Theorem fourierdlem108 32200

Description: The integral of a piecewise continuous periodic function F is unchanged if the domain is shifted by any positive value X. This lemma generalizes fourierdlem92 32184 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem108.a	|- ( ph -> A e. RR )
fourierdlem108.b	|- ( ph -> B e. RR )
fourierdlem108.t	|- T = ( B - A )
fourierdlem108.x	|- ( ph -> X e. RR+ )
fourierdlem108.p	|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
fourierdlem108.m	|- ( ph -> M e. NN )
fourierdlem108.q	|- ( ph -> Q e. ( P ` M ) )
fourierdlem108.f	|- ( ph -> F : RR --> CC )
fourierdlem108.fper	|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
fourierdlem108.fcn	|- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
fourierdlem108.r	|- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
fourierdlem108.l	|- ( ( ph /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )

Assertion

Ref	Expression
fourierdlem108	|- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )

Distinct variable groups:   A, i, x   A, m, p, i   B, i, x   B, m, p   i, F, x   x, L   i, M, x   m, M, p   Q, i, x   Q, m, p   x, R   T, i, x   T, m, p   i, X, x   m, X, p   ph, i, x

Allowed substitution hints:   ph( m, p)   P( x, i, m, p)   R( i, m, p)   F( m, p)   L( i, m, p)

