Theorem fourierdlem115 40438

Description: Fourier serier convergence, for piecewise smooth functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem115.f	|- ( ph -> F : RR --> RR )
fourierdlem115.t	|- T = ( 2 x. pi )
fourierdlem115.per	|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
fourierdlem115.g	|- G = ( ( RR _D F ) |` ( -u pi (,) pi ) )
fourierdlem115.dmdv	|- ( ph -> ( ( -u pi (,) pi ) \ dom G ) e. Fin )
fourierdlem115.dvcn	|- ( ph -> G e. ( dom G -cn-> CC ) )
fourierdlem115.rlim	|- ( ( ph /\ x e. ( ( -u pi [,) pi ) \ dom G ) ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
fourierdlem115.llim	|- ( ( ph /\ x e. ( ( -u pi (,] pi ) \ dom G ) ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
fourierdlem115.x	|- ( ph -> X e. RR )
fourierdlem115.l	|- ( ph -> L e. ( ( F |` ( -oo (,) X ) ) lim CC X ) )
fourierdlem115.r	|- ( ph -> R e. ( ( F |` ( X (,) +oo ) ) lim CC X ) )
fourierdlem115.a	|- A = ( n e. NN0 |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x / pi ) )
fourierdlem115.b	|- B = ( n e. NN |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x / pi ) )
fourierdlem115.s	|- S = ( k e. NN |-> ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) )

Assertion

Ref	Expression
fourierdlem115	|- ( ph -> ( seq 1 ( + , S ) ~~> ( ( ( L + R ) / 2 ) - ( ( A ` 0 ) / 2 ) ) /\ ( ( ( A ` 0 ) / 2 ) + sum_ n e. NN ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( ( L + R ) / 2 ) ) )

Distinct variable groups:   x, k   A, k   B, k   k, F, n, x   k, G, x   k, L   R, k   T, k, x   k, X, n, x   ph, k, x

Allowed substitution hints:   ph( n)   A( x, n)   B( x, n)   R( x, n)   S( x, k, n)   T( n)   G( n)   L( x, n)

