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Theorem List for Metamath Proof Explorer - 40601-40700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwallispilem2 40601* A first set of properties for the sequence 𝐼 that will be used in the proof of the Wallis product formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)       ((𝐼‘0) = π ∧ (𝐼‘1) = 2 ∧ (𝑁 ∈ (ℤ‘2) → (𝐼𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2)))))
 
Theoremwallispilem3 40602* I maps to real values. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)       (𝑁 ∈ ℕ0 → (𝐼𝑁) ∈ ℝ+)
 
Theoremwallispilem4 40603* 𝐹 maps to explicit expression for the ratio of two consecutive values of 𝐼. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
𝐹 = (𝑘 ∈ ℕ ↦ (((2 · 𝑘) / ((2 · 𝑘) − 1)) · ((2 · 𝑘) / ((2 · 𝑘) + 1))))    &   𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑧)↑𝑛) d𝑧)    &   𝐺 = (𝑛 ∈ ℕ ↦ ((𝐼‘(2 · 𝑛)) / (𝐼‘((2 · 𝑛) + 1))))    &   𝐻 = (𝑛 ∈ ℕ ↦ ((π / 2) · (1 / (seq1( · , 𝐹)‘𝑛))))       𝐺 = 𝐻
 
Theoremwallispilem5 40604* The sequence 𝐻 converges to 1. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
𝐹 = (𝑘 ∈ ℕ ↦ (((2 · 𝑘) / ((2 · 𝑘) − 1)) · ((2 · 𝑘) / ((2 · 𝑘) + 1))))    &   𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)    &   𝐺 = (𝑛 ∈ ℕ ↦ ((𝐼‘(2 · 𝑛)) / (𝐼‘((2 · 𝑛) + 1))))    &   𝐻 = (𝑛 ∈ ℕ ↦ ((π / 2) · (1 / (seq1( · , 𝐹)‘𝑛))))    &   𝐿 = (𝑛 ∈ ℕ ↦ (((2 · 𝑛) + 1) / (2 · 𝑛)))       𝐻 ⇝ 1
 
Theoremwallispi 40605* Wallis' formula for π : Wallis' product converges to π / 2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐹 = (𝑘 ∈ ℕ ↦ (((2 · 𝑘) / ((2 · 𝑘) − 1)) · ((2 · 𝑘) / ((2 · 𝑘) + 1))))    &   𝑊 = (𝑛 ∈ ℕ ↦ (seq1( · , 𝐹)‘𝑛))       𝑊 ⇝ (π / 2)
 
Theoremwallispi2lem1 40606 An intermediate step between the first version of the Wallis' formula for π and the second version of Wallis' formula. This second version will then be used to prove Stirling's approximation formula for the factorial. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
(𝑁 ∈ ℕ → (seq1( · , (𝑘 ∈ ℕ ↦ (((2 · 𝑘) / ((2 · 𝑘) − 1)) · ((2 · 𝑘) / ((2 · 𝑘) + 1)))))‘𝑁) = ((1 / ((2 · 𝑁) + 1)) · (seq1( · , (𝑘 ∈ ℕ ↦ (((2 · 𝑘)↑4) / (((2 · 𝑘) · ((2 · 𝑘) − 1))↑2))))‘𝑁)))
 
Theoremwallispi2lem2 40607 Two expressions are proven to be equal, and this is used to complete the proof of the second version of Wallis' formula for π . (Contributed by Glauco Siliprandi, 30-Jun-2017.)
(𝑁 ∈ ℕ → (seq1( · , (𝑘 ∈ ℕ ↦ (((2 · 𝑘)↑4) / (((2 · 𝑘) · ((2 · 𝑘) − 1))↑2))))‘𝑁) = (((2↑(4 · 𝑁)) · ((!‘𝑁)↑4)) / ((!‘(2 · 𝑁))↑2)))
 
