Theorem fourierdlem109 31952

Description: The integral of a piecewise continuous periodic function F is unchanged if the domain is shifted by any value X. This lemma generalizes fourierdlem92 31935 where the integral was shifted by the exact period (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem109.a	|- ( ph -> A e. RR )
fourierdlem109.b	|- ( ph -> B e. RR )
fourierdlem109.t	|- T = ( B - A )
fourierdlem109.x	|- ( ph -> X e. RR )
fourierdlem109.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 ) ) ) } )
fourierdlem109.m	|- ( ph -> M e. NN )
fourierdlem109.q	|- ( ph -> Q e. ( P ` M ) )
fourierdlem109.f	|- ( ph -> F : RR --> CC )
fourierdlem109.fper	|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
fourierdlem109.fcn	|- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
fourierdlem109.r	|- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
fourierdlem109.l	|- ( ( ph /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
fourierdlem109.o	|- O = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
fourierdlem109.h	|- H = ( { ( A - X ) , ( B - X ) } u. { x e. ( ( A - X ) [,] ( B - X ) ) | E. k e. ZZ ( x + ( k x. T ) ) e. ran Q } )
fourierdlem109.n	|- N = ( ( # ` H ) - 1 )
fourierdlem109.16	|- S = ( iota f f Isom < , < ( ( 0 ... N ) , H ) )
fourierdlem109.17	|- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
fourierdlem109.18	|- J = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) )
fourierdlem109.19	|- I = ( x e. RR |-> sup ( { j e. ( 0..^ M ) | ( Q ` j ) <_ ( J ` ( E ` x ) ) } , RR , < ) )

Assertion

Ref	Expression
fourierdlem109	|- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )

Distinct variable groups:   A, f, j, k, y   A, i, x, j, k, y   A, m, p, i, j   B, f, j, k, y   B, i, x   B, m, p   f, E, j, k, y   i, E, x   i, F, j, x, y   f, H, y   x, H   f, I, k, y   i, I, x   i, J, j, x, y   x, L, y   i, M, x, y, j   m, M, p   f, N, j, k, y   i, N, x   m, N, p   Q, f, j, k, y   Q, i, x   Q, m, p   x, R, y   S, f, j, k, y   S, i, x   S, m, p   T, f, j, k, y   T, i, x   T, m, p   f, X, j, y   i, X, m, p   x, X   ph, f, j, k, y   ph, i, x

Allowed substitution hints:   ph( m, p)   P( x, y, f, i, j, k, m, p)   R( f, i, j, k, m, p)   E( m, p)   F( f, k, m, p)   H( i, j, k, m, p)   I( j, m, p)   J( f, k, m, p)   L( f, i, j, k, m, p)   M( f, k)   O( x, y, f, i, j, k, m, p)   X( k)

Proof of Theorem fourierdlem109

Step	Hyp	Ref	Expression
1		fourierdlem109.a	. . . 4 |- ( ph -> A e. RR )
2	1	adantr 465	. . 3 |- ( ( ph /\ 0 < X ) -> A e. RR )
3		fourierdlem109.b	. . . 4 |- ( ph -> B e. RR )
4	3	adantr 465	. . 3 |- ( ( ph /\ 0 < X ) -> B e. RR )
5		fourierdlem109.t	. . 3 |- T = ( B - A )
6		fourierdlem109.x	. . . . 5 |- ( ph -> X e. RR )
7	6	adantr 465	. . . 4 |- ( ( ph /\ 0 < X ) -> X e. RR )
8		simpr 461	. . . 4 |- ( ( ph /\ 0 < X ) -> 0 < X )
9	7, 8	elrpd 11265	. . 3 |- ( ( ph /\ 0 < X ) -> X e. RR+ )
10		fourierdlem109.p	. . 3 |- 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 ) ) ) } )
11		fourierdlem109.m	. . . 4 |- ( ph -> M e. NN )
12	11	adantr 465	. . 3 |- ( ( ph /\ 0 < X ) -> M e. NN )
13		fourierdlem109.q	. . . 4 |- ( ph -> Q e. ( P ` M ) )
14	13	adantr 465	. . 3 |- ( ( ph /\ 0 < X ) -> Q e. ( P ` M ) )
