Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem107.a |
|- ( ph -> A e. RR ) |
2 |
|
fourierdlem107.b |
|- ( ph -> B e. RR ) |
3 |
|
fourierdlem107.t |
|- T = ( B - A ) |
4 |
|
fourierdlem107.x |
|- ( ph -> X e. RR+ ) |
5 |
|
fourierdlem107.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 ) ) ) } ) |
6 |
|
fourierdlem107.m |
|- ( ph -> M e. NN ) |
7 |
|
fourierdlem107.q |
|- ( ph -> Q e. ( P ` M ) ) |
8 |
|
fourierdlem107.f |
|- ( ph -> F : RR --> CC ) |
9 |
|
fourierdlem107.fper |
|- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) ) |
10 |
|
fourierdlem107.fcn |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) ) |
11 |
|
fourierdlem107.r |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
12 |
|
fourierdlem107.l |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
13 |
|
fourierdlem107.o |
|- O = ( 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 |
|
fourierdlem107.h |
|- H = ( { ( A - X ) , A } u. { y e. ( ( A - X ) [,] A ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q } ) |
15 |
|
fourierdlem107.n |
|- N = ( ( # ` H ) - 1 ) |
16 |
|
fourierdlem107.s |
|- S = ( iota f f Isom < , < ( ( 0 ... N ) , H ) ) |
17 |
|
fourierdlem107.e |
|- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) ) |
18 |
|
fourierdlem107.z |
|- Z = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) ) |
19 |
|
fourierdlem107.i |
|- I = ( x e. RR |-> sup ( { i e. ( 0 ..^ M ) | ( Q ` i ) <_ ( Z ` ( E ` x ) ) } , RR , < ) ) |
20 |
3
|
oveq2i |
|- ( ( A - X ) + T ) = ( ( A - X ) + ( B - A ) ) |
21 |
1
|
recnd |
|- ( ph -> A e. CC ) |
22 |
4
|
rpred |
|- ( ph -> X e. RR ) |
23 |
22
|
recnd |
|- ( ph -> X e. CC ) |
24 |
2
|
recnd |
|- ( ph -> B e. CC ) |
25 |
21 23 24 21
|
subadd4b |
|- ( ph -> ( ( A - X ) + ( B - A ) ) = ( ( A - A ) + ( B - X ) ) ) |
26 |
20 25
|
syl5eq |
|- ( ph -> ( ( A - X ) + T ) = ( ( A - A ) + ( B - X ) ) ) |
27 |
21
|
subidd |
|- ( ph -> ( A - A ) = 0 ) |
28 |
27
|
oveq1d |
|- ( ph -> ( ( A - A ) + ( B - X ) ) = ( 0 + ( B - X ) ) ) |
29 |
2 22
|
resubcld |
|- ( ph -> ( B - X ) e. RR ) |
30 |
29
|
recnd |
|- ( ph -> ( B - X ) e. CC ) |
31 |
30
|
addid2d |
|- ( ph -> ( 0 + ( B - X ) ) = ( B - X ) ) |
32 |
26 28 31
|
3eqtrd |
|- ( ph -> ( ( A - X ) + T ) = ( B - X ) ) |
33 |
3
|
oveq2i |
|- ( A + T ) = ( A + ( B - A ) ) |
34 |
21 24
|
pncan3d |
|- ( ph -> ( A + ( B - A ) ) = B ) |
35 |
33 34
|
syl5eq |
|- ( ph -> ( A + T ) = B ) |
36 |
32 35
|
oveq12d |
|- ( ph -> ( ( ( A - X ) + T ) [,] ( A + T ) ) = ( ( B - X ) [,] B ) ) |
37 |
36
|
eqcomd |
|- ( ph -> ( ( B - X ) [,] B ) = ( ( ( A - X ) + T ) [,] ( A + T ) ) ) |
38 |
37
|
itgeq1d |
|- ( ph -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x = S. ( ( ( A - X ) + T ) [,] ( A + T ) ) ( F ` x ) _d x ) |
39 |
1 22
|
resubcld |
|- ( ph -> ( A - X ) e. RR ) |
40 |
|
fveq2 |
|- ( i = j -> ( p ` i ) = ( p ` j ) ) |
41 |
|
oveq1 |
|- ( i = j -> ( i + 1 ) = ( j + 1 ) ) |
42 |
41
|
fveq2d |
|- ( i = j -> ( p ` ( i + 1 ) ) = ( p ` ( j + 1 ) ) ) |
43 |
40 42
|
breq12d |
|- ( i = j -> ( ( p ` i ) < ( p ` ( i + 1 ) ) <-> ( p ` j ) < ( p ` ( j + 1 ) ) ) ) |
44 |
43
|
cbvralvw |
|- ( A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) |
45 |
44
|
a1i |
|- ( m e. NN -> ( A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) ) |
46 |
45
|
anbi2d |
|- ( m e. NN -> ( ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = A ) /\ A. i e. ( 0 ..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) <-> ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = A ) /\ A. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) ) ) |
47 |
46
|
rabbidv |
|- ( 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 ) ) ) } = { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = A ) /\ A. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) } ) |
48 |
47
|
mpteq2ia |
|- ( 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. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) } ) |
49 |
13 48
|
eqtri |
|- O = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A - X ) /\ ( p ` m ) = A ) /\ A. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) } ) |
50 |
1 4
|
ltsubrpd |
|- ( ph -> ( A - X ) < A ) |
51 |
3 5 6 7 39 1 50 13 14 15 16
|
