Theorem fourierdlem109 40432

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 40415 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:   x, k   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 481	. . 3 |- ( ( ph /\ 0 < X ) -> A e. RR )
3		fourierdlem109.b	. . . 4 |- ( ph -> B e. RR )
4	3	adantr 481	. . 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 481	. . . 4 |- ( ( ph /\ 0 < X ) -> X e. RR )
8		simpr 477	. . . 4 |- ( ( ph /\ 0 < X ) -> 0 < X )
9	7, 8	elrpd 11869	. . 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 481	. . 3 |- ( ( ph /\ 0 < X ) -> M e. NN )
13		fourierdlem109.q	. . . 4 |- ( ph -> Q e. ( P ` M ) )
14	13	adantr 481	. . 3 |- ( ( ph /\ 0 < X ) -> Q e. ( P ` M ) )
15		fourierdlem109.f	. . . 4 |- ( ph -> F : RR --> CC )
16	15	adantr 481	. . 3 |- ( ( ph /\ 0 < X ) -> F : RR --> CC )
17		fourierdlem109.fper	. . . 4 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
18	17	adantlr 751	. . 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 751	. . 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 751	. . 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 751	. . 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 40431	. 2 |- ( ( ph /\ 0 < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
26		oveq2 6658	. . . . . . 7 |- ( X = 0 -> ( A - X ) = ( A - 0 ) )
27	1	recnd 10068	. . . . . . . 8 |- ( ph -> A e. CC )
28	27	subid1d 10381	. . . . . . 7 |- ( ph -> ( A - 0 ) = A )
29	26, 28	sylan9eqr 2678	. . . . . 6 |- ( ( ph /\ X = 0 ) -> ( A - X ) = A )
30		oveq2 6658	. . . . . . 7 |- ( X = 0 -> ( B - X ) = ( B - 0 ) )
31	3	recnd 10068	. . . . . . . 8 |- ( ph -> B e. CC )
32	31	subid1d 10381	. . . . . . 7 |- ( ph -> ( B - 0 ) = B )
33	30, 32	sylan9eqr 2678	. . . . . 6 |- ( ( ph /\ X = 0 ) -> ( B - X ) = B )
34	29, 33	oveq12d 6668	. . . . 5 |- ( ( ph /\ X = 0 ) -> ( ( A - X ) [,] ( B - X ) ) = ( A [,] B ) )
35	34	itgeq1d 40172	. . . 4 |- ( ( ph /\ X = 0 ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
36	35	adantlr 751	. . 3 |- ( ( ( ph /\ -. 0 < X ) /\ X = 0 ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
37		simpll 790	. . . 4 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> ph )
38	37, 6	syl 17	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> X e. RR )
39		0red 10041	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> 0 e. RR )
40		simpr 477	. . . . . 6 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> -. X = 0 )
41	40	neqned 2801	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> X =/= 0 )
42		simplr 792	. . . . 5 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> -. 0 < X )
43	38, 39, 41, 42	lttri5d 39513	. . . 4 |- ( ( ( ph /\ -. 0 < X ) /\ -. X = 0 ) -> X < 0 )
44	6	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> X e. CC )
45	27, 44	subcld 10392	. . . . . . . . . . 11 |- ( ph -> ( A - X ) e. CC )
46	45, 44	subnegd 10399	. . . . . . . . . 10 |- ( ph -> ( ( A - X ) - -u X ) = ( ( A - X ) + X ) )
47	27, 44	npcand 10396	. . . . . . . . . 10 |- ( ph -> ( ( A - X ) + X ) = A )
48	46, 47	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( A - X ) - -u X ) = A )
49	31, 44	subcld 10392	. . . . . . . . . . 11 |- ( ph -> ( B - X ) e. CC )
50	49, 44	subnegd 10399	. . . . . . . . . 10 |- ( ph -> ( ( B - X ) - -u X ) = ( ( B - X ) + X ) )
51	31, 44	npcand 10396	. . . . . . . . . 10 |- ( ph -> ( ( B - X ) + X ) = B )
52	50, 51	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( B - X ) - -u X ) = B )
53	48, 52	oveq12d 6668	. . . . . . . 8 |- ( ph -> ( ( ( A - X ) - -u X ) [,] ( ( B - X ) - -u X ) ) = ( A [,] B ) )