15		fourierdlem109.f	. . . 4 |- ( ph -> F : RR --> CC )
16	15	adantr 465	. . 3 |- ( ( ph /\ 0 < X ) -> F : RR --> CC )
17		fourierdlem109.fper	. . . 4 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
18	17	adantlr 714	. . 3 |- ( ( ( ph /\ 0 < X ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
19		fourierdlem109.fcn	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
20	19	adantlr 714	. . 3 |- ( ( ( ph /\ 0 < X ) /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
21		fourierdlem109.r	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
22	21	adantlr 714	. . 3 |- ( ( ( ph /\ 0 < X ) /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
23		fourierdlem109.l	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
24	23	adantlr 714	. . 3 |- ( ( ( ph /\ 0 < X ) /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
25	2, 4, 5, 9, 10, 12, 14, 16, 18, 20, 22, 24	fourierdlem108 31951	. 2 |- ( ( ph /\ 0 < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
26		oveq2 6289	. . . . . . 7 |- ( X = 0 -> ( A - X ) = ( A - 0 ) )
27	1	recnd 9625	. . . . . . . 8 |- ( ph -> A e. CC )
28	27	subid1d 9925	. . . . . . 7 |- ( ph -> ( A - 0 ) = A )
29	26, 28	sylan9eqr 2506	. . . . . 6 |- ( ( ph /\ X = 0 ) -> ( A - X ) = A )
30		oveq2 6289	. . . . . . 7 |- ( X = 0 -> ( B - X ) = ( B - 0 ) )
31	3	recnd 9625	. . . . . . . 8 |- ( ph -> B e. CC )
32	31	subid1d 9925	. . . . . . 7 |- ( ph -> ( B - 0 ) = B )
33	30, 32	sylan9eqr 2506	. . . . . 6 |- ( ( ph /\ X = 0 ) -> ( B - X ) = B )
34	29, 33	oveq12d 6299	. . . . 5 |- ( ( ph /\ X = 0 ) -> ( ( A - X ) [,] ( B - X ) ) = ( A [,] B ) )
35	34	itgeq1d 31709	. . . 4 |- ( ( ph /\ X = 0 ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
36	35	adantlr 714	. . 3 |- ( ( ( ph /\ -. 0 < X ) /\ X = 0 ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
37		simpll 753	. . . 4 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> ph )
38	37, 6	syl 16	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> X e. RR )
39		0red 9600	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> 0 e. RR )
40		simpr 461	. . . . . 6 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> -. X = 0 )
41	40	neqned 2646	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> X =/= 0 )
42		simplr 755	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> -. 0 < X )
43	38, 39, 41, 42	lttri5d 31453	. . . 4 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> X < 0 )
44	6	recnd 9625	. . . . . . . . . . . 12 |- ( ph -> X e. CC )
45	27, 44	subcld 9936	. . . . . . . . . . 11 |- ( ph -> ( A - X ) e. CC )
46	45, 44	subnegd 9943	. . . . . . . . . 10 |- ( ph -> ( ( A - X ) - -u X ) = ( ( A - X ) + X ) )
47	27, 44	npcand 9940	. . . . . . . . . 10 |- ( ph -> ( ( A - X ) + X ) = A )
48	46, 47	eqtrd 2484	. . . . . . . . 9 |- ( ph -> ( ( A - X ) - -u X ) = A )
49	31, 44	subcld 9936	. . . . . . . . . . 11 |- ( ph -> ( B - X ) e. CC )
50	49, 44	subnegd 9943	. . . . . . . . . 10 |- ( ph -> ( ( B - X ) - -u X ) = ( ( B - X ) + X ) )
51	31, 44	npcand 9940	. . . . . . . . . 10 |- ( ph -> ( ( B - X ) + X ) = B )
52	50, 51	eqtrd 2484	. . . . . . . . 9 |- ( ph -> ( ( B - X ) - -u X ) = B )