fourierdlem54 |
|- ( ph -> ( ( N e. NN /\ S e. ( O ` N ) ) /\ S Isom < , < ( ( 0 ... N ) , H ) ) ) |
52 |
51
|
simpld |
|- ( ph -> ( N e. NN /\ S e. ( O ` N ) ) ) |
53 |
52
|
simpld |
|- ( ph -> N e. NN ) |
54 |
2 1
|
resubcld |
|- ( ph -> ( B - A ) e. RR ) |
55 |
3 54
|
eqeltrid |
|- ( ph -> T e. RR ) |
56 |
52
|
simprd |
|- ( ph -> S e. ( O ` N ) ) |
57 |
39
|
adantr |
|- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> ( A - X ) e. RR ) |
58 |
1
|
adantr |
|- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> A e. RR ) |
59 |
|
simpr |
|- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> x e. ( ( A - X ) [,] A ) ) |
60 |
|
eliccre |
|- ( ( ( A - X ) e. RR /\ A e. RR /\ x e. ( ( A - X ) [,] A ) ) -> x e. RR ) |
61 |
57 58 59 60
|
syl3anc |
|- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> x e. RR ) |
62 |
61 9
|
syldan |
|- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> ( F ` ( x + T ) ) = ( F ` x ) ) |
63 |
|
fveq2 |
|- ( i = j -> ( S ` i ) = ( S ` j ) ) |
64 |
63
|
oveq1d |
|- ( i = j -> ( ( S ` i ) + T ) = ( ( S ` j ) + T ) ) |
65 |
64
|
cbvmptv |
|- ( i e. ( 0 ... N ) |-> ( ( S ` i ) + T ) ) = ( j e. ( 0 ... N ) |-> ( ( S ` j ) + T ) ) |
66 |
|
eqid |
|- ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( ( A - X ) + T ) /\ ( p ` m ) = ( A + T ) ) /\ A. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) } ) = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( ( A - X ) + T ) /\ ( p ` m ) = ( A + T ) ) /\ A. j e. ( 0 ..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) } ) |
67 |
6
|
adantr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> M e. NN ) |
68 |
7
|
adantr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> Q e. ( P ` M ) ) |
69 |
8
|
adantr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> F : RR --> CC ) |
70 |
9
|
adantlr |
|- ( ( ( ph /\ j e. ( 0 ..^ N ) ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) ) |
71 |
10
|
adantlr |
|- ( ( ( ph /\ j e. ( 0 ..^ N ) ) /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) ) |
72 |
39
|
adantr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> ( A - X ) e. RR ) |
73 |
72
|
rexrd |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> ( A - X ) e. RR* ) |
74 |
|
pnfxr |
|- +oo e. RR* |
75 |
74
|
a1i |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> +oo e. RR* ) |
76 |
1
|
adantr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> A e. RR ) |
77 |
50
|
adantr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> ( A - X ) < A ) |
78 |
1
|
ltpnfd |
|- ( ph -> A < +oo ) |
79 |
78
|
adantr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> A < +oo ) |
80 |
73 75 76 77 79
|
eliood |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> A e. ( ( A - X ) (,) +oo ) ) |
81 |
|
simpr |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> j e. ( 0 ..^ N ) ) |
82 |
|
eqid |
|- ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) = ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) |
83 |
|
eqid |
|- ( F |` ( ( Z ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) = ( F |` ( ( Z ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) |
84 |
|
eqid |
|- ( y e. ( ( ( Z ` ( E ` ( S ` j ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) (,) ( ( E ` ( S ` ( j + 1 ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) |-> ( ( F |` ( ( Z ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) ` ( y - ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) ) = ( y e. ( ( ( Z ` ( E ` ( S ` j ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) (,) ( ( E ` ( S ` ( j + 1 ) ) ) + ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) |-> ( ( F |` ( ( Z ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) ` ( y - ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) ) ) ) |
85 |
5 3 67 68 69 70 71 72 80 13 14 15 16 17 18 81 82 83 84 19
|
fourierdlem90 |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) e. ( ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) -cn-> CC ) ) |
86 |
11
|
adantlr |
|- ( ( ( ph /\ j e. ( 0 ..^ N ) ) /\ i e. ( 0 ..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
87 |
|
eqid |
|- ( i e. ( 0 ..^ M ) |-> R ) = ( i e. ( 0 ..^ M ) |-> R ) |
88 |
