Theorem fourierdlem108 40431

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 40415 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 k f g 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 2622	. 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 6657	. . . . . 6 |- ( w = y -> ( w + ( k x. T ) ) = ( y + ( k x. T ) ) )
15	14	eleq1d 2686	. . . . 5 |- ( w = y -> ( ( w + ( k x. T ) ) e. ran Q <-> ( y + ( k x. T ) ) e. ran Q ) )
16	15	rexbidv 3052	. . . 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 3199	. . 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 3764	. 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 6657	. . . . . . . . . 10 |- ( l = k -> ( l x. T ) = ( k x. T ) )
20	19	oveq2d 6666	. . . . . . . . 9 |- ( l = k -> ( w + ( l x. T ) ) = ( w + ( k x. T ) ) )
21	20	eleq1d 2686	. . . . . . . 8 |- ( l = k -> ( ( w + ( l x. T ) ) e. ran Q <-> ( w + ( k x. T ) ) e. ran Q ) )
22	21	cbvrexv 3172	. . . . . . 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 2924	. . . . . 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 3120	. . . . . 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 220	. . . . 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 3764	. . . 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 6194	. . 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 6660	. 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 6571	. . . . 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 6567	. . . 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 272	. . 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 5857	. 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 6658	. . . . . . 7 |- ( w = x -> ( B - w ) = ( B - x ) )
36	35	oveq1d 6665	. . . . . 6 |- ( w = x -> ( ( B - w ) / T ) = ( ( B - x ) / T ) )
37	36	fveq2d 6195	. . . . 5 |- ( w = x -> ( |_ ` ( ( B - w ) / T ) ) = ( |_ ` ( ( B - x ) / T ) ) )
38	37	oveq1d 6665	. . . 4 |- ( w = x -> ( ( |_ ` ( ( B - w ) / T ) ) x. T ) = ( ( |_ ` ( ( B - x ) / T ) ) x. T ) )
39	34, 38	oveq12d 6668	. . 3 |- ( w = x -> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) = ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
40	39	cbvmptv 4750	. 2 |- ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
41		eqeq1 2626	. . . 4 |- ( w = y -> ( w = B <-> y = B ) )
42		id 22	. . . 4 |- ( w = y -> w = y )
43	41, 42	ifbieq2d 4111	. . 3 |- ( w = y -> if ( w = B , A , w ) = if ( y = B , A , y ) )
44	43	cbvmptv 4750	. 2 |- ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) )
45		fveq2 6191	. . . . . . . 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 6195	. . . . . . 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 4665	. . . . . 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 3189	. . . . 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 6191	. . . . . . 7 |- ( j = i -> ( Q ` j ) = ( Q ` i ) )
50	49	breq1d 4663	. . . . . 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 3199	. . . . 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 2672	. . . 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 8352	. . 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 4750	. 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 40430	1 |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   u. cun 3572   ifcif 4086   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   iotacio 5849   -->wf 5884   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   ^m cmap 7857   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   RR+crp 11832   (,)cioo 12175   (,]cioc 12176   [,]cicc 12178   ...cfz 12326  ..^cfzo 12465   |_cfl 12591   #chash 13117   -cn->ccncf 22679   S.citg 23387   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-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-hash 13118  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-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-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:  fourierdlem109  40432