Theoremwallispi2 40608 An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to prove Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑉 = (𝑛 ∈ ℕ ↦ ((((2↑(4 · 𝑛)) · ((!‘𝑛)↑4)) / ((!‘(2 · 𝑛))↑2)) / ((2 · 𝑛) + 1)))       𝑉 ⇝ (π / 2)
 
20.32.14  Stirling's approximation formula for ` n ` factorial
 
Theoremstirlinglem1 40609 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
𝐻 = (𝑛 ∈ ℕ ↦ ((𝑛↑2) / (𝑛 · ((2 · 𝑛) + 1))))    &   𝐹 = (𝑛 ∈ ℕ ↦ (1 − (1 / ((2 · 𝑛) + 1))))    &   𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))    &   𝐿 = (𝑛 ∈ ℕ ↦ (1 / 𝑛))       𝐻 ⇝ (1 / 2)
 
Theoremstirlinglem2 40610 𝐴 maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))       (𝑁 ∈ ℕ → (𝐴𝑁) ∈ ℝ+)
 
Theoremstirlinglem3 40611 Long but simple algebraic transformations are applied to show that 𝑉, the Wallis formula for π , can be expressed in terms of 𝐴, the Stirling's approximation formula for the factorial, up to a constant factor. This will allow (in a later theorem) to determine the right constant factor to be put into the 𝐴, in order to get the exact Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))    &   𝐸 = (𝑛 ∈ ℕ ↦ ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))    &   𝑉 = (𝑛 ∈ ℕ ↦ ((((2↑(4 · 𝑛)) · ((!‘𝑛)↑4)) / ((!‘(2 · 𝑛))↑2)) / ((2 · 𝑛) + 1)))       𝑉 = (𝑛 ∈ ℕ ↦ ((((𝐴𝑛)↑4) / ((𝐷𝑛)↑2)) · ((𝑛↑2) / (𝑛 · ((2 · 𝑛) + 1)))))
 
Theoremstirlinglem4 40612* Algebraic manipulation of ((𝐵 n ) - ( B (𝑛 + 1))). It will be used in other theorems to show that 𝐵 is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴𝑛)))    &   𝐽 = (𝑛 ∈ ℕ ↦ ((((1 + (2 · 𝑛)) / 2) · (log‘((𝑛 + 1) / 𝑛))) − 1))       (𝑁 ∈ ℕ → ((𝐵𝑁) − (𝐵‘(𝑁 + 1))) = (𝐽𝑁))
 
Theoremstirlinglem5 40613* If 𝑇 is between 0 and 1, then a series (without alternating negative and positive terms) is given that converges to log (1+T)/(1-T) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐷 = (𝑗 ∈ ℕ ↦ ((-1↑(𝑗 − 1)) · ((𝑇𝑗) / 𝑗)))    &   𝐸 = (𝑗 ∈ ℕ ↦ ((𝑇𝑗) / 𝑗))    &   𝐹 = (𝑗 ∈ ℕ ↦ (((-1↑(𝑗 − 1)) · ((𝑇𝑗) / 𝑗)) + ((𝑇𝑗) / 𝑗)))    &   𝐻 = (𝑗 ∈ ℕ0 ↦ (2 · ((1 / ((2 · 𝑗) + 1)) · (𝑇↑((2 · 𝑗) + 1)))))    &   𝐺 = (𝑗 ∈ ℕ0 ↦ ((2 · 𝑗) + 1))    &   (𝜑𝑇 ∈ ℝ+)    &   (𝜑 → (abs‘𝑇) < 1)       (𝜑 → seq0( + , 𝐻) ⇝ (log‘((1 + 𝑇) / (1 − 𝑇))))
 
Theoremstirlinglem6 40614* A series that converges to log (N+1)/N. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐻 = (𝑗 ∈ ℕ0 ↦ (2 · ((1 / ((2 · 𝑗) + 1)) · ((1 / ((2 · 𝑁) + 1))↑((2 · 𝑗) + 1)))))       (𝑁 ∈ ℕ → seq0( + , 𝐻) ⇝ (log‘((𝑁 + 1) / 𝑁)))
 
