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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem107 Structured version   Visualization version   Unicode version
Theorem fourierdlem107 40430
Description: The integral of a piecewise continuous periodic function F is unchanged if the domain is shifted by any positive value X. This lemma generalizes fourierdlem92 40415 where the integral was shifted by the exact period. This lemma uses local definitions, so that the proof is more readable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem107.a |- ( ph -> A e. RR )
fourierdlem107.b |- ( ph -> B e. RR )
fourierdlem107.t |- T = ( B - A )
fourierdlem107.x |- ( ph -> X e. RR+ )
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 ) ) ) } )
fourierdlem107.m |- ( ph -> M e. NN )
fourierdlem107.q |- ( ph -> Q e. ( P ` M ) )
fourierdlem107.f |- ( ph -> F : RR --> CC )
fourierdlem107.fper |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
fourierdlem107.fcn |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
fourierdlem107.r |- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
fourierdlem107.l |- ( ( ph /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
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 ) ) ) } )
fourierdlem107.h |- H = ( { ( A - X ) , A } u. { y e. ( ( A - X ) [,] A ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q } )
fourierdlem107.n |- N = ( ( # ` H ) - 1 )
fourierdlem107.s |- S = ( iota f f Isom < , < ( ( 0 ... N ) , H ) )
fourierdlem107.e |- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
fourierdlem107.z |- Z = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) )
fourierdlem107.i |- I = ( x e. RR |-> sup ( { i e. ( 0..^ M ) | ( Q ` i ) <_ ( Z ` ( E ` x ) ) } , RR , < ) )
Assertion
Ref Expression
fourierdlem107 |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
Distinct variable groups:   x, k   A, f, k, y   A, i, x, k, y   A, m, p, i   B, f, k, y   B, i, x   B, m, p   f, E, k, y   i, E, x   i, F, x, y   f, H, y   x, H   f, I, k, y   i, I, x   x, L, y   i, M, x, y   m, M, p   f, N, k, y   i, N, x   m, N, p   Q, f, k, y   Q, i, x   Q, m, p   x, R, y   S, f, k, y   S, i, x   S, p   T, f, k, y   T, i, x   T, m, p   f, X, y   i, X, m, p   x, X   i, Z, x, y   ph, f, k, y   ph, i, x
Allowed substitution hints:   ph( m, p)   P( x, y, f, i, k, m, p)   R( f, i, k, m, p)   S( m)   E( m, p)   F( f, k, m, p)   H( i, k, m, p)   I( m, p)   L( f, i, k, m, p)   M( f, k)   O( x, y, f, i, k, m, p)   X( k)   Z( f, k, m, p)
Proof of Theorem fourierdlem107
Dummy variable j is distinct from all other variables.
StepHypRef Expression
1 fourierdlem107.t . . . . . . . . . . . . . . . . 17 |- T = ( B - A )
21oveq2i 6661 . . . . . . . . . . . . . . . 16 |- ( ( A - X ) + T ) = ( ( A - X ) + ( B - A ) )
3 fourierdlem107.a . . . . . . . . . . . . . . . . . 18 |- ( ph -> A e. RR )
43recnd 10068 . . . . . . . . . . . . . . . . 17 |- ( ph -> A e. CC )
5 fourierdlem107.x . . . . . . . . . . . . . . . . . . 19 |- ( ph -> X e. RR+ )
65rpred 11872 . . . . . . . . . . . . . . . . . 18 |- ( ph -> X e. RR )
76recnd 10068 . . . . . . . . . . . . . . . . 17 |- ( ph -> X e. CC )
8 fourierdlem107.b . . . . . . . . . . . . . . . . . 18 |- ( ph -> B e. RR )
98recnd 10068 . . . . . . . . . . . . . . . . 17 |- ( ph -> B e. CC )
104, 7, 9, 4subadd4b 39494 . . . . . . . . . . . . . . . 16 |- ( ph -> ( ( A - X ) + ( B - A ) ) = ( ( A - A ) + ( B - X ) ) )
112, 10syl5eq 2668 . . . . . . . . . . . . . . 15 |- ( ph -> ( ( A - X ) + T ) = ( ( A - A ) + ( B - X ) ) )
124subidd 10380 . . . . . . . . . . . . . . . 16 |- ( ph -> ( A - A ) = 0 )
1312oveq1d 6665 . . . . . . . . . . . . . . 15 |- ( ph -> ( ( A - A ) + ( B - X ) ) = ( 0 + ( B - X ) ) )
148, 6resubcld 10458 . . . . . . . . . . . . . . . . 17 |- ( ph -> ( B - X ) e. RR )
1514recnd 10068 . . . . . . . . . . . . . . . 16 |- ( ph -> ( B - X ) e. CC )
1615addid2d 10237 . . . . . . . . . . . . . . 15 |- ( ph -> ( 0 + ( B - X ) ) = ( B - X ) )
1711, 13, 163eqtrd 2660 . . . . . . . . . . . . . 14 |- ( ph -> ( ( A - X ) + T ) = ( B - X ) )
181oveq2i 6661 . . . . . . . . . . . . . . 15 |- ( A + T ) = ( A + ( B - A ) )
194, 9pncan3d 10395 . . . . . . . . . . . . . . 15 |- ( ph -> ( A + ( B - A ) ) = B )
2018, 19syl5eq 2668 . . . . . . . . . . . . . 14 |- ( ph -> ( A + T ) = B )
2117, 20oveq12d 6668 . . . . . . . . . . . . 13 |- ( ph -> ( ( ( A - X ) + T ) [,] ( A + T ) ) = ( ( B - X ) [,] B ) )
2221eqcomd 2628 . . . . . . . . . . . 12 |- ( ph -> ( ( B - X ) [,] B ) = ( ( ( A - X ) + T ) [,] ( A + T ) ) )