5 3 67 68 69 70 71 86 72 80 13 14 15 16 17 18 81 82 19 87
|
fourierdlem89 |
|- ( ( ph /\ j e. ( 0 ..^ N ) ) -> if ( ( Z ` ( E ` ( S ` j ) ) ) = ( Q ` ( I ` ( S ` j ) ) ) , ( ( i e. ( 0 ..^ M ) |-> R ) ` ( I ` ( S ` j ) ) ) , ( F ` ( Z ` ( E ` ( S ` j ) ) ) ) ) e. ( ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) limCC ( S ` j ) ) ) |
89 |
12
|
adantlr |
|- ( ( ( ph /\ j e. ( 0 ..^ N ) ) /\ i e. ( 0 ..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
90 |
|
eqid |
|- ( i e. ( 0 ..^ M ) |-> L ) = ( i e. ( 0 ..^ M ) |-> L ) |
91 |
5 3 67 68 69 70 71 89 72 80 13 14 15 16 17 18 81 82 19 90
|
fourierdlem91 |
|- ( ( 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 ) ) ) ) limCC ( S ` ( j + 1 ) ) ) ) |
92 |
39 1 49 53 55 56 62 65 66 8 85 88 91
|
fourierdlem92 |
|- ( ph -> S. ( ( ( A - X ) + T ) [,] ( A + T ) ) ( F ` x ) _d x = S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) |
93 |
38 92
|
eqtrd |
|- ( ph -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x = S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) |
94 |
8
|
adantr |
|- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> F : RR --> CC ) |
95 |
29
|
adantr |
|- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> ( B - X ) e. RR ) |
96 |
2
|
adantr |
|- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> B e. RR ) |
97 |
|
simpr |
|- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> x e. ( ( B - X ) [,] B ) ) |
98 |
|
eliccre |
|- ( ( ( B - X ) e. RR /\ B e. RR /\ x e. ( ( B - X ) [,] B ) ) -> x e. RR ) |
99 |
95 96 97 98
|
syl3anc |
|- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> x e. RR ) |
100 |
94 99
|
ffvelrnd |
|- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> ( F ` x ) e. CC ) |
101 |
29
|
rexrd |
|- ( ph -> ( B - X ) e. RR* ) |
102 |
74
|
a1i |
|- ( ph -> +oo e. RR* ) |
103 |
2 4
|
ltsubrpd |
|- ( ph -> ( B - X ) < B ) |
104 |
2
|
ltpnfd |
|- ( ph -> B < +oo ) |
105 |
101 102 2 103 104
|
eliood |
|- ( ph -> B e. ( ( B - X ) (,) +oo ) ) |
106 |
5 3 6 7 8 9 10 11 12 29 105
|
fourierdlem105 |
|- ( ph -> ( x e. ( ( B - X ) [,] B ) |-> ( F ` x ) ) e. L^1 ) |
107 |
100 106
|
itgcl |
|- ( ph -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x e. CC ) |
108 |
93 107
|
eqeltrrd |
|- ( ph -> S. ( ( A - X ) [,] A ) ( F ` x ) _d x e. CC ) |
109 |
108
|
subidd |
|- ( ph -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) = 0 ) |
110 |
109
|
eqcomd |
|- ( ph -> 0 = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) |
111 |
110
|
adantr |
|- ( ( ph /\ T < X ) -> 0 = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) |
112 |
39
|
adantr |
|- ( ( ph /\ T < X ) -> ( A - X ) e. RR ) |
113 |
1
|
adantr |
|- ( ( ph /\ T < X ) -> A e. RR ) |
114 |
29
|
adantr |
|- ( ( ph /\ T < X ) -> ( B - X ) e. RR ) |
115 |
5 6 7
|
fourierdlem11 |
|- ( ph -> ( A e. RR /\ B e. RR /\ A < B ) ) |
116 |
115
|
simp3d |
|- ( ph -> A < B ) |
117 |
1 2 116
|
ltled |
|- ( ph -> A <_ B ) |
118 |
117
|
adantr |
|- ( ( ph /\ T < X ) -> A <_ B ) |
119 |
1 2 22
|
lesub1d |
|- ( ph -> ( A <_ B <-> ( A - X ) <_ ( B - X ) ) ) |
120 |
119
|
adantr |
|- ( ( ph /\ T < X ) -> ( A <_ B <-> ( A - X ) <_ ( B - X ) ) ) |
121 |
118 120
|
mpbid |
|- ( ( ph /\ T < X ) -> ( A - X ) <_ ( B - X ) ) |
122 |
2
|
adantr |
|- ( ( ph /\ T < X ) -> B e. RR ) |
123 |
22
|
adantr |
|- ( ( ph /\ T < X ) -> X e. RR ) |
124 |
|
simpr |
|- ( ( ph /\ T < X ) -> T < X ) |
125 |
3 124
|
eqbrtrrid |
|- ( ( ph /\ T < X ) -> ( B - A ) < X ) |
126 |
122 113 123 125
|
ltsub23d |
|- ( ( ph /\ T < X ) -> ( B - X ) < A ) |
127 |
114 113 126
|
ltled |
|- ( ( ph /\ T < X ) -> ( B - X ) <_ A ) |
128 |
112 113 114 121 127
|
eliccd |
|- ( ( ph /\ T < X ) -> ( B - X ) e. ( ( A - X ) [,] A ) ) |
129 |
8
|
adantr |
|- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> F : RR --> CC ) |
130 |
129 61
|
ffvelrnd |
|- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> ( F ` x ) e. CC ) |
131 |
130
|
adantlr |
|- ( ( ( ph /\ T < X ) /\ x e. ( ( A - X ) [,] A ) ) -> ( F ` x ) e. CC ) |
132 |
39
|
rexrd |
|- ( ph -> ( A - X ) e. RR* ) |
133 |
1 2 22 116
|
ltsub1dd |
|- ( ph -> ( A - X ) < ( B - X ) ) |
134 |
29
|
ltpnfd |
|- ( ph -> ( B - X ) < +oo ) |
135 |
132 102 29 133 134