Theoremstirlinglem7 40615* Algebraic manipulation of the formula for J(n). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐽 = (𝑛 ∈ ℕ ↦ ((((1 + (2 · 𝑛)) / 2) · (log‘((𝑛 + 1) / 𝑛))) − 1))    &   𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘))))    &   𝐻 = (𝑘 ∈ ℕ0 ↦ (2 · ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑((2 · 𝑘) + 1)))))       (𝑁 ∈ ℕ → seq1( + , 𝐾) ⇝ (𝐽𝑁))
 
Theoremstirlinglem8 40616 If 𝐴 converges to 𝐶, then 𝐹 converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑛𝜑    &   𝑛𝐴    &   𝑛𝐷    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))    &   (𝜑𝐴:ℕ⟶ℝ+)    &   𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛)↑4) / ((𝐷𝑛)↑2)))    &   𝐿 = (𝑛 ∈ ℕ ↦ ((𝐴𝑛)↑4))    &   𝑀 = (𝑛 ∈ ℕ ↦ ((𝐷𝑛)↑2))    &   ((𝜑𝑛 ∈ ℕ) → (𝐷𝑛) ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴𝐶)       (𝜑𝐹 ⇝ (𝐶↑2))
 
Theoremstirlinglem9 40617* ((𝐵𝑁) − (𝐵‘(𝑁 + 1))) is expressed as a limit of a series. This result will be used both to prove that 𝐵 is decreasing and to prove that 𝐵 is bounded (below). It will follow that 𝐵 converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴𝑛)))    &   𝐽 = (𝑛 ∈ ℕ ↦ ((((1 + (2 · 𝑛)) / 2) · (log‘((𝑛 + 1) / 𝑛))) − 1))    &   𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘))))       (𝑁 ∈ ℕ → seq1( + , 𝐾) ⇝ ((𝐵𝑁) − (𝐵‘(𝑁 + 1))))
 
Theoremstirlinglem10 40618* A bound for any B(N)-B(N + 1) that will allow to find a lower bound for the whole 𝐵 sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴𝑛)))    &   𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘))))    &   𝐿 = (𝑘 ∈ ℕ ↦ ((1 / (((2 · 𝑁) + 1)↑2))↑𝑘))       (𝑁 ∈ ℕ → ((𝐵𝑁) − (𝐵‘(𝑁 + 1))) ≤ ((1 / 4) · (1 / (𝑁 · (𝑁 + 1)))))
 
Theoremstirlinglem11 40619* 𝐵 is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴𝑛)))    &   𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘))))       (𝑁 ∈ ℕ → (𝐵‘(𝑁 + 1)) < (𝐵𝑁))
 
Theoremstirlinglem12 40620* The sequence 𝐵 is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴𝑛)))    &   𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))       (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵𝑁))
 
Theoremstirlinglem13 40621* 𝐵 is decreasing and has a lower bound, then it converges. Since 𝐵 is log𝐴, in another theorem it is proven that 𝐴 converges as well. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴𝑛)))       𝑑 ∈ ℝ 𝐵𝑑
 
Theoremstirlinglem14 40622* The sequence 𝐴 converges to a positive real. This proves that the Stirling's formula converges to the factorial, up to a constant. In another theorem, using Wallis' formula for π& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴𝑛)))       𝑐 ∈ ℝ+ 𝐴𝑐
 