54	53	eqcomd 2628	. . . . . . 7 |- ( ph -> ( A [,] B ) = ( ( ( A - X ) - -u X ) [,] ( ( B - X ) - -u X ) ) )
55	54	itgeq1d 40172	. . . . . 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 481	. . . . 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 10458	. . . . . . 7 |- ( ph -> ( A - X ) e. RR )
58	57	adantr 481	. . . . . 6 |- ( ( ph /\ X < 0 ) -> ( A - X ) e. RR )
59	3, 6	resubcld 10458	. . . . . . 7 |- ( ph -> ( B - X ) e. RR )
60	59	adantr 481	. . . . . 6 |- ( ( ph /\ X < 0 ) -> ( B - X ) e. RR )
61		eqid 2622	. . . . . 6 |- ( ( B - X ) - ( A - X ) ) = ( ( B - X ) - ( A - X ) )
62	6	renegcld 10457	. . . . . . . 8 |- ( ph -> -u X e. RR )
63	62	adantr 481	. . . . . . 7 |- ( ( ph /\ X < 0 ) -> -u X e. RR )
64	6	lt0neg1d 10597	. . . . . . . 8 |- ( ph -> ( X < 0 <-> 0 < -u X ) )
65	64	biimpa 501	. . . . . . 7 |- ( ( ph /\ X < 0 ) -> 0 < -u X )
66	63, 65	elrpd 11869	. . . . . 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 6191	. . . . . . . . . . . . 13 |- ( i = j -> ( p ` i ) = ( p ` j ) )
69		oveq1 6657	. . . . . . . . . . . . . 14 |- ( i = j -> ( i + 1 ) = ( j + 1 ) )
70	69	fveq2d 6195	. . . . . . . . . . . . 13 |- ( i = j -> ( p ` ( i + 1 ) ) = ( p ` ( j + 1 ) ) )
71	68, 70	breq12d 4666	. . . . . . . . . . . 12 |- ( i = j -> ( ( p ` i ) < ( p ` ( i + 1 ) ) <-> ( p ` j ) < ( p ` ( j + 1 ) ) ) )
72	71	cbvralv 3171	. . . . . . . . . . 11 |- ( A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) )
73	72	anbi2i 730	. . . . . . . . . 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 3185	. . . . . . . 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 4741	. . . . . . 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 2644	. . . . . 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 40335	. . . . . . . . . . . 12 |- ( ph -> ( A e. RR /\ B e. RR /\ A < B ) )
79	78	simp3d 1075	. . . . . . . . . . 11 |- ( ph -> A < B )
80	1, 3, 6, 79	ltsub1dd 10639	. . . . . . . . . 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 40377	. . . . . . . . 9 |- ( ph -> ( ( N e. NN /\ S e. ( O ` N ) ) /\ S Isom < , < ( ( 0 ... N ) , H ) ) )
85	84	simpld 475	. . . . . . . 8 |- ( ph -> ( N e. NN /\ S e. ( O ` N ) ) )
86	85	simpld 475	. . . . . . 7 |- ( ph -> N e. NN )
87	86	adantr 481	. . . . . 6 |- ( ( ph /\ X < 0 ) -> N e. NN )
88	85	simprd 479	. . . . . . 7 |- ( ph -> S e. ( O ` N ) )
89	88	adantr 481	. . . . . 6 |- ( ( ph /\ X < 0 ) -> S e. ( O ` N ) )
90	15	adantr 481	. . . . . 6 |- ( ( ph /\ X < 0 ) -> F : RR --> CC )
91	31, 27, 44	nnncan2d 10427	. . . . . . . . . . . 12 |- ( ph -> ( ( B - X ) - ( A - X ) ) = ( B - A ) )
92	91, 5	syl6eqr 2674	. . . . . . . . . . 11 |- ( ph -> ( ( B - X ) - ( A - X ) ) = T )
93	92	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( x + ( ( B - X ) - ( A - X ) ) ) = ( x + T ) )
94	93	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x e. RR ) -> ( x + ( ( B - X ) - ( A - X ) ) ) = ( x + T ) )
95	94	fveq2d 6195	. . . . . . . 8 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + ( ( B - X ) - ( A - X ) ) ) ) = ( F ` ( x + T ) ) )
96	95, 17	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + ( ( B - X ) - ( A - X ) ) ) ) = ( F ` x ) )
97	96	adantlr 751	. . . . . 6 |- ( ( ( ph /\ X < 0 ) /\ x e. RR ) -> ( F ` ( x + ( ( B - X ) - ( A - X ) ) ) ) = ( F ` x ) )
98	11	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> M e. NN )
99	13	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> Q e. ( P ` M ) )
100	15	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> F : RR --> CC )
101	17	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