2322itgeq1d 40172 . . . . . . . . . . 11 |- ( ph -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x = S. ( ( ( A - X ) + T ) [,] ( A + T ) ) ( F ` x ) _d x )
243, 6resubcld 10458 . . . . . . . . . . . 12 |- ( ph -> ( A - X ) e. RR )
25 fourierdlem107.o . . . . . . . . . . . . 13 |- 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 ) ) ) } )
26 fveq2 6191 . . . . . . . . . . . . . . . . . . 19 |- ( i = j -> ( p ` i ) = ( p ` j ) )
27 oveq1 6657 . . . . . . . . . . . . . . . . . . . 20 |- ( i = j -> ( i + 1 ) = ( j + 1 ) )
2827fveq2d 6195 . . . . . . . . . . . . . . . . . . 19 |- ( i = j -> ( p ` ( i + 1 ) ) = ( p ` ( j + 1 ) ) )
2926, 28breq12d 4666 . . . . . . . . . . . . . . . . . 18 |- ( i = j -> ( ( p ` i ) < ( p ` ( i + 1 ) ) <-> ( p ` j ) < ( p ` ( j + 1 ) ) ) )
3029cbvralv 3171 . . . . . . . . . . . . . . . . 17 |- ( A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) )
3130a1i 11 . . . . . . . . . . . . . . . 16 |- ( m e. NN -> ( A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) <-> A. j e. ( 0..^ m ) ( p ` j ) < ( p ` ( j + 1 ) ) ) )
3231anbi2d 740 . . . . . . . . . . . . . . 15 |- ( 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 ) ) ) ) )
3332rabbidv 3189 . . . . . . . . . . . . . 14 |- ( 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 ) ) ) } )
3433mpteq2ia 4740 . . . . . . . . . . . . 13 |- ( 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 ) ) ) } )
3525, 34eqtri 2644 . . . . . . . . . . . 12 |- 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 ) ) ) } )
36 fourierdlem107.p . . . . . . . . . . . . . . 15 |- 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 ) ) ) } )
37 fourierdlem107.m . . . . . . . . . . . . . . 15 |- ( ph -> M e. NN )
38 fourierdlem107.q . . . . . . . . . . . . . . 15 |- ( ph -> Q e. ( P ` M ) )
393, 5ltsubrpd 11904 . . . . . . . . . . . . . . 15 |- ( ph -> ( A - X ) < A )
40 fourierdlem107.h . . . . . . . . . . . . . . 15 |- H = ( { ( A - X ) , A } u. { y e. ( ( A - X ) [,] A ) | E. k e. ZZ ( y + ( k x. T ) ) e. ran Q } )
41 fourierdlem107.n . . . . . . . . . . . . . . 15 |- N = ( ( # ` H ) - 1 )
42 fourierdlem107.s . . . . . . . . . . . . . . 15 |- S = ( iota f f Isom < , < ( ( 0 ... N ) , H ) )
431, 36, 37, 38, 24, 3, 39, 25, 40, 41, 42fourierdlem54 40377 . . . . . . . . . . . . . 14 |- ( ph -> ( ( N e. NN /\ S e. ( O ` N ) ) /\ S Isom < , < ( ( 0 ... N ) , H ) ) )
4443simpld 475 . . . . . . . . . . . . 13 |- ( ph -> ( N e. NN /\ S e. ( O ` N ) ) )
4544simpld 475 . . . . . . . . . . . 12 |- ( ph -> N e. NN )
468, 3resubcld 10458 . . . . . . . . . . . . 13 |- ( ph -> ( B - A ) e. RR )
471, 46syl5eqel 2705 . . . . . . . . . . . 12 |- ( ph -> T e. RR )
4844simprd 479 . . . . . . . . . . . 12 |- ( ph -> S e. ( O ` N ) )
4924adantr 481 . . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> ( A - X ) e. RR )
503adantr 481 . . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> A e. RR )
51 simpr 477 . . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> x e. ( ( A - X ) [,] A ) )
52 eliccre 39728 . . . . . . . . . . . . . 14 |- ( ( ( A - X ) e. RR /\ A e. RR /\ x e. ( ( A - X ) [,] A ) ) -> x e. RR )
5349, 50, 51, 52syl3anc 1326 . . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> x e. RR )
54 fourierdlem107.fper . . . . . . . . . . . . 13 |- ( ( ph /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
5553, 54syldan 487 . . . . . . . . . . . 12 |- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> ( F ` ( x + T ) ) = ( F ` x ) )
56 fveq2 6191 . . . . . . . . . . . . . 14 |- ( i = j -> ( S ` i ) = ( S ` j ) )
5756oveq1d 6665 . . . . . . . . . . . . 13 |- ( i = j -> ( ( S ` i ) + T ) = ( ( S ` j ) + T ) )
5857cbvmptv 4750 . . . . . . . . . . . 12 |- ( i e. ( 0 ... N ) |-> ( ( S ` i ) + T ) ) = ( j e. ( 0 ... N ) |-> ( ( S ` j ) + T ) )
59 eqid 2622 . . . . . . . . . . . 12 |- ( 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 ) ) ) } )
60 fourierdlem107.f . . . . . . . . . . . 12 |- ( ph -> F : RR --> CC )
6137adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 0..^ N ) ) -> M e. NN )
6238adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 0..^ N ) ) -> Q e. ( P ` M ) )
6360adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 0..^ N ) ) -> F : RR --> CC )
6454adantlr 751 . . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
65 fourierdlem107.fcn . . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
6665adantlr 751 . . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
6724adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( A - X ) e. RR )