Theoremstirlinglem15 40623* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 40624 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑛𝜑    &   𝑆 = (𝑛 ∈ ℕ0 ↦ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))    &   𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛))))    &   𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛)))    &   𝐸 = (𝑛 ∈ ℕ ↦ ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))    &   𝑉 = (𝑛 ∈ ℕ ↦ ((((2↑(4 · 𝑛)) · ((!‘𝑛)↑4)) / ((!‘(2 · 𝑛))↑2)) / ((2 · 𝑛) + 1)))    &   𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴𝑛)↑4) / ((𝐷𝑛)↑2)))    &   𝐻 = (𝑛 ∈ ℕ ↦ ((𝑛↑2) / (𝑛 · ((2 · 𝑛) + 1))))    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴𝐶)       (𝜑 → (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆𝑛))) ⇝ 1)
 
Theoremstirling 40624 Stirling's approximation formula for 𝑛 factorial. The proof follows two major steps: first it is proven that 𝑆 and 𝑛 factorial are asymptotically equivalent, up to an unknown constant. Then, using Wallis' formula for π it is proven that the unknown constant is the square root of π and then the exact Stirling's formula is established. This is Metamath 100 proof #90. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑆 = (𝑛 ∈ ℕ0 ↦ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))       (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆𝑛))) ⇝ 1
 
Theoremstirlingr 40625 Stirling's approximation formula for 𝑛 factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 40624 is proven for convergence in the topology of complex numbers. The variable 𝑅 is used to denote convergence with respect to the standard topology on the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑆 = (𝑛 ∈ ℕ0 ↦ ((√‘((2 · π) · 𝑛)) · ((𝑛 / e)↑𝑛)))    &   𝑅 = (⇝𝑡‘(topGen‘ran (,)))       (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆𝑛)))𝑅1
 
20.32.15  Dirichlet kernel
 
Theoremdirkerval 40626* The Nth Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))       (𝑁 ∈ ℕ → (𝐷𝑁) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
 
Theoremdirker2re 40627 The Dirchlet Kernel value is a real if the argument is not a multiple of π . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ ¬ (𝑆 mod (2 · π)) = 0) → ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2)))) ∈ ℝ)
 
Theoremdirkerdenne0 40628 The Dirchlet Kernel denominator is never 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑆 ∈ ℝ ∧ ¬ (𝑆 mod (2 · π)) = 0) → ((2 · π) · (sin‘(𝑆 / 2))) ≠ 0)
 
Theoremdirkerval2 40629* The Nth Dirichlet Kernel evaluated at a specific point 𝑆. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))       ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) → ((𝐷𝑁)‘𝑆) = if((𝑆 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2))))))
 
Theoremdirkerre 40630* The Dirichlet Kernel at any point evaluates to a real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))       ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) → ((𝐷𝑁)‘𝑆) ∈ ℝ)
 
Theoremdirkerper 40631* the Dirichlet Kernel has period . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))    &   𝑇 = (2 · π)       ((𝑁 ∈ ℕ ∧ 𝑥 ∈ ℝ) → ((𝐷𝑁)‘(𝑥 + 𝑇)) = ((𝐷𝑁)‘𝑥))
 
Theoremdirkerf 40632* For any natural number 𝑁, the Dirichlet Kernel (𝐷𝑁) is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))       (𝑁 ∈ ℕ → (𝐷𝑁):ℝ⟶ℝ)
 
Theoremdirkertrigeqlem1 40633* Sum of an even number of alternating cos values. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ ℕ → Σ𝑛 ∈ (1...(2 · 𝐾))(cos‘(𝑛 · π)) = 0)
 
Theoremdirkertrigeqlem2 40634* Trigonomic equality lemma for the Dirichlet Kernel trigonomic equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → (sin‘𝐴) ≠ 0)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (((1 / 2) + Σ𝑛 ∈ (1...𝑁)(cos‘(𝑛 · 𝐴))) / π) = ((sin‘((𝑁 + (1 / 2)) · 𝐴)) / ((2 · π) · (sin‘(𝐴 / 2)))))
 
