Theorem fourierdlem108 38190

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 38174 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 2471	. 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 6315	. . . . . 6 |- ( w = y -> ( w + ( k x. T ) ) = ( y + ( k x. T ) ) )
15	14	eleq1d 2533	. . . . 5 |- ( w = y -> ( ( w + ( k x. T ) ) e. ran Q <-> ( y + ( k x. T ) ) e. ran Q ) )
16	15	rexbidv 2892	. . . 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 3030	. . 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 3576	. 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 6315	. . . . . . . . . 10 |- ( l = k -> ( l x. T ) = ( k x. T ) )
20	19	oveq2d 6324	. . . . . . . . 9 |- ( l = k -> ( w + ( l x. T ) ) = ( w + ( k x. T ) ) )
21	20	eleq1d 2533	. . . . . . . 8 |- ( l = k -> ( ( w + ( l x. T ) ) e. ran Q <-> ( w + ( k x. T ) ) e. ran Q ) )
22	21	cbvrexv 3006	. . . . . . 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 2768	. . . . . 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 2955	. . . . . 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 213	. . . . 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 3576	. . . 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 5882	. . 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 6318	. 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 6232	. . . . 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 6228	. . . 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 265	. . 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 5559	. 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 6316	. . . . . . 7 |- ( w = x -> ( B - w ) = ( B - x ) )
36	35	oveq1d 6323	. . . . . 6 |- ( w = x -> ( ( B - w ) / T ) = ( ( B - x ) / T ) )
37	36	fveq2d 5883	. . . . 5 |- ( w = x -> ( |_ ` ( ( B - w ) / T ) ) = ( |_ ` ( ( B - x ) / T ) ) )
38	37	oveq1d 6323	. . . 4 |- ( w = x -> ( ( |_ ` ( ( B - w ) / T ) ) x. T ) = ( ( |_ ` ( ( B - x ) / T ) ) x. T ) )
39	34, 38	oveq12d 6326	. . 3 |- ( w = x -> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) = ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
40	39	cbvmptv 4488	. 2 |- ( w e. RR |-> ( w + ( ( |_ ` ( ( B - w ) / T ) ) x. T ) ) ) = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
41		eqeq1 2475	. . . 4 |- ( w = y -> ( w = B <-> y = B ) )
42		id 22	. . . 4 |- ( w = y -> w = y )
43	41, 42	ifbieq2d 3897	. . 3 |- ( w = y -> if ( w = B , A , w ) = if ( y = B , A , y ) )
44	43	cbvmptv 4488	. 2 |- ( w e. ( A (,] B ) |-> if ( w = B , A , w ) ) = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) )
45		fveq2 5879	. . . . . . . 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 5883	. . . . . . 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 4407	. . . . . 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 3022	. . . . 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 5879	. . . . . . 7 |- ( j = i -> ( Q ` j ) = ( Q ` i ) )
50	49	breq1d 4405	. . . . . 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 3030	. . . . 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 2521	. . . 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 7978	. . 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 4488	. 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 38189	1 |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   u. cun 3388   ifcif 3872   {cpr 3961   class class class wbr 4395   |-> cmpt 4454   ran crn 4840   |` cres 4841   iotacio 5551   -->wf 5585   ` cfv 5589   Isom wiso 5590  (class class class)co 6308   ^m cmap 7490   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   < clt 9693   <_ cle 9694   - cmin 9880   / cdiv 10291   NNcn 10631   ZZcz 10961   RR+crp 11325   (,)cioo 11660   (,]cioc 11661   [,]cicc 11663   ...cfz 11810  ..^cfzo 11942   |_cfl 12059   #chash 12553   -cn->ccncf 21986   S.citg 22655   lim CC climc 22896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-itg2 22658  df-ibl 22659  df-itg 22660  df-0p 22707  df-ditg 22881  df-limc 22900  df-dv 22901

This theorem is referenced by:  fourierdlem109  38191