Proof of Theorem fourierdlem115

Dummy variables z f g w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem115.f	. . . 4 |- ( ph -> F : RR --> RR )
2		fourierdlem115.t	. . . 4 |- T = ( 2 x. pi )
3		fourierdlem115.per	. . . 4 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
4		fourierdlem115.g	. . . 4 |- G = ( ( RR _D F ) |` ( -u pi (,) pi ) )
5		fourierdlem115.dmdv	. . . 4 |- ( ph -> ( ( -u pi (,) pi ) \ dom G ) e. Fin )
6		fourierdlem115.dvcn	. . . 4 |- ( ph -> G e. ( dom G -cn-> CC ) )
7		fourierdlem115.rlim	. . . 4 |- ( ( ph /\ x e. ( ( -u pi [,) pi ) \ dom G ) ) -> ( ( G |` ( x (,) +oo ) ) lim CC x ) =/= (/) )
8		fourierdlem115.llim	. . . 4 |- ( ( ph /\ x e. ( ( -u pi (,] pi ) \ dom G ) ) -> ( ( G |` ( -oo (,) x ) ) lim CC x ) =/= (/) )
9		fourierdlem115.x	. . . 4 |- ( ph -> X e. RR )
10		fourierdlem115.l	. . . 4 |- ( ph -> L e. ( ( F |` ( -oo (,) X ) ) lim CC X ) )
11		fourierdlem115.r	. . . 4 |- ( ph -> R e. ( ( F |` ( X (,) +oo ) ) lim CC X ) )
12		fourierdlem115.a	. . . . 5 |- A = ( n e. NN0 |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x / pi ) )
13		oveq1 6657	. . . . . . . . . . 11 |- ( n = k -> ( n x. x ) = ( k x. x ) )
14	13	fveq2d 6195	. . . . . . . . . 10 |- ( n = k -> ( cos ` ( n x. x ) ) = ( cos ` ( k x. x ) ) )
15	14	oveq2d 6666	. . . . . . . . 9 |- ( n = k -> ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) = ( ( F ` x ) x. ( cos ` ( k x. x ) ) ) )
16	15	adantr 481	. . . . . . . 8 |- ( ( n = k /\ x e. ( -u pi (,) pi ) ) -> ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) = ( ( F ` x ) x. ( cos ` ( k x. x ) ) ) )
17	16	itgeq2dv 23548	. . . . . . 7 |- ( n = k -> S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x = S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( k x. x ) ) ) _d x )
18	17	oveq1d 6665	. . . . . 6 |- ( n = k -> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x / pi ) = ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( k x. x ) ) ) _d x / pi ) )
19	18	cbvmptv 4750	. . . . 5 |- ( n e. NN0 |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x / pi ) ) = ( k e. NN0 |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( k x. x ) ) ) _d x / pi ) )
20	12, 19	eqtri 2644	. . . 4 |- A = ( k e. NN0 |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( k x. x ) ) ) _d x / pi ) )
21		fourierdlem115.b	. . . . 5 |- B = ( n e. NN |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x / pi ) )
22	13	fveq2d 6195	. . . . . . . . . 10 |- ( n = k -> ( sin ` ( n x. x ) ) = ( sin ` ( k x. x ) ) )
23	22	oveq2d 6666	. . . . . . . . 9 |- ( n = k -> ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) = ( ( F ` x ) x. ( sin ` ( k x. x ) ) ) )
24	23	adantr 481	. . . . . . . 8 |- ( ( n = k /\ x e. ( -u pi (,) pi ) ) -> ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) = ( ( F ` x ) x. ( sin ` ( k x. x ) ) ) )
25	24	itgeq2dv 23548	. . . . . . 7 |- ( n = k -> S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x = S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( k x. x ) ) ) _d x )
26	25	oveq1d 6665	. . . . . 6 |- ( n = k -> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x / pi ) = ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( k x. x ) ) ) _d x / pi ) )
27	26	cbvmptv 4750	. . . . 5 |- ( n e. NN |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x / pi ) ) = ( k e. NN |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( k x. x ) ) ) _d x / pi ) )
28	21, 27	eqtri 2644	. . . 4 |- B = ( k e. NN |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( k x. x ) ) ) _d x / pi ) )
29		fourierdlem115.s	. . . 4 |- S = ( k e. NN |-> ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) )
30		eqid 2622	. . . 4 |- ( k e. NN |-> { w e. ( RR ^m ( 0 ... k ) ) | ( ( ( w ` 0 ) = -u pi /\ ( w ` k ) = pi ) /\ A. z e. ( 0..^ k ) ( w ` z ) < ( w ` ( z + 1 ) ) ) } ) = ( k e. NN |-> { w e. ( RR ^m ( 0 ... k ) ) | ( ( ( w ` 0 ) = -u pi /\ ( w ` k ) = pi ) /\ A. z e. ( 0..^ k ) ( w ` z ) < ( w ` ( z + 1 ) ) ) } )
31		id 22	. . . . . 6 |- ( y = x -> y = x )
32		oveq2 6658	. . . . . . . . 9 |- ( y = x -> ( pi - y ) = ( pi - x ) )
33	32	oveq1d 6665	. . . . . . . 8 |- ( y = x -> ( ( pi - y ) / T ) = ( ( pi - x ) / T ) )
34	33	fveq2d 6195	. . . . . . 7 |- ( y = x -> ( |_ ` ( ( pi - y ) / T ) ) = ( |_ ` ( ( pi - x ) / T ) ) )
35	34	oveq1d 6665	. . . . . 6 |- ( y = x -> ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) = ( ( |_ ` ( ( pi - x ) / T ) ) x. T ) )
36	31, 35	oveq12d 6668	. . . . 5 |- ( y = x -> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) = ( x + ( ( |_ ` ( ( pi - x ) / T ) ) x. T ) ) )
37	36	cbvmptv 4750	. . . 4 |- ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) = ( x e. RR |-> ( x + ( ( |_ ` ( ( pi - x ) / T ) ) x. T ) ) )
38		eqid 2622	. . . 4 |- ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) = ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) )
39		eqid 2622	. . . 4 |- ( ( # ` ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) - 1 ) = ( ( # ` ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) - 1 )
40		isoeq1 6567	. . . . 5 |- ( g = f -> ( g Isom < , < ( ( 0 ... ( ( # ` ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) - 1 ) ) , ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) <-> f Isom < , < ( ( 0 ... ( ( # ` ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) - 1 ) ) , ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) ) )