Theoremdirkertrigeqlem3 40635* Trigonometric equality lemma for the Dirichlet Kernel trigonometric equality. Here we handle the case for an angle that's an odd multiple of π. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℤ)    &   𝐴 = (((2 · 𝐾) + 1) · π)       (𝜑 → (((1 / 2) + Σ𝑛 ∈ (1...𝑁)(cos‘(𝑛 · 𝐴))) / π) = ((sin‘((𝑁 + (1 / 2)) · 𝐴)) / ((2 · π) · (sin‘(𝐴 / 2)))))
 
Theoremdirkertrigeq 40636* Trigonometric equality for the Dirichlet kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))    &   (𝜑𝑁 ∈ ℕ)    &   𝐹 = (𝐷𝑁)    &   𝐻 = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))       (𝜑𝐹 = 𝐻)
 
Theoremdirkeritg 40637* The definite integral of the Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝑥 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑥)) / ((2 · π) · (sin‘(𝑥 / 2)))))))    &   (𝜑𝑁 ∈ ℕ)    &   𝐹 = (𝐷𝑁)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) / π))       (𝜑 → ∫(𝐴(,)𝐵)(𝐹𝑥) d𝑥 = ((𝐺𝐵) − (𝐺𝐴)))
 
Theoremdirkercncflem1 40638* If 𝑌 is a multiple of π then it belongs to an open inerval (𝐴(,)𝐵) such that for any other point 𝑦 in the interval, cos y/2 and sin y/2 are non zero. Such an interval is needed to apply De L'Hopital theorem. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (𝑌 − π)    &   𝐵 = (𝑌 + π)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑 → (𝑌 mod (2 · π)) = 0)       (𝜑 → (𝑌 ∈ (𝐴(,)𝐵) ∧ ∀𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})((sin‘(𝑦 / 2)) ≠ 0 ∧ (cos‘(𝑦 / 2)) ≠ 0)))
 
Theoremdirkercncflem2 40639* Lemma used to prove that the Dirichlet Kernel is continuous at 𝑌 points that are multiples of (2 · π). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))    &   𝐹 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑦)))    &   𝐺 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((2 · π) · (sin‘(𝑦 / 2))))    &   ((𝜑𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (sin‘(𝑦 / 2)) ≠ 0)    &   𝐻 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ ((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑦))))    &   𝐼 = (𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌}) ↦ (π · (cos‘(𝑦 / 2))))    &   𝐿 = (𝑤 ∈ (𝐴(,)𝐵) ↦ (((𝑁 + (1 / 2)) · (cos‘((𝑁 + (1 / 2)) · 𝑤))) / (π · (cos‘(𝑤 / 2)))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑌 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝑌 mod (2 · π)) = 0)    &   ((𝜑𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (cos‘(𝑦 / 2)) ≠ 0)       (𝜑 → ((𝐷𝑁)‘𝑌) ∈ (((𝐷𝑁) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) lim 𝑌))
 
Theoremdirkercncflem3 40640* The Dirichlet Kernel is continuous at 𝑌 points that are multiples of (2 · π). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))    &   𝐴 = (𝑌 − π)    &   𝐵 = (𝑌 + π)    &   𝐹 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))    &   𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((2 · π) · (sin‘(𝑦 / 2))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑 → (𝑌 mod (2 · π)) = 0)       (𝜑 → ((𝐷𝑁)‘𝑌) ∈ ((𝐷𝑁) lim 𝑌))
 
Theoremdirkercncflem4 40641* The Dirichlet Kernel is continuos at points that are not multiple of 2 π . This is the easier condition, for the proof of the continuity of the Dirichlet kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑 → (𝑌 mod (2 · π)) ≠ 0)    &   𝐴 = (⌊‘(𝑌 / (2 · π)))    &   𝐵 = (𝐴 + 1)    &   𝐶 = (𝐴 · (2 · π))    &   𝐸 = (𝐵 · (2 · π))       (𝜑 → (𝐷𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑌))
 
