antrr 762	. . . . . . . 8 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( D ` N ) : RR --> CC )
7		oveq1 6657	. . . . . . . . . . . . . 14 |- ( y = w -> ( y mod ( 2 x. pi ) ) = ( w mod ( 2 x. pi ) ) )
8	7	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( y = w -> ( ( y mod ( 2 x. pi ) ) = 0 <-> ( w mod ( 2 x. pi ) ) = 0 ) )
9		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( y = w -> ( ( n + ( 1 / 2 ) ) x. y ) = ( ( n + ( 1 / 2 ) ) x. w ) )
10	9	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( y = w -> ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) = ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) )
11		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( y = w -> ( y / 2 ) = ( w / 2 ) )
12	11	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( y = w -> ( sin ` ( y / 2 ) ) = ( sin ` ( w / 2 ) ) )
13	12	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( y = w -> ( ( 2 x. pi ) x. ( sin ` ( y / 2 ) ) ) = ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) )
14	10, 13	oveq12d 6668	. . . . . . . . . . . . 13 |- ( y = w -> ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. pi ) x. ( sin ` ( y / 2 ) ) ) ) = ( ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) / ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) )
15	8, 14	ifbieq2d 4111	. . . . . . . . . . . 12 |- ( y = w -> if ( ( y mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. pi ) x. ( sin ` ( y / 2 ) ) ) ) ) = if ( ( w mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) / ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) ) )
16	15	cbvmptv 4750	. . . . . . . . . . 11 |- ( y e. RR |-> if ( ( y mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) = ( w e. RR |-> if ( ( w mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) / ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) ) )
17	16	mpteq2i 4741	. . . . . . . . . 10 |- ( n e. NN |-> ( y e. RR |-> if ( ( y mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. y ) ) / ( ( 2 x. pi ) x. ( sin ` ( y / 2 ) ) ) ) ) ) ) = ( n e. NN |-> ( w e. RR |-> if ( ( w mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) / ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) ) ) )
18	1, 17	eqtri 2644	. . . . . . . . 9 |- D = ( n e. NN |-> ( w e. RR |-> if ( ( w mod ( 2 x. pi ) ) = 0 , ( ( ( 2 x. n ) + 1 ) / ( 2 x. pi ) ) , ( ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) / ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) ) ) )
19		eqid 2622	. . . . . . . . 9 |- ( y - pi ) = ( y - pi )
20		eqid 2622	. . . . . . . . 9 |- ( y + pi ) = ( y + pi )
21		eqid 2622	. . . . . . . . 9 |- ( w e. ( ( y - pi ) (,) ( y + pi ) ) |-> ( ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) / ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) ) = ( w e. ( ( y - pi ) (,) ( y + pi ) ) |-> ( ( sin ` ( ( n + ( 1 / 2 ) ) x. w ) ) / ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) )
22		eqid 2622	. . . . . . . . 9 |- ( w e. ( ( y - pi ) (,) ( y + pi ) ) |-> ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) ) = ( w e. ( ( y - pi ) (,) ( y + pi ) ) |-> ( ( 2 x. pi ) x. ( sin ` ( w / 2 ) ) ) )
23		simpll 790	. . . . . . . . 9 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> N e. NN )
24		simplr 792	. . . . . . . . 9 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> y e. RR )
25		simpr 477	. . . . . . . . 9 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( y mod ( 2 x. pi ) ) = 0 )
26	18, 19, 20, 21, 22, 23, 24, 25	dirkercncflem3 40322	. . . . . . . 8 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( ( D ` N ) ` y ) e. ( ( D ` N ) lim CC y ) )
27	3	jctl 564	. . . . . . . . . 10 |- ( y e. RR -> ( RR C_ CC /\ y e. RR ) )
28	27	ad2antlr 763	. . . . . . . . 9 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( RR C_ CC /\ y e. RR ) )
29		eqid 2622	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
30	29	tgioo2 22606	. . . . . . . . . 10 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
31	29, 30	cnplimc 23651	. . . . . . . . 9 |- ( ( RR C_ CC /\ y e. RR ) -> ( ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( TopOpen ` ℂfld ) ) ` y ) <-> ( ( D ` N ) : RR --> CC /\ ( ( D ` N ) ` y ) e. ( ( D ` N ) lim CC y ) ) ) )
32	28, 31	syl 17	. . . . . . . 8 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( TopOpen ` ℂfld ) ) ` y ) <-> ( ( D ` N ) : RR --> CC /\ ( ( D ` N ) ` y ) e. ( ( D ` N ) lim CC y ) ) ) )
33	6, 26, 32	mpbir2and 957	. . . . . . 7 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( TopOpen ` ℂfld ) ) ` y ) )
34	29	cnfldtop 22587	. . . . . . . . 9 |- ( TopOpen ` ℂfld ) e. Top
35	34	a1i 11	. . . . . . . 8 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( TopOpen ` ℂfld ) e. Top )
36	2	ad2antrr 762	. . . . . . . 8 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( D ` N ) : RR --> RR )