6867rexrd 10089 . . . . . . . . . . . . . 14 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( A - X ) e. RR* )
69 pnfxr 10092 . . . . . . . . . . . . . . 15 |- +oo e. RR*
7069a1i 11 . . . . . . . . . . . . . 14 |- ( ( ph /\ j e. ( 0..^ N ) ) -> +oo e. RR* )
713adantr 481 . . . . . . . . . . . . . 14 |- ( ( ph /\ j e. ( 0..^ N ) ) -> A e. RR )
7239adantr 481 . . . . . . . . . . . . . 14 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( A - X ) < A )
733ltpnfd 11955 . . . . . . . . . . . . . . 15 |- ( ph -> A < +oo )
7473adantr 481 . . . . . . . . . . . . . 14 |- ( ( ph /\ j e. ( 0..^ N ) ) -> A < +oo )
7568, 70, 71, 72, 74eliood 39720 . . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 0..^ N ) ) -> A e. ( ( A - X ) (,) +oo ) )
76 fourierdlem107.e . . . . . . . . . . . . 13 |- E = ( x e. RR |-> ( x + ( ( |_ ` ( ( B - x ) / T ) ) x. T ) ) )
77 fourierdlem107.z . . . . . . . . . . . . 13 |- Z = ( y e. ( A (,] B ) |-> if ( y = B , A , y ) )
78 simpr 477 . . . . . . . . . . . . 13 |- ( ( ph /\ j e. ( 0..^ N ) ) -> j e. ( 0..^ N ) )
79 eqid 2622 . . . . . . . . . . . . 13 |- ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) ) = ( ( S ` ( j + 1 ) ) - ( E ` ( S ` ( j + 1 ) ) ) )
80 eqid 2622 . . . . . . . . . . . . 13 |- ( F |` ( ( Z ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) ) = ( F |` ( ( Z ` ( E ` ( S ` j ) ) ) (,) ( E ` ( S ` ( j + 1 ) ) ) ) )
81 eqid 2622 . . . . . . . . . . . . 13 |- ( 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 ) ) ) ) ) ) )
82 fourierdlem107.i . . . . . . . . . . . . 13 |- I = ( x e. RR |-> sup ( { i e. ( 0..^ M ) | ( Q ` i ) <_ ( Z ` ( E ` x ) ) } , RR , < ) )
8336, 1, 61, 62, 63, 64, 66, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 80, 81, 82fourierdlem90 40413 . . . . . . . . . . . 12 |- ( ( ph /\ j e. ( 0..^ N ) ) -> ( F |` ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) ) e. ( ( ( S ` j ) (,) ( S ` ( j + 1 ) ) ) -cn-> CC ) )
84 fourierdlem107.r . . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
8584adantlr 751 . . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
86 eqid 2622 . . . . . . . . . . . . 13 |- ( i e. ( 0..^ M ) |-> R ) = ( i e. ( 0..^ M ) |-> R )
8736, 1, 61, 62, 63, 64, 66, 85, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 82, 86fourierdlem89 40412 . . . . . . . . . . . 12 |- ( ( 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 ) ) ) ) lim CC ( S ` j ) ) )
88 fourierdlem107.l . . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
8988adantlr 751 . . . . . . . . . . . . 13 |- ( ( ( ph /\ j e. ( 0..^ N ) ) /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
90 eqid 2622 . . . . . . . . . . . . 13 |- ( i e. ( 0..^ M ) |-> L ) = ( i e. ( 0..^ M ) |-> L )
9136, 1, 61, 62, 63, 64, 66, 89, 67, 75, 25, 40, 41, 42, 76, 77, 78, 79, 82, 90fourierdlem91 40414 . . . . . . . . . . . 12 |- ( ( 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 ) ) ) )
9224, 3, 35, 45, 47, 48, 55, 58, 59, 60, 83, 87, 91fourierdlem92 40415 . . . . . . . . . . 11 |- ( ph -> S. ( ( ( A - X ) + T ) [,] ( A + T ) ) ( F ` x ) _d x = S. ( ( A - X ) [,] A ) ( F ` x ) _d x )
9323, 92eqtrd 2656 . . . . . . . . . 10 |- ( ph -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x = S. ( ( A - X ) [,] A ) ( F ` x ) _d x )
9460adantr 481 . . . . . . . . . . . 12 |- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> F : RR --> CC )
9514adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> ( B - X ) e. RR )
968adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> B e. RR )
97 simpr 477 . . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> x e. ( ( B - X ) [,] B ) )
98 eliccre 39728 . . . . . . . . . . . . 13 |- ( ( ( B - X ) e. RR /\ B e. RR /\ x e. ( ( B - X ) [,] B ) ) -> x e. RR )
9995, 96, 97, 98syl3anc 1326 . . . . . . . . . . . 12 |- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> x e. RR )
10094, 99ffvelrnd 6360 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( B - X ) [,] B ) ) -> ( F ` x ) e. CC )
10114rexrd 10089 . . . . . . . . . . . . 13 |- ( ph -> ( B - X ) e. RR* )
10269a1i 11 . . . . . . . . . . . . 13 |- ( ph -> +oo e. RR* )
1038, 5ltsubrpd 11904 . . . . . . . . . . . . 13 |- ( ph -> ( B - X ) < B )
1048ltpnfd 11955 . . . . . . . . . . . . 13 |- ( ph -> B < +oo )
105101, 102, 8, 103, 104eliood 39720 . . . . . . . . . . . 12 |- ( ph -> B e. ( ( B - X ) (,) +oo ) )
10636, 1, 37, 38, 60, 54, 65, 84, 88, 14, 105fourierdlem105 40428 . . . . . . . . . . 11 |- ( ph -> ( x e. ( ( B - X ) [,] B ) |-> ( F ` x ) ) e. L^1 )