Theoremdirkercncf 40642* For any natural number 𝑁, the Dirichlet Kernel (𝐷𝑁) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ ℕ ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑦)) / ((2 · π) · (sin‘(𝑦 / 2)))))))       (𝑁 ∈ ℕ → (𝐷𝑁) ∈ (ℝ–cn→ℝ))
 
20.32.16  Fourier Series
 
Theoremfourierdlem1 40643 A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))    &   (𝜑𝐼 ∈ (0..^𝑀))    &   (𝜑𝑋 ∈ ((𝑄𝐼)[,](𝑄‘(𝐼 + 1))))       (𝜑𝑋 ∈ (𝐴[,]𝐵))
 
Theoremfourierdlem2 40644* Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})       (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
 
Theoremfourierdlem3 40645* Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ ((-π[,]π) ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})       (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ ((-π[,]π) ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
 
Theoremfourierdlem4 40646* 𝐸 is a function that maps any point to a periodic corresponding point in (𝐴, 𝐵]. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))       (𝜑𝐸:ℝ⟶(𝐴(,]𝐵))
 
Theoremfourierdlem5 40647* 𝑆 is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑆 = (𝑥 ∈ (-π[,]π) ↦ (sin‘((𝑋 + (1 / 2)) · 𝑥)))       (𝑋 ∈ ℝ → 𝑆:(-π[,]π)⟶ℝ)
 
Theoremfourierdlem6 40648 𝑋 is in the periodic partition, when the considered interval is centered at 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐼 ∈ ℤ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑𝐼 < 𝐽)    &   (𝜑 → (𝑋 + (𝐼 · 𝑇)) ∈ (𝐴[,]𝐵))    &   (𝜑 → (𝑋 + (𝐽 · 𝑇)) ∈ (𝐴[,]𝐵))       (𝜑𝐽 = (𝐼 + 1))
 
Theoremfourierdlem7 40649* The difference between a point and it's periodic image in the interval, is decreasing. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑋𝑌)       (𝜑 → ((𝐸𝑌) − 𝑌) ≤ ((𝐸𝑋) − 𝑋))
 
Theoremfourierdlem8 40650 A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))    &   (𝜑𝐼 ∈ (0..^𝑀))       (𝜑 → ((𝑄𝐼)[,](𝑄‘(𝐼 + 1))) ⊆ (𝐴[,]𝐵))
 
Theoremfourierdlem9 40651* 𝐻 is a complex function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))       (𝜑𝐻:(-π[,]π)⟶ℝ)
 
Theoremfourierdlem10 40652 Condition on the bounds of a non empty subinterval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))       (𝜑 → (𝐴𝐶𝐷𝐵))
 
Theoremfourierdlem11 40653* If there is a partition, than the lower bound is strictly less than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))       (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵))
 
Theoremfourierdlem12 40654* A point of a partition is not an element of any open interval determined by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝑋 ∈ ran 𝑄)       ((𝜑𝑖 ∈ (0..^𝑀)) → ¬ 𝑋 ∈ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
 
Theoremfourierdlem13 40655* Value of 𝑉 in terms of value of 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   (𝜑𝐼 ∈ (0...𝑀))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))       (𝜑 → ((𝑄𝐼) = ((𝑉𝐼) − 𝑋) ∧ (𝑉𝐼) = (𝑋 + (𝑄𝐼))))
 
Theoremfourierdlem14 40656* Given the partition 𝑉, 𝑄 is the partition shifted to the left by 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))       (𝜑𝑄 ∈ (𝑂𝑀))
 
Theoremfourierdlem15 40657* The range of the partition is between its starting point and its ending point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))       (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))
 
Theoremfourierdlem16 40658* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐶 = (-π(,)π)    &   (𝜑 → (𝐹𝐶) ∈ 𝐿1)    &   𝐴 = (𝑛 ∈ ℕ0 ↦ (∫𝐶((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (((𝐴𝑁) ∈ ℝ ∧ (𝑥𝐶 ↦ (𝐹𝑥)) ∈ 𝐿1) ∧ ∫𝐶((𝐹𝑥) · (cos‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ))
 