|
eliood |
|- ( ph -> ( B - X ) e. ( ( A - X ) (,) +oo ) ) |
136 |
5 3 6 7 8 9 10 11 12 39 135
|
fourierdlem105 |
|- ( ph -> ( x e. ( ( A - X ) [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 ) |
137 |
136
|
adantr |
|- ( ( ph /\ T < X ) -> ( x e. ( ( A - X ) [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 ) |
138 |
6
|
adantr |
|- ( ( ph /\ T < X ) -> M e. NN ) |
139 |
7
|
adantr |
|- ( ( ph /\ T < X ) -> Q e. ( P ` M ) ) |
140 |
8
|
adantr |
|- ( ( ph /\ T < X ) -> F : RR --> CC ) |
141 |
9
|
adantlr |
|- ( ( ( ph /\ T < X ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) ) |
142 |
10
|
adantlr |
|- ( ( ( ph /\ T < X ) /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) ) |
143 |
11
|
adantlr |
|- ( ( ( ph /\ T < X ) /\ i e. ( 0 ..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
144 |
12
|
adantlr |
|- ( ( ( ph /\ T < X ) /\ i e. ( 0 ..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
145 |
101
|
adantr |
|- ( ( ph /\ T < X ) -> ( B - X ) e. RR* ) |
146 |
74
|
a1i |
|- ( ( ph /\ T < X ) -> +oo e. RR* ) |
147 |
113
|
ltpnfd |
|- ( ( ph /\ T < X ) -> A < +oo ) |
148 |
145 146 113 126 147
|
eliood |
|- ( ( ph /\ T < X ) -> A e. ( ( B - X ) (,) +oo ) ) |
149 |
5 3 138 139 140 141 142 143 144 114 148
|
fourierdlem105 |
|- ( ( ph /\ T < X ) -> ( x e. ( ( B - X ) [,] A ) |-> ( F ` x ) ) e. L^1 ) |
150 |
112 113 128 131 137 149
|
itgspliticc |
|- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] A ) ( F ` x ) _d x = ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) ) |
151 |
150
|
oveq1d |
|- ( ( ph /\ T < X ) -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) = ( ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) |
152 |
8
|
adantr |
|- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> F : RR --> CC ) |
153 |
39
|
adantr |
|- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( A - X ) e. RR ) |
154 |
29
|
adantr |
|- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( B - X ) e. RR ) |
155 |
|
simpr |
|- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> x e. ( ( A - X ) [,] ( B - X ) ) ) |
156 |
|
eliccre |
|- ( ( ( A - X ) e. RR /\ ( B - X ) e. RR /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> x e. RR ) |
157 |
153 154 155 156
|
syl3anc |
|- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> x e. RR ) |
158 |
152 157
|
ffvelrnd |
|- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( F ` x ) e. CC ) |
159 |
158 136
|
itgcl |
|- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x e. CC ) |
160 |
159
|
adantr |
|- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x e. CC ) |
161 |
8
|
adantr |
|- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> F : RR --> CC ) |
162 |
29
|
adantr |
|- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> ( B - X ) e. RR ) |
163 |
1
|
adantr |
|- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> A e. RR ) |
164 |
|
simpr |
|- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> x e. ( ( B - X ) [,] A ) ) |
165 |
|
eliccre |
|- ( ( ( B - X ) e. RR /\ A e. RR /\ x e. ( ( B - X ) [,] A ) ) -> x e. RR ) |
166 |
162 163 164 165
|
syl3anc |
|- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> x e. RR ) |
167 |
161 166
|
ffvelrnd |
|- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> ( F ` x ) e. CC ) |
168 |
167
|
adantlr |
|- ( ( ( ph /\ T < X ) /\ x e. ( ( B - X ) [,] A ) ) -> ( F ` x ) e. CC ) |
169 |
168 149
|
itgcl |
|- ( ( ph /\ T < X ) -> S. ( ( B - X ) [,] A ) ( F ` x ) _d x e. CC ) |
170 |
108
|
adantr |
|- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] A ) ( F ` x ) _d x e. CC ) |
171 |
160 169 170
|
addsubassd |
|- ( ( ph /\ T < X ) -> ( ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) = ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) ) |
172 |
111 151 171
|
3eqtrd |
|- ( ( ph /\ T < X ) -> 0 = ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) ) |
173 |
172
|
oveq2d |
|- ( ( ph /\ T < X ) -> ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - 0 ) = ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) ) ) |
174 |
160
|
subid1d |
|- ( ( ph /\ T < X ) -> ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - 0 ) = S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x ) |