Proof of Theorem fourierdlem108

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

Step	Hyp	Ref	Expression
1		fourierdlem108.a	. 2 |- ( ph -> A e. RR )
2		fourierdlem108.b	. 2 |- ( ph -> B e. RR )
3		fourierdlem108.t	. 2 |- T = ( B - A )
4		fourierdlem108.x	. 2 |- ( ph -> X e. RR+ )
5		fourierdlem108.p	. 2 |- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
6		fourierdlem108.m	. 2 |- ( ph -> M e. NN )
7		fourierdlem108.q	. 2 |- ( ph -> Q e. ( P ` M ) )
8		fourierdlem108.f	. 2 |- ( ph -> F : RR --> CC )
9		fourierdlem108.fper	. 2 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
10		fourierdlem108.fcn	. 2 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
11		fourierdlem108.r	. 2 |- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
12		fourierdlem108.l	. 2 |- ( ( ph /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
13		eqid 2457	. 2 |- ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = A ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } ) = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = A ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
14		oveq1 6303	. . . . . 6 |- ( w = y -> ( w + ( k x. T ) ) = ( y + ( k x. T ) ) )
15	14	eleq1d 2526	. . . . 5 |- ( w = y -> ( ( w + ( k x. T ) ) e. ran Q <-> ( y + ( k x. T ) ) e. ran Q ) )
16	15	rexbidv 2968	. . . 4 |- ( w = y -> ( E. k e. ZZ ( w + ( k x. T ) ) e. ran Q <-> E. k e. ZZ ( y + ( k x. T ) ) e. ran Q ) )
17	16	cbvrabv 3108	. . 3 |- { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } = { y e. ( ( A - X ) [,] A ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q }
18	17	uneq2i 3651	. 2 |- ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) = ( { ( A - X ) , A } u. { y e. ( ( A - X ) [,] A ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q } )
19		oveq1 6303	. . . . . . . . . 10 |- ( l = k -> ( l x. T ) = ( k x. T ) )
20	19	oveq2d 6312	. . . . . . . . 9 |- ( l = k -> ( w + ( l x. T ) ) = ( w + ( k x. T ) ) )
21	20	eleq1d 2526	. . . . . . . 8 |- ( l = k -> ( ( w + ( l x. T ) ) e. ran Q <-> ( w + ( k x. T ) ) e. ran Q ) )
22	21	cbvrexv 3085	. . . . . . 7 |- ( E. l e. ZZ ( w + ( l x. T ) ) e. ran Q <-> E. k e. ZZ ( w + ( k x. T ) ) e. ran Q )
23	22	rgenw 2818	. . . . . 6 |- A. w e. ( ( A - X ) [,] A ) ( E. l e. ZZ ( w + ( l x. T ) ) e. ran Q <-> E. k e. ZZ ( w + ( k x. T ) ) e. ran Q )
24		rabbi 3036	. . . . . 6 |- ( A. w e. ( ( A - X ) [,] A ) ( E. l e. ZZ ( w + ( l x. T ) ) e. ran Q <-> E. k e. ZZ ( w + ( k x. T ) ) e. ran Q ) <-> { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } = { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } )
25	23, 24	mpbi 208	. . . . 5 |- { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } = { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q }
26	25	uneq2i 3651	. . . 4 |- ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) = ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } )
27	26	fveq2i 5875	. . 3 |- ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) = ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) )
28	27	oveq1i 6306	. 2 |- ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) = ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) ) - 1 )
29		isoeq5 6220	. . . . 5 |- ( ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) = ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) -> ( g Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) <-> g Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) ) ) )
30	26, 29	ax-mp 5	. . . 4 |- ( g Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) <-> g Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) ) )
31		isoeq1 6216	. . . 4 |- ( g = f -> ( g Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) ) <-> f Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) ) ) )
32	30, 31	syl5bb 257	. . 3 |- ( g = f -> ( g Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) <-> f Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) ) ) )
33	32	cbviotav 5563	. 2 |- ( iota g g Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) ) = ( iota f f Isom < , < ( ( 0 ... ( ( # ` ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. l e. ZZ ( w + ( l x. T ) ) e. ran Q } ) ) - 1 ) ) , ( { ( A - X ) , A } u. { w e. ( ( A - X ) [,] A ) | E. k e. ZZ ( w + ( k x. T ) ) e. ran Q } ) ) )
34		id 22	. . . 4 |- ( w = x -> w = x )
35		oveq2 6304	. . . . . . 7 |- ( w = x -> ( B - w ) = ( B - x ) )
36	35	oveq1d 6311	. . . . . 6 |- ( w = x -> ( ( B - w ) / T ) = ( ( B - x ) / T ) )
37	36	fveq2d 5876	. . . . 5 |- ( w = x -> ( |_ ` ( ( B - w ) / T ) ) = ( |_ ` ( ( B - x ) / T ) ) )
38	37	oveq1d 6311	. . . 4 |- ( w = x -> ( ( |_ ` ( ( B - w ) / T ) ) x. T ) = ( ( |_ ` ( ( B - x ) / T ) ) x. T ) )
39	34, 38	oveq12d 6314	. . 3 |- ( w = x -> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) = ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
40	39	cbvmptv 4548	. 2 |- ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
41		eqeq1 2461	. . . 4 |- ( w = y -> ( w = B <-> y = B ) )
42		id 22	. . . 4 |- ( w = y -> w = y )
43	41, 42	ifbieq2d 3969	. . 3 |- ( w = y -> if ( w = B , A , w ) = if ( y = B , A , y ) )
44	43	cbvmptv 4548	. 2 |- ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) )
45		fveq2 5872	. . . . . . . 8 |- ( z = x -> ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` z ) = ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) )
46	45	fveq2d 5876	. . . . . . 7 |- ( z = x -> ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` z ) ) = ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) )
47	46	breq2d 4468	. . . . . 6 |- ( z = x -> ( ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` z ) ) <-> ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) ) )
48	47	rabbidv 3101	. . . . 5 |- ( z = x -> { j e. ( 0..^ M ) | ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` z ) ) } = { j e. ( 0..^ M ) | ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) } )
49		fveq2 5872	. . . . . . 7 |- ( j = i -> ( Q ` j ) = ( Q ` i ) )
50	49	breq1d 4466	. . . . . 6 |- ( j = i -> ( ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) <-> ( Q ` i ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) ) )
51	50	cbvrabv 3108	. . . . 5 |- { j e. ( 0..^ M ) | ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) } = { i e. ( 0..^ M ) | ( Q ` i ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) }
52	48, 51	syl6eq 2514	. . . 4 |- ( z = x -> { j e. ( 0..^ M ) | ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` z ) ) } = { i e. ( 0..^ M ) | ( Q ` i ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) } )
53	52	supeq1d 7923	. . 3 |- ( z = x -> sup ( { j e. ( 0..^ M ) | ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` z ) ) } , RR , < ) = sup ( { i e. ( 0..^ M ) | ( Q ` i ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) } , RR , < ) )
54	53	cbvmptv 4548	. 2 |- ( z e. RR |-> sup ( { j e. ( 0..^ M ) | ( Q ` j ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` z ) ) } , RR , < ) ) = ( x e. RR |-> sup ( { i e. ( 0..^ M ) | ( Q ` i ) <_ ( ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) ` ( ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) ` x ) ) } , RR , < ) )
55	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 28, 33, 40, 44, 54	fourierdlem107 32199	1 |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   u. cun 3469   ifcif 3944   {cpr 4034   class class class wbr 4456   |-> cmpt 4515   ran crn 5009   |` cres 5010   iotacio 5555   -->wf 5590   ` cfv 5594   Isom wiso 5595  (class class class)co 6296   ^m cmap 7438   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   < clt 9645   <_ cle 9646   - cmin 9824   / cdiv 10227   NNcn 10556   ZZcz 10885   RR+crp 11245   (,)cioo 11554   (,]cioc 11555   [,]cicc 11557   ...cfz 11697  ..^cfzo 11821   |_cfl 11930   #chash 12408   -cn->ccncf 21506   S.citg 22153   lim CC climc 22392

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-inf2 8075  ax-cc 8832  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  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  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-disj 4428  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-se 4848  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-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  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-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-cmp 20014  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-ovol 22002  df-vol 22003  df-mbf 22154  df-itg1 22155  df-itg2 22156  df-ibl 22157  df-itg 22158  df-0p 22203  df-ditg 22377  df-limc 22396  df-dv 22397

This theorem is referenced by:  fourierdlem109  32201