53	48, 52	oveq12d 6299	. . . . . . . 8 |- ( ph -> ( ( ( A - X ) - -u X ) [,] ( ( B - X ) - -u X ) ) = ( A [,] B ) )
54	53	eqcomd 2451	. . . . . . 7 |- ( ph -> ( A [,] B ) = ( ( ( A - X ) - -u X ) [,] ( ( B - X ) - -u X ) ) )
55	54	itgeq1d 31709	. . . . . 6 |- ( ph -> S. ( A [,] B ) ( F ` x ) _d x = S. ( ( ( A - X ) - -u X ) [,] ( ( B - X ) - -u X ) ) ( F ` x ) _d x )
56	55	adantr 465	. . . . 5 |- ( ( ph /\ X < 0 ) -> S. ( A [,] B ) ( F ` x ) _d x = S. ( ( ( A - X ) - -u X ) [,] ( ( B - X ) - -u X ) ) ( F ` x ) _d x )
57	1, 6	resubcld 9994	. . . . . . 7 |- ( ph -> ( A - X ) e. RR )
58	57	adantr 465	. . . . . 6 |- ( ( ph /\ X < 0 ) -> ( A - X ) e. RR )
59	3, 6	resubcld 9994	. . . . . . 7 |- ( ph -> ( B - X ) e. RR )
60	59	adantr 465	. . . . . 6 |- ( ( ph /\ X < 0 ) -> ( B - X ) e. RR )
61		eqid 2443	. . . . . 6 |- ( ( B - X ) - ( A - X ) ) = ( ( B - X ) - ( A - X ) )
62	6	renegcld 9993	. . . . . . . 8 |- ( ph -> -u X e. RR )
63	62	adantr 465	. . . . . . 7 |- ( ( ph /\ X < 0 ) -> -u X e. RR )
64	6	lt0neg1d 10129	. . . . . . . 8 |- ( ph -> ( X < 0 <-> 0 < -u X ) )
65	64	biimpa 484	. . . . . . 7 |- ( ( ph /\ X < 0 ) -> 0 < -u X )
66	63, 65	elrpd 11265	. . . . . 6 |- ( ( ph /\ X < 0 ) -> -u X e. RR+ )
67		fourierdlem109.o	. . . . . . 7 |- O = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
68		fveq2 5856	. . . . . . . . . . . . 13 |- ( i = j -> ( p ` i ) = ( p ` j ) )
69		oveq1 6288	. . . . . . . . . . . . . 14 |- ( i = j -> ( i + 1 ) = ( j + 1 ) )
70	69	fveq2d 5860	. . . . . . . . . . . . 13 |- ( i = j -> ( p ` ( i + 1 ) ) = ( p ` ( j + 1 ) ) )
71	68, 70	breq12d 4450	. . . . . . . . . . . 12 |- ( i = j -> ( ( p ` i ) < ( p ` ( i + 1 ) ) <-> ( p ` j ) < ( p ` ( j + 1 ) ) ) )
72	71	cbvralv 3070	. . . . . . . . . . 11 |- ( A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) )
73	72	anbi2i 694	. . . . . . . . . 10 |- ( ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) <-> ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) )
74	73	a1i 11	. . . . . . . . 9 |- ( p e. ( RR ^m ( 0 ... m ) ) -> ( ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) <-> ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) ) )
75	74	rabbiia 3084	. . . . . . . 8 |- { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } = { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) }
76	75	mpteq2i 4520	. . . . . . 7 |- ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ 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 ) = ( B - X ) ) /\ A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) } )
77	67, 76	eqtri 2472	. . . . . 6 |- O = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = ( B - X ) ) /\ A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) } )
78	10, 11, 13	fourierdlem11 31854	. . . . . . . . . . . 12 |- ( ph -> ( A e. RR /\ B e. RR /\ A < B ) )
79	78	simp3d 1011	. . . . . . . . . . 11 |- ( ph -> A < B )
80	1, 3, 6, 79	ltsub1dd 10171	. . . . . . . . . 10 |- ( ph -> ( A - X ) < ( B - X ) )
81		fourierdlem109.h	. . . . . . . . . 10 |- H = ( { ( A - X ) , ( B - X ) } u. { x e. ( ( A - X ) [,] ( B - X ) ) | E. k e. ZZ ( x + ( k x. T ) ) e. ran Q } )
82		fourierdlem109.n	. . . . . . . . . 10 |- N = ( ( # ` H ) - 1 )