107100, 106itgcl 23550 . . . . . . . . . 10 |- ( ph -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x e. CC )
10893, 107eqeltrrd 2702 . . . . . . . . 9 |- ( ph -> S. ( ( A - X ) [,] A ) ( F ` x ) _d x e. CC )
109108subidd 10380 . . . . . . . 8 |- ( ph -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) = 0 )
110109eqcomd 2628 . . . . . . 7 |- ( ph -> 0 = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) )
111110adantr 481 . . . . . 6 |- ( ( ph /\ T < X ) -> 0 = ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) )
11224adantr 481 . . . . . . . 8 |- ( ( ph /\ T < X ) -> ( A - X ) e. RR )
1133adantr 481 . . . . . . . 8 |- ( ( ph /\ T < X ) -> A e. RR )
11414adantr 481 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> ( B - X ) e. RR )
11536, 37, 38fourierdlem11 40335 . . . . . . . . . . . . 13 |- ( ph -> ( A e. RR /\ B e. RR /\ A < B ) )
116115simp3d 1075 . . . . . . . . . . . 12 |- ( ph -> A < B )
1173, 8, 116ltled 10185 . . . . . . . . . . 11 |- ( ph -> A <_ B )
118117adantr 481 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> A <_ B )
1193, 8, 6lesub1d 10634 . . . . . . . . . . 11 |- ( ph -> ( A <_ B <-> ( A - X ) <_ ( B - X ) ) )
120119adantr 481 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> ( A <_ B <-> ( A - X ) <_ ( B - X ) ) )
121118, 120mpbid 222 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> ( A - X ) <_ ( B - X ) )
1228adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ T < X ) -> B e. RR )
1236adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ T < X ) -> X e. RR )
124 simpr 477 . . . . . . . . . . . 12 |- ( ( ph /\ T < X ) -> T < X )
1251, 124syl5eqbrr 4689 . . . . . . . . . . 11 |- ( ( ph /\ T < X ) -> ( B - A ) < X )
126122, 113, 123, 125ltsub23d 10632 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> ( B - X ) < A )
127114, 113, 126ltled 10185 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> ( B - X ) <_ A )
128112, 113, 114, 121, 127eliccd 39726 . . . . . . . 8 |- ( ( ph /\ T < X ) -> ( B - X ) e. ( ( A - X ) [,] A ) )
12960adantr 481 . . . . . . . . . 10 |- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> F : RR --> CC )
130129, 53ffvelrnd 6360 . . . . . . . . 9 |- ( ( ph /\ x e. ( ( A - X ) [,] A ) ) -> ( F ` x ) e. CC )
131130adantlr 751 . . . . . . . 8 |- ( ( ( ph /\ T < X ) /\ x e. ( ( A - X ) [,] A ) ) -> ( F ` x ) e. CC )
13224rexrd 10089 . . . . . . . . . . 11 |- ( ph -> ( A - X ) e. RR* )
1333, 8, 6, 116ltsub1dd 10639 . . . . . . . . . . 11 |- ( ph -> ( A - X ) < ( B - X ) )
13414ltpnfd 11955 . . . . . . . . . . 11 |- ( ph -> ( B - X ) < +oo )
135132, 102, 14, 133, 134eliood 39720 . . . . . . . . . 10 |- ( ph -> ( B - X ) e. ( ( A - X ) (,) +oo ) )
13636, 1, 37, 38, 60, 54, 65, 84, 88, 24, 135fourierdlem105 40428 . . . . . . . . 9 |- ( ph -> ( x e. ( ( A - X ) [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 )
137136adantr 481 . . . . . . . 8 |- ( ( ph /\ T < X ) -> ( x e. ( ( A - X ) [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 )
13837adantr 481 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> M e. NN )
13938adantr 481 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> Q e. ( P ` M ) )
14060adantr 481 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> F : RR --> CC )
14154adantlr 751 . . . . . . . . 9 |- ( ( ( ph /\ T < X ) /\ x e. RR ) -> ( F ` ( x + T ) ) = ( F ` x ) )
14265adantlr 751 . . . . . . . . 9 |- ( ( ( ph /\ T < X ) /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
14384adantlr 751 . . . . . . . . 9 |- ( ( ( ph /\ T < X ) /\ i e. ( 0..^ M ) ) -> R e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
14488adantlr 751 . . . . . . . . 9 |- ( ( ( ph /\ T < X ) /\ i e. ( 0..^ M ) ) -> L e. ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
145101adantr 481 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> ( B - X ) e. RR* )
14669a1i 11 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> +oo e. RR* )
147113ltpnfd 11955 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> A < +oo )
148145, 146, 113, 126, 147eliood 39720 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> A e. ( ( B - X ) (,) +oo ) )
14936, 1, 138, 139, 140, 141, 142, 143, 144, 114, 148fourierdlem105 40428 . . . . . . . 8 |- ( ( ph /\ T < X ) -> ( x e. ( ( B - X ) [,] A ) |-> ( F ` x ) ) e. L^1 )
150112, 113, 128, 131, 137, 149itgspliticc 23603 . . . . . . 7 |- ( ( 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 ) )
151150oveq1d 6665 . . . . . 6 |- ( ( 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 ) )
15260adantr 481 . . . . . . . . . 10 |- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> F : RR --> CC )