175 |
159
|
subidd |
|- ( ph -> ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x ) = 0 ) |
176 |
175
|
oveq1d |
|- ( ph -> ( ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x ) - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) = ( 0 - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) ) |
177 |
176
|
adantr |
|- ( ( ph /\ T < X ) -> ( ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x ) - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) = ( 0 - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) ) |
178 |
169 170
|
subcld |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) e. CC ) |
179 |
160 160 178
|
subsub4d |
|- ( ( ph /\ T < X ) -> ( ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x ) - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) = ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) ) ) |
180 |
|
df-neg |
|- -u ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) = ( 0 - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) |
181 |
169 170
|
negsubdi2d |
|- ( ( ph /\ T < X ) -> -u ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) ) |
182 |
180 181
|
eqtr3id |
|- ( ( ph /\ T < X ) -> ( 0 - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) ) |
183 |
177 179 182
|
3eqtr3d |
|- ( ( ph /\ T < X ) -> ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) ) = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) ) |
184 |
173 174 183
|
3eqtr3d |
|- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) ) |
185 |
107
|
subidd |
|- ( ph -> ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = 0 ) |
186 |
185
|
eqcomd |
|- ( ph -> 0 = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) |
187 |
186
|
oveq2d |
|- ( ph -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + 0 ) = ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) ) |
188 |
187
|
adantr |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + 0 ) = ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) ) |
189 |
169
|
addid1d |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + 0 ) = S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) |
190 |
114 122 113 127 118
|
eliccd |
|- ( ( ph /\ T < X ) -> A e. ( ( B - X ) [,] B ) ) |
191 |
100
|
adantlr |
|- ( ( ( ph /\ T < X ) /\ x e. ( ( B - X ) [,] B ) ) -> ( F ` x ) e. CC ) |
192 |
1 2
|
iccssred |
|- ( ph -> ( A [,] B ) C_ RR ) |
193 |
8 192
|
feqresmpt |
|- ( ph -> ( F |` ( A [,] B ) ) = ( x e. ( A [,] B ) |-> ( F ` x ) ) ) |
194 |
8 192
|
fssresd |
|- ( ph -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> CC ) |
195 |
|
ioossicc |
|- ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) |
196 |
1
|
rexrd |
|- ( ph -> A e. RR* ) |
197 |
196
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> A e. RR* ) |
198 |
2
|
rexrd |
|- ( ph -> B e. RR* ) |
199 |
198
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> B e. RR* ) |
200 |
5 6 7
|
fourierdlem15 |
|- ( ph -> Q : ( 0 ... M ) --> ( A [,] B ) ) |
201 |
200
|
adantr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> Q : ( 0 ... M ) --> ( A [,] B ) ) |
202 |
|
simpr |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> i e. ( 0 ..^ M ) ) |
203 |
197 199 201 202
|
fourierdlem8 |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) C_ ( A [,] B ) ) |
204 |
195 203
|
sstrid |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( A [,] B ) ) |
205 |
204
|
resabs1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
206 |
205 10
|
eqeltrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) ) |
207 |
205
|
eqcomd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) ) |
208 |
207
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) = ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
209 |
11 208
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> R e. ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` i ) ) ) |
210 |
207
|
oveq1d |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) = ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
211 |
12 210
|
eleqtrd |
|- ( ( ph /\ i e. ( 0 ..^ M ) ) -> L e. ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) limCC ( Q ` ( i + 1 ) ) ) ) |
212 |
5 6 7 194 206 209 211