83		fourierdlem109.16	. . . . . . . . . 10 |- S = ( iota f f Isom < , < ( ( 0 ... N ) , H ) )
84	5, 10, 11, 13, 57, 59, 80, 67, 81, 82, 83	fourierdlem54 31897	. . . . . . . . 9 |- ( ph -> ( ( N e. NN /\ S e. ( O ` N ) ) /\ S Isom < , < ( ( 0 ... N ) , H ) ) )
85	84	simpld 459	. . . . . . . 8 |- ( ph -> ( N e. NN /\ S e. ( O ` N ) ) )
86	85	simpld 459	. . . . . . 7 |- ( ph -> N e. NN )
87	86	adantr 465	. . . . . 6 |- ( ( ph /\ X < 0 ) -> N e. NN )
88	85	simprd 463	. . . . . . 7 |- ( ph -> S e. ( O ` N ) )
89	88	adantr 465	. . . . . 6 |- ( ( ph /\ X < 0 ) -> S e. ( O ` N ) )
90	15	adantr 465	. . . . . 6 |- ( ( ph /\ X < 0 ) -> F : RR --> CC )
91	31, 27, 44	nnncan2d 9971	. . . . . . . . . . . 12 |- ( ph -> ( ( B - X ) - ( A - X ) ) = ( B - A ) )
92	91, 5	syl6eqr 2502	. . . . . . . . . . 11 |- ( ph -> ( ( B - X ) - ( A - X ) ) = T )
93	92	oveq2d 6297	. . . . . . . . . 10 |- ( ph -> ( x + ( ( B - X ) - ( A - X ) ) ) = ( x + T ) )
94	93	adantr 465	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> ( x + ( ( B - X ) - ( A - X ) ) ) = ( x + T ) )
95	94	fveq2d 5860	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + ( ( B - X ) - ( A - X ) ) ) ) = ( F ` ( x + T ) ) )
96	95, 17	eqtrd 2484	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + ( ( B - X ) - ( A - X ) ) ) ) = ( F ` x ) )
97	96	adantlr 714	. . . . . 6 |- ( ( ( ph /\ X < 0 ) /\ x e. RR ) -> ( F ` ( x + ( ( B - X ) - ( A - X ) ) ) ) = ( F ` x ) )
98	11	adantr 465	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> M e. NN )
99	13	adantr 465	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> Q e. ( P ` M ) )
100	15	adantr 465	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> F : RR --> CC )
101	17	adantlr 714	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
102	19	adantlr 714	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
103	57	adantr 465	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( A - X ) e. RR )
104	57	rexrd 9646	. . . . . . . . . 10 |- ( ph -> ( A - X ) e. RR* )
105		pnfxr 11332	. . . . . . . . . . 11 |- +oo e. RR*
106	105	a1i 11	. . . . . . . . . 10 |- ( ph -> +oo e. RR* )
107	59	ltpnfd 31434	. . . . . . . . . 10 |- ( ph -> ( B - X ) < +oo )
108	104, 106, 59, 80, 107	eliood 31485	. . . . . . . . 9 |- ( ph -> ( B - X ) e. ( ( A - X ) (,) +oo ) )
109	108	adantr 465	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( B - X ) e. ( ( A - X ) (,) +oo ) )
110		oveq1 6288	. . . . . . . . . . . . 13 |- ( x = y -> ( x + ( k x. T ) ) = ( y + ( k x. T ) ) )
111	110	eleq1d 2512	. . . . . . . . . . . 12 |- ( x = y -> ( ( x + ( k x. T ) ) e. ran Q <-> ( y + ( k x. T ) ) e. ran Q ) )
112	111	rexbidv 2954	. . . . . . . . . . 11 |- ( x = y -> ( E. k e. ZZ ( x + ( k x. T ) ) e. ran Q <-> E. k e. ZZ ( y + ( k x. T ) ) e. ran Q ) )
113	112	cbvrabv 3094	. . . . . . . . . 10 |- { x e. ( ( A - X ) [,] ( B - X ) ) | E. k e. ZZ ( x + ( k x. T ) ) e. ran Q } = { y e. ( ( A - X ) [,] ( B - X ) ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q }
114	113	uneq2i 3640	. . . . . . . . 9 |- ( { ( A - X ) , ( B - X ) } u. { x e. ( ( A - X ) [,] ( B - X ) ) | E. k e. ZZ ( x + ( k x. T ) ) e. ran Q } ) = ( { ( A - X ) , ( B - X ) } u. { y e. ( ( A - X ) [,] ( B - X ) ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q } )