15324adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( A - X ) e. RR )
15414adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( B - X ) e. RR )
155 simpr 477 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> x e. ( ( A - X ) [,] ( B - X ) ) )
156 eliccre 39728 . . . . . . . . . . 11 |- ( ( ( A - X ) e. RR /\ ( B - X ) e. RR /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> x e. RR )
157153, 154, 155, 156syl3anc 1326 . . . . . . . . . 10 |- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> x e. RR )
158152, 157ffvelrnd 6360 . . . . . . . . 9 |- ( ( ph /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( F ` x ) e. CC )
159158, 136itgcl 23550 . . . . . . . 8 |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x e. CC )
160159adantr 481 . . . . . . 7 |- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x e. CC )
16160adantr 481 . . . . . . . . . 10 |- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> F : RR --> CC )
16214adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> ( B - X ) e. RR )
1633adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> A e. RR )
164 simpr 477 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> x e. ( ( B - X ) [,] A ) )
165 eliccre 39728 . . . . . . . . . . 11 |- ( ( ( B - X ) e. RR /\ A e. RR /\ x e. ( ( B - X ) [,] A ) ) -> x e. RR )
166162, 163, 164, 165syl3anc 1326 . . . . . . . . . 10 |- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> x e. RR )
167161, 166ffvelrnd 6360 . . . . . . . . 9 |- ( ( ph /\ x e. ( ( B - X ) [,] A ) ) -> ( F ` x ) e. CC )
168167adantlr 751 . . . . . . . 8 |- ( ( ( ph /\ T < X ) /\ x e. ( ( B - X ) [,] A ) ) -> ( F ` x ) e. CC )
169168, 149itgcl 23550 . . . . . . 7 |- ( ( ph /\ T < X ) -> S. ( ( B - X ) [,] A ) ( F ` x ) _d x e. CC )
170108adantr 481 . . . . . . 7 |- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] A ) ( F ` x ) _d x e. CC )
171160, 169, 170addsubassd 10412 . . . . . 6 |- ( ( 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 ) ) )
172111, 151, 1713eqtrd 2660 . . . . 5 |- ( ( 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 ) ) )
173172oveq2d 6666 . . . 4 |- ( ( 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 ) ) ) )
174160subid1d 10381 . . . 4 |- ( ( ph /\ T < X ) -> ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - 0 ) = S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x )
175159subidd 10380 . . . . . . 7 |- ( ph -> ( S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x - S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x ) = 0 )
176175oveq1d 6665 . . . . . 6 |- ( 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 ) ) )
177176adantr 481 . . . . 5 |- ( ( 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 ) ) )
178169, 170subcld 10392 . . . . . 6 |- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( A - X ) [,] A ) ( F ` x ) _d x ) e. CC )
179160, 160, 178subsub4d 10423 . . . . 5 |- ( ( 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 10269 . . . . . 6 |- -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 ) )
181169, 170negsubdi2d 10408 . . . . . 6 |- ( ( 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 ) )
182180, 181syl5eqr 2670 . . . . 5 |- ( ( 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 ) )
183177, 179, 1823eqtr3d 2664 . . . 4 |- ( ( 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 ) )
184173, 174, 1833eqtr3d 2664 . . 3 |- ( ( 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 ) )
185107subidd 10380 . . . . . . . 8 |- ( ph -> ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = 0 )
186185eqcomd 2628 . . . . . . 7 |- ( ph -> 0 = ( S. ( ( B - X ) [,] B ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) )
187186oveq2d 6666 . . . . . 6 |- ( 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 ) ) )
188187adantr 481 . . . . 5 |- ( ( 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 ) ) )
189169addid1d 10236 . . . . 5 |- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + 0 ) = S. ( ( B - X ) [,] A ) ( F ` x ) _d x )
190114, 122, 113, 127, 118eliccd 39726 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> A e. ( ( B - X ) [,] B ) )
191100adantlr 751 . . . . . . . . 9 |- ( ( ( ph /\ T < X ) /\ x e. ( ( B - X ) [,] B ) ) -> ( F ` x ) e. CC )
1923, 8iccssred 39727 . . . . . . . . . . . 12 |- ( ph -> ( A [,] B ) C_ RR )
19360, 192feqresmpt 6250 . . . . . . . . . . 11 |- ( ph -> ( F |` ( A [,] B ) ) = ( x e. ( A [,] B ) |-> ( F ` x ) ) )
19460, 192fssresd 6071 . . . . . . . . . . . 12 |- ( ph -> ( F |` ( A [,] B ) ) : ( A [,] B ) --> CC )