37	3	a1i 11	. . . . . . . 8 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> RR C_ CC )
38		retopon 22567	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
39	38	toponunii 20721	. . . . . . . . 9 |- RR = U. ( topGen ` ran (,) )
40	29	cnfldtopon 22586	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
41	40	toponunii 20721	. . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
42	39, 41	cnprest2 21094	. . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( D ` N ) : RR --> RR /\ RR C_ CC ) -> ( ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( TopOpen ` ℂfld ) ) ` y ) <-> ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( ( TopOpen ` ℂfld ) ↾t RR ) ) ` y ) ) )
43	35, 36, 37, 42	syl3anc 1326	. . . . . . 7 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( TopOpen ` ℂfld ) ) ` y ) <-> ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( ( TopOpen ` ℂfld ) ↾t RR ) ) ` y ) ) )
44	33, 43	mpbid 222	. . . . . 6 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( ( TopOpen ` ℂfld ) ↾t RR ) ) ` y ) )
45	30	eqcomi 2631	. . . . . . . . 9 |- ( ( TopOpen ` ℂfld ) ↾t RR ) = ( topGen ` ran (,) )
46	45	a1i 11	. . . . . . . 8 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( ( TopOpen ` ℂfld ) ↾t RR ) = ( topGen ` ran (,) ) )
47	46	oveq2d 6666	. . . . . . 7 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( ( topGen ` ran (,) ) CnP ( ( TopOpen ` ℂfld ) ↾t RR ) ) = ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) )
48	47	fveq1d 6193	. . . . . 6 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( ( ( topGen ` ran (,) ) CnP ( ( TopOpen ` ℂfld ) ↾t RR ) ) ` y ) = ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` y ) )
49	44, 48	eleqtrd 2703	. . . . 5 |- ( ( ( N e. NN /\ y e. RR ) /\ ( y mod ( 2 x. pi ) ) = 0 ) -> ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` y ) )
50		simpll 790	. . . . . 6 |- ( ( ( N e. NN /\ y e. RR ) /\ -. ( y mod ( 2 x. pi ) ) = 0 ) -> N e. NN )
51		simplr 792	. . . . . 6 |- ( ( ( N e. NN /\ y e. RR ) /\ -. ( y mod ( 2 x. pi ) ) = 0 ) -> y e. RR )
52		neqne 2802	. . . . . . 7 |- ( -. ( y mod ( 2 x. pi ) ) = 0 -> ( y mod ( 2 x. pi ) ) =/= 0 )
53	52	adantl 482	. . . . . 6 |- ( ( ( N e. NN /\ y e. RR ) /\ -. ( y mod ( 2 x. pi ) ) = 0 ) -> ( y mod ( 2 x. pi ) ) =/= 0 )
54		eqid 2622	. . . . . 6 |- ( |_ ` ( y / ( 2 x. pi ) ) ) = ( |_ ` ( y / ( 2 x. pi ) ) )
55		eqid 2622	. . . . . 6 |- ( ( |_ ` ( y / ( 2 x. pi ) ) ) + 1 ) = ( ( |_ ` ( y / ( 2 x. pi ) ) ) + 1 )
56		eqid 2622	. . . . . 6 |- ( ( |_ ` ( y / ( 2 x. pi ) ) ) x. ( 2 x. pi ) ) = ( ( |_ ` ( y / ( 2 x. pi ) ) ) x. ( 2 x. pi ) )
57		eqid 2622	. . . . . 6 |- ( ( ( |_ ` ( y / ( 2 x. pi ) ) ) + 1 ) x. ( 2 x. pi ) ) = ( ( ( |_ ` ( y / ( 2 x. pi ) ) ) + 1 ) x. ( 2 x. pi ) )
58	18, 50, 51, 53, 54, 55, 56, 57	dirkercncflem4 40323	. . . . 5 |- ( ( ( N e. NN /\ y e. RR ) /\ -. ( y mod ( 2 x. pi ) ) = 0 ) -> ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` y ) )
59	49, 58	pm2.61dan 832	. . . 4 |- ( ( N e. NN /\ y e. RR ) -> ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` y ) )
60	59	ralrimiva 2966	. . 3 |- ( N e. NN -> A. y e. RR ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` y ) )
61		cncnp 21084	. . . 4 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ ( topGen ` ran (,) ) e. (TopOn ` RR ) ) -> ( ( D ` N ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) <-> ( ( D ` N ) : RR --> RR /\ A. y e. RR ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` y ) ) ) )
62	38, 38, 61	mp2an 708	. . 3 |- ( ( D ` N ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) <-> ( ( D ` N ) : RR --> RR /\ A. y e. RR ( D ` N ) e. ( ( ( topGen ` ran (,) ) CnP ( topGen ` ran (,) ) ) ` y ) ) )
63	2, 60, 62	sylanbrc 698	. 2 |- ( N e. NN -> ( D ` N ) e. ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
64	29, 30, 30	cncfcn 22712	. . 3 |- ( ( RR C_ CC /\ RR C_ CC ) -> ( RR -cn-> RR ) = ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) ) )
65	3, 3, 64	mp2an 708	. 2 |- ( RR -cn-> RR ) = ( ( topGen ` ran (,) ) Cn ( topGen ` ran (,) ) )
66	63, 65	syl6eleqr 2712	1 |- ( N e. NN -> ( D ` N ) e. ( RR -cn-> RR ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   ifcif 4086   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   (,)cioo 12175   |_cfl 12591   mod cmo 12668   sincsin 14794   picpi 14797   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   CnP ccnp 21029   -cn->ccncf 22679   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-inf2 8538  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  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  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-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  fourierdlem83  40406  fourierdlem95  40418  fourierdlem101  40424  fourierdlem103  40426  fourierdlem104  40427  fourierdlem111  40434  fourierdlem112  40435