115	81, 114	eqtri 2472	. . . . . . . 8 |- H = ( { ( A - X ) , ( B - X ) } u. { y e. ( ( A - X ) [,] ( B - X ) ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q } )
116		fourierdlem109.17	. . . . . . . 8 |- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
117		fourierdlem109.18	. . . . . . . 8 |- J = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) )
118		simpr 461	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> j e. ( 0..^ N ) )
119		eqid 2443	. . . . . . . 8 |- ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) = ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) )
120		eqid 2443	. . . . . . . 8 |- ( F |` ( ( J ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) = ( F |` ( ( J ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) )
121		eqid 2443	. . . . . . . 8 |- ( y e. ( ( ( J ` ( E ` ( S ` j ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) (,) ( ( E ` ( S ` ( j + 1 ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) |-> ( ( F |` ( ( J ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) ` ( y - ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) ) = ( y e. ( ( ( J ` ( E ` ( S ` j ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) (,) ( ( E ` ( S ` ( j + 1 ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) |-> ( ( F |` ( ( J ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) ` ( y - ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) )
122		fourierdlem109.19	. . . . . . . . 9 |- I = ( x e. RR |-> sup ( { j e. ( 0..^ M ) | ( Q ` j ) <_ ( J ` ( E ` x ) ) } , RR , < ) )
123		fveq2 5856	. . . . . . . . . . . . 13 |- ( j = i -> ( Q ` j ) = ( Q ` i ) )
124	123	breq1d 4447	. . . . . . . . . . . 12 |- ( j = i -> ( ( Q ` j ) <_ ( J ` ( E ` x ) ) <-> ( Q ` i ) <_ ( J ` ( E ` x ) ) ) )
125	124	cbvrabv 3094	. . . . . . . . . . 11 |- { j e. ( 0..^ M ) | ( Q ` j ) <_ ( J ` ( E ` x ) ) } = { i e. ( 0..^ M ) | ( Q ` i ) <_ ( J ` ( E ` x ) ) }
126	125	supeq1i 7909	. . . . . . . . . 10 |- sup ( { j e. ( 0..^ M ) | ( Q ` j ) <_ ( J ` ( E ` x ) ) } , RR , < ) = sup ( { i e. ( 0..^ M ) | ( Q ` i ) <_ ( J ` ( E ` x ) ) } , RR , < )
127	126	mpteq2i 4520	. . . . . . . . 9 |- ( x e. RR |-> sup ( { j e. ( 0..^ M ) | ( Q ` j ) <_ ( J ` ( E ` x ) ) } , RR , < ) ) = ( x e. RR |-> sup ( { i e. ( 0..^ M ) | ( Q ` i ) <_ ( J ` ( E ` x ) ) } , RR , < ) )
128	122, 127	eqtri 2472	. . . . . . . 8 |- I = ( x e. RR |-> sup ( { i e. ( 0..^ M ) | ( Q ` i ) <_ ( J ` ( E ` x ) ) } , RR , < ) )
129	10, 5, 98, 99, 100, 101, 102, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 120, 121, 128	fourierdlem90 31933	. . . . . . 7 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) e. ( ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) -cn-> CC ) )
130	129	adantlr 714	. . . . . 6 |- ( ( ( ph /\ X < 0 ) /\ j e. ( 0..^ N ) ) -> ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) e. ( ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) -cn-> CC ) )
131	21	adantlr 714	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
132		eqid 2443	. . . . . . . 8 |- ( i e. ( 0..^ M ) |-> R ) = ( i e. ( 0..^ M ) |-> R )