|
fourierdlem69 |
|- ( ph -> ( F |` ( A [,] B ) ) e. L^1 ) |
213 |
193 212
|
eqeltrrd |
|- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. L^1 ) |
214 |
213
|
adantr |
|- ( ( ph /\ T < X ) -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. L^1 ) |
215 |
114 122 190 191 149 214
|
itgspliticc |
|- ( ( ph /\ T < X ) -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x = ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) |
216 |
215
|
oveq2d |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) ) |
217 |
216
|
oveq2d |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) = ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) ) ) |
218 |
107
|
adantr |
|- ( ( ph /\ T < X ) -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x e. CC ) |
219 |
215 218
|
eqeltrrd |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) e. CC ) |
220 |
169 218 219
|
addsub12d |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) ) = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) ) ) |
221 |
8
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> F : RR --> CC ) |
222 |
1
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR ) |
223 |
2
|
adantr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR ) |
224 |
|
simpr |
|- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) ) |
225 |
|
eliccre |
|- ( ( A e. RR /\ B e. RR /\ x e. ( A [,] B ) ) -> x e. RR ) |
226 |
222 223 224 225
|
syl3anc |
|- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR ) |
227 |
221 226
|
ffvelrnd |
|- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC ) |
228 |
227 213
|
itgcl |
|- ( ph -> S. ( A [,] B ) ( F ` x ) _d x e. CC ) |
229 |
228
|
adantr |
|- ( ( ph /\ T < X ) -> S. ( A [,] B ) ( F ` x ) _d x e. CC ) |
230 |
169 169 229
|
subsub4d |
|- ( ( ph /\ T < X ) -> ( ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( A [,] B ) ( F ` x ) _d x ) = ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) ) |
231 |
230
|
eqcomd |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) = ( ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( A [,] B ) ( F ` x ) _d x ) ) |
232 |
231
|
oveq2d |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) ) = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x + ( ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( A [,] B ) ( F ` x ) _d x ) ) ) |
233 |
169
|
subidd |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) = 0 ) |
234 |
233
|
oveq1d |
|- ( ( ph /\ T < X ) -> ( ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( A [,] B ) ( F ` x ) _d x ) = ( 0 - S. ( A [,] B ) ( F ` x ) _d x ) ) |
235 |
|
df-neg |
|- -u S. ( A [,] B ) ( F ` x ) _d x = ( 0 - S. ( A [,] B ) ( F ` x ) _d x ) |
236 |
234 235
|
eqtr4di |
|- ( ( ph /\ T < X ) -> ( ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( A [,] B ) ( F ` x ) _d x ) = -u S. ( A [,] B ) ( F ` x ) _d x ) |
237 |
236
|
oveq2d |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x + ( ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) - S. ( A [,] B ) ( F ` x ) _d x ) ) = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x + -u S. ( A [,] B ) ( F ` x ) _d x ) ) |
238 |
218 229
|
negsubd |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x + -u S. ( A [,] B ) ( F ` x ) _d x ) = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) |
239 |
232 237 238
|
3eqtrd |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) ) ) = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) |
240 |
217 220 239
|
3eqtrd |
|- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) |
241 |
188 189 240
|
3eqtr3d |
|- ( ( ph /\ T < X ) -> S. ( ( B - X ) [,] A ) ( F ` x ) _d x = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) |
242 |
241
|
oveq2d |
|- ( ( ph /\ T < X ) -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) ) |
243 |
108 107 228
|
subsubd |
|- ( ph -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) = ( ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) + S. ( A [,] B ) ( F ` x ) _d x ) ) |
244 |
93
|
oveq2d |
|- ( ph -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) |
245 |
244 109
|
eqtrd |
|- ( ph -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = 0 ) |
246 |
245
|
oveq1d |
