Metamath Proof Explorer


Theorem fourierdlem108

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 where the integral was shifted by the exact period. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses fourierdlem108.a φ A
fourierdlem108.b φ B
fourierdlem108.t T = B A
fourierdlem108.x φ X +
fourierdlem108.p P = m p 0 m | p 0 = A p m = B i 0 ..^ m p i < p i + 1
fourierdlem108.m φ M
fourierdlem108.q φ Q P M
fourierdlem108.f φ F :
fourierdlem108.fper φ x F x + T = F x
fourierdlem108.fcn φ i 0 ..^ M F Q i Q i + 1 : Q i Q i + 1 cn
fourierdlem108.r φ i 0 ..^ M R F Q i Q i + 1 lim Q i
fourierdlem108.l φ i 0 ..^ M L F Q i Q i + 1 lim Q i + 1
Assertion fourierdlem108 φ A X B X F x dx = A B F x dx

Proof

Step Hyp Ref Expression
1 fourierdlem108.a φ A
2 fourierdlem108.b φ B
3 fourierdlem108.t T = B A
4 fourierdlem108.x φ X +
5 fourierdlem108.p P = m p 0 m | p 0 = A p m = B i 0 ..^ m p i < p i + 1
6 fourierdlem108.m φ M
7 fourierdlem108.q φ Q P M
8 fourierdlem108.f φ F :
9 fourierdlem108.fper φ x F x + T = F x
10 fourierdlem108.fcn φ i 0 ..^ M F Q i Q i + 1 : Q i Q i + 1 cn
11 fourierdlem108.r φ i 0 ..^ M R F Q i Q i + 1 lim Q i
12 fourierdlem108.l φ i 0 ..^ M L F Q i Q i + 1 lim Q i + 1
13 eqid m p 0 m | p 0 = A X p m = A i 0 ..^ m p i < p i + 1 = m p 0 m | p 0 = A X p m = A i 0 ..^ m p i < p i + 1
14 oveq1 w = y w + k T = y + k T
15 14 eleq1d w = y w + k T ran Q y + k T ran Q
16 15 rexbidv w = y k w + k T ran Q k y + k T ran Q
17 16 cbvrabv w A X A | k w + k T ran Q = y A X A | k y + k T ran Q
18 17 uneq2i A X A w A X A | k w + k T ran Q = A X A y A X A | k y + k T ran Q
19 oveq1 l = k l T = k T
20 19 oveq2d l = k w + l T = w + k T
21 20 eleq1d l = k w + l T ran Q w + k T ran Q
22 21 cbvrexvw l w + l T ran Q k w + k T ran Q
23 22 rgenw w A X A l w + l T ran Q k w + k T ran Q
24 rabbi w A X A l w + l T ran Q k w + k T ran Q w A X A | l w + l T ran Q = w A X A | k w + k T ran Q
25 23 24 mpbi w A X A | l w + l T ran Q = w A X A | k w + k T ran Q
26 25 uneq2i A X A w A X A | l w + l T ran Q = A X A w A X A | k w + k T ran Q
27 26 fveq2i A X A w A X A | l w + l T ran Q = A X A w A X A | k w + k T ran Q
28 27 oveq1i A X A w A X A | l w + l T ran Q 1 = A X A w A X A | k w + k T ran Q 1
29 isoeq5 A X A w A X A | l w + l T ran Q = A X A w A X A | k w + k T ran Q g Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | l w + l T ran Q g Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | k w + k T ran Q
30 26 29 ax-mp g Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | l w + l T ran Q g Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | k w + k T ran Q
31 isoeq1 g = f g Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | k w + k T ran Q f Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | k w + k T ran Q
32 30 31 syl5bb g = f g Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | l w + l T ran Q f Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | k w + k T ran Q
33 32 cbviotavw ι g | g Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | l w + l T ran Q = ι f | f Isom < , < 0 A X A w A X A | l w + l T ran Q 1 A X A w A X A | k w + k T ran Q
34 id w = x w = x
35 oveq2 w = x B w = B x
36 35 oveq1d w = x B w T = B x T
37 36 fveq2d w = x B w T = B x T
38 37 oveq1d w = x B w T T = B x T T
39 34 38 oveq12d w = x w + B w T T = x + B x T T
40 39 cbvmptv w w + B w T T = x x + B x T T
41 eqeq1 w = y w = B y = B
42 id w = y w = y
43 41 42 ifbieq2d w = y if w = B A w = if y = B A y
44 43 cbvmptv w A B if w = B A w = y A B if y = B A y
45 fveq2 z = x w w + B w T T z = w w + B w T T x
46 45 fveq2d z = x w A B if w = B A w w w + B w T T z = w A B if w = B A w w w + B w T T x
47 46 breq2d z = x Q j w A B if w = B A w w w + B w T T z Q j w A B if w = B A w w w + B w T T x
48 47 rabbidv z = x j 0 ..^ M | Q j w A B if w = B A w w w + B w T T z = j 0 ..^ M | Q j w A B if w = B A w w w + B w T T x
49 fveq2 j = i Q j = Q i
50 49 breq1d j = i Q j w A B if w = B A w w w + B w T T x Q i w A B if w = B A w w w + B w T T x
51 50 cbvrabv j 0 ..^ M | Q j w A B if w = B A w w w + B w T T x = i 0 ..^ M | Q i w A B if w = B A w w w + B w T T x
52 48 51 eqtrdi z = x j 0 ..^ M | Q j w A B if w = B A w w w + B w T T z = i 0 ..^ M | Q i w A B if w = B A w w w + B w T T x
53 52 supeq1d z = x sup j 0 ..^ M | Q j w A B if w = B A w w w + B w T T z < = sup i 0 ..^ M | Q i w A B if w = B A w w w + B w T T x <
54 53 cbvmptv z sup j 0 ..^ M | Q j w A B if w = B A w w w + B w T T z < = x sup i 0 ..^ M | Q i w A B if w = B A w w w + B w T T x <
55 1 2 3 4 5 6 7 8 9 10 11 12 13 18 28 33 40 44 54 fourierdlem107 φ A X B X F x dx = A B F x dx