133	10, 5, 98, 99, 100, 101, 102, 131, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 132	fourierdlem89 31932	. . . . . . 7 |- ( ( ph /\ j e. ( 0..^ N ) ) -> if ( ( J ` ( E ` ( S ` j ) ) ) = ( Q ` ( I ` ( S ` j ) ) ) , ( ( i e. ( 0..^ M ) |-> R ) ` ( I ` ( S ` j ) ) ) , ( F ` ( J ` ( E ` ( S ` j ) ) ) ) ) e. ( ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) lim CC ( S ` j ) ) )
134	133	adantlr 714	. . . . . 6 |- ( ( ( ph /\ X < 0 ) /\ j e. ( 0..^ N ) ) -> if ( ( J ` ( E ` ( S ` j ) ) ) = ( Q ` ( I ` ( S ` j ) ) ) , ( ( i e. ( 0..^ M ) |-> R ) ` ( I ` ( S ` j ) ) ) , ( F ` ( J ` ( E ` ( S ` j ) ) ) ) ) e. ( ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) lim CC ( S ` j ) ) )
135	23	adantlr 714	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
136		eqid 2443	. . . . . . . 8 |- ( i e. ( 0..^ M ) |-> L ) = ( i e. ( 0..^ M ) |-> L )
137	10, 5, 98, 99, 100, 101, 102, 135, 103, 109, 67, 115, 82, 83, 116, 117, 118, 119, 128, 136	fourierdlem91 31934	. . . . . . 7 |- ( ( ph /\ j e. ( 0..^ N ) ) -> if ( ( E ` ( S ` ( j + 1 ) ) ) = ( Q ` ( ( I ` ( S ` j ) ) + 1 ) ) , ( ( i e. ( 0..^ M ) |-> L ) ` ( I ` ( S ` j ) ) ) , ( F ` ( E ` ( S ` ( j + 1 ) ) ) ) ) e. ( ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) lim CC ( S ` ( j + 1 ) ) ) )
138	137	adantlr 714	. . . . . 6 |- ( ( ( ph /\ X < 0 ) /\ j e. ( 0..^ N ) ) -> if ( ( E ` ( S ` ( j + 1 ) ) ) = ( Q ` ( ( I ` ( S ` j ) ) + 1 ) ) , ( ( i e. ( 0..^ M ) |-> L ) ` ( I ` ( S ` j ) ) ) , ( F ` ( E ` ( S ` ( j + 1 ) ) ) ) ) e. ( ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) lim CC ( S ` ( j + 1 ) ) ) )
139	58, 60, 61, 66, 77, 87, 89, 90, 97, 130, 134, 138	fourierdlem108 31951	. . . . 5 |- ( ( ph /\ X < 0 ) -> S. ( ( ( A - X ) - -u X ) [,] ( ( B - X ) - -u X ) ) ( F ` x ) _d x = S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x )
140	56, 139	eqtr2d 2485	. . . 4 |- ( ( ph /\ X < 0 ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
141	37, 43, 140	syl2anc 661	. . 3 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
142	36, 141	pm2.61dan 791	. 2 |- ( ( ph /\ -. 0 < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
143	25, 142	pm2.61dan 791	1 |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   A.wral 2793   E.wrex 2794   {crab 2797   u. cun 3459   ifcif 3926   {cpr 4016   class class class wbr 4437   |-> cmpt 4495   ran crn 4990   |` cres 4991   iotacio 5539   -->wf 5574   ` cfv 5578   Isom wiso 5579  (class class class)co 6281   ^m cmap 7422   supcsup 7902   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   + caddc 9498   x. cmul 9500   +oocpnf 9628   RR*cxr 9630   < clt 9631   <_ cle 9632   - cmin 9810   -ucneg 9811   / cdiv 10213   NNcn 10543   ZZcz 10871   (,)cioo 11540   (,]cioc 11541   [,]cicc 11543   ...cfz 11683  ..^cfzo 11806   |_cfl 11909   #chash 12387   -cn->ccncf 21358   S.citg 22005   lim CC climc 22244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cc 8818  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-cmp 19865  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-ovol 21854  df-vol 21855  df-mbf 22006  df-itg1 22007  df-itg2 22008  df-ibl 22009  df-itg 22010  df-0p 22055  df-ditg 22229  df-limc 22248  df-dv 22249

This theorem is referenced by:  fourierdlem110  31953