102	19	adantlr 751	. . . . . . . 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 481	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( A - X ) e. RR )
104	57	rexrd 10089	. . . . . . . . . 10 |- ( ph -> ( A - X ) e. RR* )
105		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
106	105	a1i 11	. . . . . . . . . 10 |- ( ph -> +oo e. RR* )
107	59	ltpnfd 11955	. . . . . . . . . 10 |- ( ph -> ( B - X ) < +oo )
108	104, 106, 59, 80, 107	eliood 39720	. . . . . . . . 9 |- ( ph -> ( B - X ) e. ( ( A - X ) (,) +oo ) )
109	108	adantr 481	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( B - X ) e. ( ( A - X ) (,) +oo ) )
110		oveq1 6657	. . . . . . . . . . . . 13 |- ( x = y -> ( x + ( k x. T ) ) = ( y + ( k x. T ) ) )
111	110	eleq1d 2686	. . . . . . . . . . . 12 |- ( x = y -> ( ( x + ( k x. T ) ) e. ran Q <-> ( y + ( k x. T ) ) e. ran Q ) )
112	111	rexbidv 3052	. . . . . . . . . . 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 3199	. . . . . . . . . 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 3764	. . . . . . . . 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 2644	. . . . . . . 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 477	. . . . . . . 8 |- ( ( ph /\ j e. ( 0..^ N ) ) -> j e. ( 0..^ N ) )
119		eqid 2622	. . . . . . . 8 |- ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) = ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) )
120		eqid 2622	. . . . . . . 8 |- ( F |` ( ( J ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) = ( F |` ( ( J ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) )
121		eqid 2622	. . . . . . . 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 6191	. . . . . . . . . . . . 13 |- ( j = i -> ( Q ` j ) = ( Q ` i ) )
124	123	breq1d 4663	. . . . . . . . . . . 12 |- ( j = i -> ( ( Q ` j ) <_ ( J ` ( E ` x ) ) <-> ( Q ` i ) <_ ( J ` ( E ` x ) ) ) )
125	124	cbvrabv 3199	. . . . . . . . . . 11 |- { j e. ( 0..^ M ) | ( Q ` j ) <_ ( J ` ( E ` x ) ) } = { i e. ( 0..^ M ) | ( Q ` i ) <_ ( J ` ( E ` x ) ) }
126	125	supeq1i 8353	. . . . . . . . . 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 4741	. . . . . . . . 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 2644	. . . . . . . 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 40413	. . . . . . 7 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) e. ( ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) -cn-> CC ) )
130	129	adantlr 751	. . . . . 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 751	. . . . . . . 8 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
132		eqid 2622	. . . . . . . 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 40412	. . . . . . 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 751	. . . . . 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 751	. . . . . . . 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 2622	. . . . . . . 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 40414	. . . . . . 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 751	. . . . . 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 40431	. . . . 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 2657	. . . 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 693	. . 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 832	. 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 832	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 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   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   ZZcz 11377   (,)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:  fourierdlem110  40433