41	40	cbviotav 5857	. . . 4 |- ( iota g g Isom < , < ( ( 0 ... ( ( # ` ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) - 1 ) ) , ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) ) = ( iota f f Isom < , < ( ( 0 ... ( ( # ` ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) - 1 ) ) , ( { -u pi , pi , ( ( y e. RR |-> ( y + ( ( |_ ` ( ( pi - y ) / T ) ) x. T ) ) ) ` X ) } u. ( ( -u pi [,] pi ) \ dom G ) ) ) )
42	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 28, 29, 30, 37, 38, 39, 41	fourierdlem114 40437	. . 3 |- ( ph -> ( seq 1 ( + , S ) ~~> ( ( ( L + R ) / 2 ) - ( ( A ` 0 ) / 2 ) ) /\ ( ( ( A ` 0 ) / 2 ) + sum_ k e. NN ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) ) = ( ( L + R ) / 2 ) ) )
43	42	simpld 475	. 2 |- ( ph -> seq 1 ( + , S ) ~~> ( ( ( L + R ) / 2 ) - ( ( A ` 0 ) / 2 ) ) )
44		nfcv 2764	. . . . 5 |- F/_ k ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) )
45		nfmpt1 4747	. . . . . . . . 9 |- F/_ n ( n e. NN0 |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( cos ` ( n x. x ) ) ) _d x / pi ) )
46	12, 45	nfcxfr 2762	. . . . . . . 8 |- F/_ n A
47		nfcv 2764	. . . . . . . 8 |- F/_ n k
48	46, 47	nffv 6198	. . . . . . 7 |- F/_ n ( A ` k )
49		nfcv 2764	. . . . . . 7 |- F/_ n x.
50		nfcv 2764	. . . . . . 7 |- F/_ n ( cos ` ( k x. X ) )
51	48, 49, 50	nfov 6676	. . . . . 6 |- F/_ n ( ( A ` k ) x. ( cos ` ( k x. X ) ) )
52		nfcv 2764	. . . . . 6 |- F/_ n +
53		nfmpt1 4747	. . . . . . . . 9 |- F/_ n ( n e. NN |-> ( S. ( -u pi (,) pi ) ( ( F ` x ) x. ( sin ` ( n x. x ) ) ) _d x / pi ) )
54	21, 53	nfcxfr 2762	. . . . . . . 8 |- F/_ n B
55	54, 47	nffv 6198	. . . . . . 7 |- F/_ n ( B ` k )
56		nfcv 2764	. . . . . . 7 |- F/_ n ( sin ` ( k x. X ) )
57	55, 49, 56	nfov 6676	. . . . . 6 |- F/_ n ( ( B ` k ) x. ( sin ` ( k x. X ) ) )
58	51, 52, 57	nfov 6676	. . . . 5 |- F/_ n ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) )
59		fveq2 6191	. . . . . . 7 |- ( n = k -> ( A ` n ) = ( A ` k ) )
60		oveq1 6657	. . . . . . . 8 |- ( n = k -> ( n x. X ) = ( k x. X ) )
61	60	fveq2d 6195	. . . . . . 7 |- ( n = k -> ( cos ` ( n x. X ) ) = ( cos ` ( k x. X ) ) )
62	59, 61	oveq12d 6668	. . . . . 6 |- ( n = k -> ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) = ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) )
63		fveq2 6191	. . . . . . 7 |- ( n = k -> ( B ` n ) = ( B ` k ) )
64	60	fveq2d 6195	. . . . . . 7 |- ( n = k -> ( sin ` ( n x. X ) ) = ( sin ` ( k x. X ) ) )
65	63, 64	oveq12d 6668	. . . . . 6 |- ( n = k -> ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) = ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) )
66	62, 65	oveq12d 6668	. . . . 5 |- ( n = k -> ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) = ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) )
67	44, 58, 66	cbvsumi 14427	. . . 4 |- sum_ n e. NN ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) = sum_ k e. NN ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) )
68	67	oveq2i 6661	. . 3 |- ( ( ( A ` 0 ) / 2 ) + sum_ n e. NN ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( ( ( A ` 0 ) / 2 ) + sum_ k e. NN ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) )
69	42	simprd 479	. . 3 |- ( ph -> ( ( ( A ` 0 ) / 2 ) + sum_ k e. NN ( ( ( A ` k ) x. ( cos ` ( k x. X ) ) ) + ( ( B ` k ) x. ( sin ` ( k x. X ) ) ) ) ) = ( ( L + R ) / 2 ) )
70	68, 69	syl5eq 2668	. 2 |- ( ph -> ( ( ( A ` 0 ) / 2 ) + sum_ n e. NN ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( ( L + R ) / 2 ) )
71	43, 70	jca 554	1 |- ( ph -> ( seq 1 ( + , S ) ~~> ( ( ( L + R ) / 2 ) - ( ( A ` 0 ) / 2 ) ) /\ ( ( ( A ` 0 ) / 2 ) + sum_ n e. NN ( ( ( A ` n ) x. ( cos ` ( n x. X ) ) ) + ( ( B ` n ) x. ( sin ` ( n x. X ) ) ) ) ) = ( ( L + R ) / 2 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   \ cdif 3571   u. cun 3572   (/)c0 3915   {ctp 4181   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   iotacio 5849   -->wf 5884   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   ^m cmap 7857   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   < clt 10074   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   (,)cioo 12175   (,]cioc 12176   [,)cico 12177   [,]cicc 12178   ...cfz 12326  ..^cfzo 12465   |_cfl 12591   seqcseq 12801   #chash 13117   ~~> cli 14215   sum_csu 14416   sincsin 14794   cosccos 14795   picpi 14797   -cn->ccncf 22679   S.citg 23387   lim CC climc 23626   _D cdv 23627

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-cc 9257  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-disj 4621  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-ofr 6898  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-omul 7565  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-acn 8768  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-xnn0 11364  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-ovol 23233  df-vol 23234  df-mbf 23388  df-itg1 23389  df-itg2 23390  df-ibl 23391  df-itg 23392  df-0p 23437  df-ditg 23611  df-limc 23630  df-dv 23631

This theorem is referenced by:  fourierd  40439  fourierclimd  40440