195 ioossicc 12259 . . . . . . . . . . . . . . 15 |- ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) )
1963rexrd 10089 . . . . . . . . . . . . . . . . 17 |- ( ph -> A e. RR* )
197196adantr 481 . . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 0..^ M ) ) -> A e. RR* )
1988rexrd 10089 . . . . . . . . . . . . . . . . 17 |- ( ph -> B e. RR* )
199198adantr 481 . . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 0..^ M ) ) -> B e. RR* )
20036, 37, 38fourierdlem15 40339 . . . . . . . . . . . . . . . . 17 |- ( ph -> Q : ( 0 ... M ) --> ( A [,] B ) )
201200adantr 481 . . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q : ( 0 ... M ) --> ( A [,] B ) )
202 simpr 477 . . . . . . . . . . . . . . . 16 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i e. ( 0..^ M ) )
203197, 199, 201, 202fourierdlem8 40332 . . . . . . . . . . . . . . 15 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) [,] ( Q ` ( i + 1 ) ) ) C_ ( A [,] B ) )
204195, 203syl5ss 3614 . . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) C_ ( A [,] B ) )
205204resabs1d 5428 . . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
206205, 65eqeltrd 2701 . . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) e. ( ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) -cn-> CC ) )
207205eqcomd 2628 . . . . . . . . . . . . . 14 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) = ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) )
208207oveq1d 6665 . . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) = ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
20984, 208eleqtrd 2703 . . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 0..^ M ) ) -> R e. ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` i ) ) )
210207oveq1d 6665 . . . . . . . . . . . . 13 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( F |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) = ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
21188, 210eleqtrd 2703 . . . . . . . . . . . 12 |- ( ( ph /\ i e. ( 0..^ M ) ) -> L e. ( ( ( F |` ( A [,] B ) ) |` ( ( Q ` i ) (,) ( Q ` ( i + 1 ) ) ) ) lim CC ( Q ` ( i + 1 ) ) ) )
21236, 37, 38, 194, 206, 209, 211fourierdlem69 40392 . . . . . . . . . . 11 |- ( ph -> ( F |` ( A [,] B ) ) e. L^1 )
213193, 212eqeltrrd 2702 . . . . . . . . . 10 |- ( ph -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. L^1 )
214213adantr 481 . . . . . . . . 9 |- ( ( ph /\ T < X ) -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. L^1 )
215114, 122, 190, 191, 149, 214itgspliticc 23603 . . . . . . . 8 |- ( ( 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 ) )
216215oveq2d 6666 . . . . . . 7 |- ( ( 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 ) ) )
217216oveq2d 6666 . . . . . 6 |- ( ( 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 ) ) ) )
218107adantr 481 . . . . . . 7 |- ( ( ph /\ T < X ) -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x e. CC )
219215, 218eqeltrrd 2702 . . . . . . 7 |- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x + S. ( A [,] B ) ( F ` x ) _d x ) e. CC )
220169, 218, 219addsub12d 10415 . . . . . 6 |- ( ( 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 ) ) ) )
22160adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> F : RR --> CC )
2223adantr 481 . . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> A e. RR )
2238adantr 481 . . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> B e. RR )
224 simpr 477 . . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. ( A [,] B ) )
225 eliccre 39728 . . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR /\ x e. ( A [,] B ) ) -> x e. RR )
226222, 223, 224, 225syl3anc 1326 . . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. RR )
227221, 226ffvelrnd 6360 . . . . . . . . . . . 12 |- ( ( ph /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
228227, 213itgcl 23550 . . . . . . . . . . 11 |- ( ph -> S. ( A [,] B ) ( F ` x ) _d x e. CC )
229228adantr 481 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> S. ( A [,] B ) ( F ` x ) _d x e. CC )
230169, 169, 229subsub4d 10423 . . . . . . . . 9 |- ( ( 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 ) ) )
231230eqcomd 2628 . . . . . . . 8 |- ( ( 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 ) )
232231oveq2d 6666 . . . . . . 7 |- ( ( 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 ) ) )
233169subidd 10380 . . . . . . . . . 10 |- ( ( ph /\ T < X ) -> ( S. ( ( B - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] A ) ( F ` x ) _d x ) = 0 )