|- ( ph -> ( ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) + S. ( A [,] B ) ( F ` x ) _d x ) = ( 0 + S. ( A [,] B ) ( F ` x ) _d x ) ) |
247 |
228
|
addid2d |
|- ( ph -> ( 0 + S. ( A [,] B ) ( F ` x ) _d x ) = S. ( A [,] B ) ( F ` x ) _d x ) |
248 |
243 246 247
|
3eqtrd |
|- ( ph -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) = S. ( A [,] B ) ( F ` x ) _d x ) |
249 |
248
|
adantr |
|- ( ( ph /\ T < X ) -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( A [,] B ) ( F ` x ) _d x ) ) = S. ( A [,] B ) ( F ` x ) _d x ) |
250 |
184 242 249
|
3eqtrd |
|- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) |
251 |
39
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( A - X ) e. RR ) |
252 |
29
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( B - X ) e. RR ) |
253 |
1
|
adantr |
|- ( ( ph /\ X <_ T ) -> A e. RR ) |
254 |
39 1 50
|
ltled |
|- ( ph -> ( A - X ) <_ A ) |
255 |
254
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( A - X ) <_ A ) |
256 |
22
|
adantr |
|- ( ( ph /\ X <_ T ) -> X e. RR ) |
257 |
2
|
adantr |
|- ( ( ph /\ X <_ T ) -> B e. RR ) |
258 |
|
id |
|- ( X <_ T -> X <_ T ) |
259 |
258 3
|
breqtrdi |
|- ( X <_ T -> X <_ ( B - A ) ) |
260 |
259
|
adantl |
|- ( ( ph /\ X <_ T ) -> X <_ ( B - A ) ) |
261 |
256 257 253 260
|
lesubd |
|- ( ( ph /\ X <_ T ) -> A <_ ( B - X ) ) |
262 |
251 252 253 255 261
|
eliccd |
|- ( ( ph /\ X <_ T ) -> A e. ( ( A - X ) [,] ( B - X ) ) ) |
263 |
158
|
adantlr |
|- ( ( ( ph /\ X <_ T ) /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( F ` x ) e. CC ) |
264 |
132 102 1 50 78
|
eliood |
|- ( ph -> A e. ( ( A - X ) (,) +oo ) ) |
265 |
5 3 6 7 8 9 10 11 12 39 264
|
fourierdlem105 |
|- ( ph -> ( x e. ( ( A - X ) [,] A ) |-> ( F ` x ) ) e. L^1 ) |
266 |
265
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( x e. ( ( A - X ) [,] A ) |-> ( F ` x ) ) e. L^1 ) |
267 |
1
|
leidd |
|- ( ph -> A <_ A ) |
268 |
4
|
rpge0d |
|- ( ph -> 0 <_ X ) |
269 |
2 22
|
subge02d |
|- ( ph -> ( 0 <_ X <-> ( B - X ) <_ B ) ) |
270 |
268 269
|
mpbid |
|- ( ph -> ( B - X ) <_ B ) |
271 |
|
iccss |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ A /\ ( B - X ) <_ B ) ) -> ( A [,] ( B - X ) ) C_ ( A [,] B ) ) |
272 |
1 2 267 270 271
|
syl22anc |
|- ( ph -> ( A [,] ( B - X ) ) C_ ( A [,] B ) ) |
273 |
|
iccmbl |
|- ( ( A e. RR /\ ( B - X ) e. RR ) -> ( A [,] ( B - X ) ) e. dom vol ) |
274 |
1 29 273
|
syl2anc |
|- ( ph -> ( A [,] ( B - X ) ) e. dom vol ) |
275 |
272 274 227 213
|
iblss |
|- ( ph -> ( x e. ( A [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 ) |
276 |
275
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( x e. ( A [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 ) |
277 |
251 252 262 263 266 276
|
itgspliticc |
|- ( ( ph /\ X <_ T ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] ( B - X ) ) ( F ` x ) _d x ) ) |
278 |
268
|
adantr |
|- ( ( ph /\ X <_ T ) -> 0 <_ X ) |
279 |
269
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( 0 <_ X <-> ( B - X ) <_ B ) ) |
280 |
278 279
|
mpbid |
|- ( ( ph /\ X <_ T ) -> ( B - X ) <_ B ) |
281 |
253 257 252 261 280
|
eliccd |
|- ( ( ph /\ X <_ T ) -> ( B - X ) e. ( A [,] B ) ) |
282 |
227
|
adantlr |
|- ( ( ( ph /\ X <_ T ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC ) |
283 |
2
|
leidd |
|- ( ph -> B <_ B ) |
284 |
283
|
adantr |
|- ( ( ph /\ X <_ T ) -> B <_ B ) |
285 |
|
iccss |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ ( B - X ) /\ B <_ B ) ) -> ( ( B - X ) [,] B ) C_ ( A [,] B ) ) |
286 |
253 257 261 284 285
|
syl22anc |
|- ( ( ph /\ X <_ T ) -> ( ( B - X ) [,] B ) C_ ( A [,] B ) ) |
287 |
|
iccmbl |
|- ( ( ( B - X ) e. RR /\ B e. RR ) -> ( ( B - X ) [,] B ) e. dom vol ) |
288 |
29 2 287
|
syl2anc |
|- ( ph -> ( ( B - X ) [,] B ) e. dom vol ) |
289 |
288
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( ( B - X ) [,] B ) e. dom vol ) |
290 |
213
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. L^1 ) |
291 |
286 289 282 290
|
iblss |
|- ( ( ph /\ X <_ T ) -> ( x e. ( ( B - X ) [,] B ) |-> ( F ` x ) ) e. L^1 ) |
292 |
253 257 281 282 276 291
|
itgspliticc |
|- ( ( ph /\ X <_ T ) -> S. ( A [,] B ) ( F ` x ) _d x = ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) |