234233oveq1d 6665 . . . . . . . . 9 |- ( ( 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 10269 . . . . . . . . 9 |- -u S. ( A [,] B ) ( F ` x ) _d x = ( 0 - S. ( A [,] B ) ( F ` x ) _d x )
236234, 235syl6eqr 2674 . . . . . . . 8 |- ( ( 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 )
237236oveq2d 6666 . . . . . . 7 |- ( ( 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 ) )
238218, 229negsubd 10398 . . . . . . 7 |- ( ( 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 ) )
239232, 237, 2383eqtrd 2660 . . . . . 6 |- ( ( 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 ) )
240217, 220, 2393eqtrd 2660 . . . . 5 |- ( ( 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 ) )
241188, 189, 2403eqtr3d 2664 . . . 4 |- ( ( 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 ) )
242241oveq2d 6666 . . 3 |- ( ( 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 ) ) )
243108, 107, 228subsubd 10420 . . . . 5 |- ( 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 ) )
24493oveq2d 6666 . . . . . . 7 |- ( 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 ) )
245244, 109eqtrd 2656 . . . . . 6 |- ( ph -> ( S. ( ( A - X ) [,] A ) ( F ` x ) _d x - S. ( ( B - X ) [,] B ) ( F ` x ) _d x ) = 0 )
246245oveq1d 6665 . . . . 5 |- ( 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 ) )
247228addid2d 10237 . . . . 5 |- ( ph -> ( 0 + S. ( A [,] B ) ( F ` x ) _d x ) = S. ( A [,] B ) ( F ` x ) _d x )
248243, 246, 2473eqtrd 2660 . . . 4 |- ( 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 )
249248adantr 481 . . 3 |- ( ( 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 )
250184, 242, 2493eqtrd 2660 . 2 |- ( ( ph /\ T < X ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
25124adantr 481 . . . . 5 |- ( ( ph /\ X <_ T ) -> ( A - X ) e. RR )
25214adantr 481 . . . . 5 |- ( ( ph /\ X <_ T ) -> ( B - X ) e. RR )
2533adantr 481 . . . . . 6 |- ( ( ph /\ X <_ T ) -> A e. RR )
25424, 3, 39ltled 10185 . . . . . . 7 |- ( ph -> ( A - X ) <_ A )
255254adantr 481 . . . . . 6 |- ( ( ph /\ X <_ T ) -> ( A - X ) <_ A )
2566adantr 481 . . . . . . 7 |- ( ( ph /\ X <_ T ) -> X e. RR )
2578adantr 481 . . . . . . 7 |- ( ( ph /\ X <_ T ) -> B e. RR )
258 id 22 . . . . . . . . 9 |- ( X <_ T -> X <_ T )
259258, 1syl6breq 4694 . . . . . . . 8 |- ( X <_ T -> X <_ ( B - A ) )
260259adantl 482 . . . . . . 7 |- ( ( ph /\ X <_ T ) -> X <_ ( B - A ) )
261256, 257, 253, 260lesubd 10631 . . . . . 6 |- ( ( ph /\ X <_ T ) -> A <_ ( B - X ) )
262251, 252, 253, 255, 261eliccd 39726 . . . . 5 |- ( ( ph /\ X <_ T ) -> A e. ( ( A - X ) [,] ( B - X ) ) )
263158adantlr 751 . . . . 5 |- ( ( ( ph /\ X <_ T ) /\ x e. ( ( A - X ) [,] ( B - X ) ) ) -> ( F ` x ) e. CC )
264132, 102, 3, 39, 73eliood 39720 . . . . . . 7 |- ( ph -> A e. ( ( A - X ) (,) +oo ) )
26536, 1, 37, 38, 60, 54, 65, 84, 88, 24, 264fourierdlem105 40428 . . . . . 6 |- ( ph -> ( x e. ( ( A - X ) [,] A ) |-> ( F ` x ) ) e. L^1 )
266265adantr 481 . . . . 5 |- ( ( ph /\ X <_ T ) -> ( x e. ( ( A - X ) [,] A ) |-> ( F ` x ) ) e. L^1 )
2673leidd 10594 . . . . . . . 8 |- ( ph -> A <_ A )
2685rpge0d 11876 . . . . . . . . 9 |- ( ph -> 0 <_ X )
2698, 6subge02d 10619 . . . . . . . . 9 |- ( ph -> ( 0 <_ X <-> ( B - X ) <_ B ) )
270268, 269mpbid 222 . . . . . . . 8 |- ( ph -> ( B - X ) <_ B )
271 iccss 12241 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ A /\ ( B - X ) <_ B ) ) -> ( A [,] ( B - X ) ) C_ ( A [,] B ) )
2723, 8, 267, 270, 271syl22anc 1327 . . . . . . 7 |- ( ph -> ( A [,] ( B - X ) ) C_ ( A [,] B ) )
273 iccmbl 23334 . . . . . . . 8 |- ( ( A e. RR /\ ( B - X ) e. RR ) -> ( A [,] ( B - X ) ) e. dom vol )
2743, 14, 273syl2anc 693 . . . . . . 7 |- ( ph -> ( A [,] ( B - X ) ) e. dom vol )
275272, 274, 227, 213iblss 23571 . . . . . 6 |- ( ph -> ( x e. ( A [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 )
276275adantr 481 . . . . 5 |- ( ( ph /\ X <_ T ) -> ( x e. ( A [,] ( B - X ) ) |-> ( F ` x ) ) e. L^1 )
277251, 252, 262, 263, 266, 276itgspliticc 23603 . . . 4 |- ( ( 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 ) )
278268adantr 481 . . . . . . . . . 10 |- ( ( ph /\ X <_ T ) -> 0 <_ X )
279269adantr 481 . . . . . . . . . 10 |- ( ( ph /\ X <_ T ) -> ( 0 <_ X <-> ( B - X ) <_ B ) )
280278, 279mpbid 222 . . . . . . . . 9 |- ( ( ph /\ X <_ T ) -> ( B - X ) <_ B )
281253, 257, 252, 261, 280eliccd 39726 . . . . . . . 8 |- ( ( ph /\ X <_ T ) -> ( B - X ) e. ( A [,] B ) )
282227adantlr 751 . . . . . . . 8 |- ( ( ( ph /\ X <_ T ) /\ x e. ( A [,] B ) ) -> ( F ` x ) e. CC )
2838leidd 10594 . . . . . . . . . . 11 |- ( ph -> B <_ B )
284283adantr 481 . . . . . . . . . 10 |- ( ( ph /\ X <_ T ) -> B <_ B )
285 iccss 12241 . . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A <_ ( B - X ) /\ B <_ B ) ) -> ( ( B - X ) [,] B ) C_ ( A [,] B ) )
286253, 257, 261, 284, 285syl22anc 1327 . . . . . . . . 9 |- ( ( ph /\ X <_ T ) -> ( ( B - X ) [,] B ) C_ ( A [,] B ) )
287 iccmbl 23334 . . . . . . . . . . 11 |- ( ( ( B - X ) e. RR /\ B e. RR ) -> ( ( B - X ) [,] B ) e. dom vol )
28814, 8, 287syl2anc 693 . . . . . . . . . 10 |- ( ph -> ( ( B - X ) [,] B ) e. dom vol )
289288adantr 481 . . . . . . . . 9 |- ( ( ph /\ X <_ T ) -> ( ( B - X ) [,] B ) e. dom vol )
290213adantr 481 . . . . . . . . 9 |- ( ( ph /\ X <_ T ) -> ( x e. ( A [,] B ) |-> ( F ` x ) ) e. L^1 )
291286, 289, 282, 290iblss 23571 . . . . . . . 8 |- ( ( ph /\ X <_ T ) -> ( x e. ( ( B - X ) [,] B ) |-> ( F ` x ) ) e. L^1 )
292253, 257, 281, 282, 276, 291itgspliticc 23603 . . . . . . 7 |- ( ( 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 ) )
293292oveq1d 6665 . . . . . 6 |- ( ( 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 ) )
29460adantr 481 . . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> F : RR --> CC )
2953adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> A e. RR )
29614adantr 481 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> ( B - X ) e. RR )
297 simpr 477 . . . . . . . . . . 11 |- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> x e. ( A [,] ( B - X ) ) )
298 eliccre 39728 . . . . . . . . . . 11 |- ( ( A e. RR /\ ( B - X ) e. RR /\ x e. ( A [,] ( B - X ) ) ) -> x e. RR )
299295, 296, 297, 298syl3anc 1326 . . . . . . . . . 10 |- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> x e. RR )
300294, 299ffvelrnd 6360 . . . . . . . . 9 |- ( ( ph /\ x e. ( A [,] ( B - X ) ) ) -> ( F ` x ) e. CC )
301300, 275itgcl 23550 . . . . . . . 8 |- ( ph -> S. ( A [,] ( B - X ) ) ( F ` x ) _d x e. CC )
302301, 107, 107addsubassd 10412 . . . . . . 7 |- ( 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 ) ) )
303302adantr 481 . . . . . 6 |- ( ( 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 ) ) )
304185oveq2d 6666 . . . . . . . 8 |- ( 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 ) )
305301addid1d 10236 . . . . . . . 8 |- ( ph -> ( S. ( A [,] ( B - X ) ) ( F ` x ) _d x + 0 ) = S. ( A [,] ( B - X ) ) ( F ` x ) _d x )
306304, 305eqtrd 2656 . . . . . . 7 |- ( 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 )
307306adantr 481 . . . . . 6 |- ( ( 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 )
308293, 303, 3073eqtrrd 2661 . . . . 5 |- ( ( 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 ) )
309308oveq2d 6666 . . . 4 |- ( ( 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 ) ) )
31093adantr 481 . . . . . . 7 |- ( ( ph /\ X <_ T ) -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x = S. ( ( A - X ) [,] A ) ( F ` x ) _d x )
311107adantr 481 . . . . . . 7 |- ( ( ph /\ X <_ T ) -> S. ( ( B - X ) [,] B ) ( F ` x ) _d x e. CC )
312310, 311eqeltrrd 2702 . . . . . 6 |- ( ( ph /\ X <_ T ) -> S. ( ( A - X ) [,] A ) ( F ` x ) _d x e. CC )
313282, 290itgcl 23550 . . . . . 6 |- ( ( ph /\ X <_ T ) -> S. ( A [,] B ) ( F ` x ) _d x e. CC )
314312, 313, 311addsub12d 10415 . . . . 5 |- ( ( 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 ) ) )
315313, 312, 311addsubassd 10412 . . . . 5 |- ( ( 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 ) ) )
316314, 315eqtr4d 2659 . . . 4 |- ( ( 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 ) )
317277, 309, 3163eqtrd 2660 . . 3 |- ( ( 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 ) )
318310oveq2d 6666 . . 3 |- ( ( 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 ) )
319313, 312pncand 10393 . . 3 |- ( ( 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 )
320317, 318, 3193eqtrd 2660 . 2 |- ( ( ph /\ X <_ T ) -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
321250, 320, 47, 6ltlecasei 10145 1 |- ( ph -> S. ( ( A - X ) [,] ( B - X ) ) ( F ` x ) _d x = S. ( A [,] B ) ( F ` x ) _d x )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   u. cun 3572   C_ wss 3574   ifcif 4086   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   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   RR+crp 11832   (,)cioo 12175   (,]cioc 12176   [,]cicc 12178   ...cfz 12326  ..^cfzo 12465   |_cfl 12591   #chash 13117   -cn->ccncf 22679   volcvol 23232   L^1cibl 23386   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:  fourierdlem108  40431
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