293 |
292
|
oveq1d |
|- ( ( ph /\ X <_ T ) -> ( S. ( A [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = ( ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) |
294 |
8
|
adantr |
|- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> F : RR --> CC ) |
295 |
1
|
adantr |
|- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> A e. RR ) |
296 |
29
|
adantr |
|- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> ( B - X ) e. RR ) |
297 |
|
simpr |
|- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> x e. ( A [,] ( B - X ) ) ) |
298 |
|
eliccre |
|- ( ( A e. RR /\ ( B - X ) e. RR /\ x e. ( A [,] ( B - X ) ) ) -> x e. RR ) |
299 |
295 296 297 298
|
syl3anc |
|- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> x e. RR ) |
300 |
294 299
|
ffvelrnd |
|- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> ( F ` x ) e. CC ) |
301 |
300 275
|
itgcl |
|- ( ph -> S. ( A [,] ( B - X ) ) ( F ` x ) _d x e. CC ) |
302 |
301 107 107
|
addsubassd |
|- ( ph -> ( ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) ) |
303 |
302
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) ) |
304 |
185
|
oveq2d |
|- ( ph -> ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) = ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + 0 ) ) |
305 |
301
|
addid1d |
|- ( ph -> ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + 0 ) = S. ( A [,] ( B - X ) ) ( F ` x ) _d x ) |
306 |
304 305
|
eqtrd |
|- ( ph -> ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) = S. ( A [,] ( B - X ) ) ( F ` x ) _d x ) |
307 |
306
|
adantr |
|- ( ( ph /\ X <_ T ) -> ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) = S. ( A [,] ( B - X ) ) ( F ` x ) _d x ) |
308 |
293 303 307
|
3eqtrrd |
|- ( ( ph /\ X <_ T ) -> S. ( A [,] ( B - X ) ) ( F ` x ) _d x = ( S. ( A [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) |
309 |
308
|
oveq2d |
|- ( ( ph /\ X <_ T ) -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] ( B - X ) ) ( F ` x ) _d x ) = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x + ( S. ( A [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) ) |
310 |
93
|
adantr |
|- ( ( ph /\ X <_ T ) -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x = S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) |
311 |
107
|
adantr |
|- ( ( ph /\ X <_ T ) -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x e. CC ) |
312 |
310 311
|
eqeltrrd |
|- ( ( ph /\ X <_ T ) -> S. ( ( A - X ) [,] A ) ( F ` x ) _d x e. CC ) |
313 |
282 290
|
itgcl |
|- ( ( ph /\ X <_ T ) -> S. ( A [,] B ) ( F ` x ) _d x e. CC ) |
314 |
312 313 311
|
addsub12d |
|- ( ( ph /\ X <_ T ) -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x + ( S. ( A [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) = ( S. ( A [,] B ) ( F ` x ) _d x + ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) ) |
315 |
313 312 311
|
addsubassd |
|- ( ( ph /\ X <_ T ) -> ( ( S. ( A [,] B ) ( F ` x ) _d x + S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = ( S. ( A [,] B ) ( F ` x ) _d x + ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) ) |
316 |
314 315
|
eqtr4d |
|- ( ( ph /\ X <_ T ) -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x + ( S. ( A [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) = ( ( S. ( A [,] B ) ( F ` x ) _d x + S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) |
317 |
277 309 316
|
3eqtrd |
|- ( ( ph /\ X <_ T ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = ( ( S. ( A [,] B ) ( F ` x ) _d x + S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) ) |
318 |
310
|
oveq2d |
|- ( ( ph /\ X <_ T ) -> ( ( S. ( A [,] B ) ( F ` x ) _d x + S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = ( ( S. ( A [,] B ) ( F ` x ) _d x + S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) ) |
319 |
313 312
|
pncand |
|- ( ( ph /\ X <_ T ) -> ( ( S. ( A [,] B ) ( F ` x ) _d x + S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) = S. ( A [,] B ) ( F ` x ) _d x ) |
320 |
317 318 319
|
3eqtrd |
|- ( ( ph /\ X <_ T ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) |
321 |
250 320 55 22
|
ltlecasei |
|- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x ) |