Theoremfourierdlem17 40659* The defined 𝐿 is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝐿 = (𝑥 ∈ (𝐴(,]𝐵) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))       (𝜑𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
 
Theoremfourierdlem18 40660* The function 𝑆 is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑁 ∈ ℝ)    &   𝑆 = (𝑠 ∈ (-π[,]π) ↦ (sin‘((𝑁 + (1 / 2)) · 𝑠)))       (𝜑𝑆 ∈ ((-π[,]π)–cn→ℝ))
 
Theoremfourierdlem19 40661* If two elements of 𝐷 have the same periodic image in (𝐴(,]𝐵) then they are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝑋 ∈ ℝ)    &   𝐷 = {𝑦 ∈ ((𝐴 + 𝑋)(,](𝐵 + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ 𝐶}    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑊𝐷)    &   (𝜑𝑍𝐷)    &   (𝜑 → (𝐸𝑍) = (𝐸𝑊))       (𝜑 → ¬ 𝑊 < 𝑍)
 
Theoremfourierdlem20 40662* Every interval in the partition 𝑆 is included in an interval of the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝑄:(0...𝑀)⟶ℝ)    &   (𝜑 → (𝑄‘0) ≤ 𝐴)    &   (𝜑𝐵 ≤ (𝑄𝑀))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))    &   (𝜑𝑆 Isom < , < ((0...𝑁), 𝑇))    &   𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) ≤ (𝑆𝐽)}, ℝ, < )       (𝜑 → ∃𝑖 ∈ (0..^𝑀)((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))
 
Theoremfourierdlem21 40663* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐶 = (-π(,)π)    &   (𝜑 → (𝐹𝐶) ∈ 𝐿1)    &   𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (((𝐵𝑁) ∈ ℝ ∧ (𝑥𝐶 ↦ ((𝐹𝑥) · (sin‘(𝑁 · 𝑥)))) ∈ 𝐿1) ∧ ∫𝐶((𝐹𝑥) · (sin‘(𝑁 · 𝑥))) d𝑥 ∈ ℝ))
 
Theoremfourierdlem22 40664* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   𝐶 = (-π(,)π)    &   (𝜑 → (𝐹𝐶) ∈ 𝐿1)    &   𝐴 = (𝑛 ∈ ℕ0 ↦ (∫𝐶((𝐹𝑥) · (cos‘(𝑛 · 𝑥))) d𝑥 / π))    &   𝐵 = (𝑛 ∈ ℕ ↦ (∫𝐶((𝐹𝑥) · (sin‘(𝑛 · 𝑥))) d𝑥 / π))       (𝜑 → ((𝑛 ∈ ℕ0 → (𝐴𝑛) ∈ ℝ) ∧ (𝑛 ∈ ℕ → (𝐵𝑛) ∈ ℝ)))
 
Theoremfourierdlem23 40665* If 𝐹 is continuous and 𝑋 is constant, then (𝐹‘(𝑋 + 𝑠)) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹 ∈ (𝐴cn→ℂ))    &   (𝜑𝐵 ⊆ ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑠𝐵) → (𝑋 + 𝑠) ∈ 𝐴)       (𝜑 → (𝑠𝐵 ↦ (𝐹‘(𝑋 + 𝑠))) ∈ (𝐵cn→ℂ))
 
Theoremfourierdlem24 40666 A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ((-π[,]π) ∖ {0}) → (𝐴 mod (2 · π)) ≠ 0)
 
Theoremfourierdlem25 40667* If 𝐶 is not in the range of the partition, then it is in an open interval induced by the partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄:(0...𝑀)⟶ℝ)    &   (𝜑𝐶 ∈ ((𝑄‘0)[,](𝑄𝑀)))    &   (𝜑 → ¬ 𝐶 ∈ ran 𝑄)    &   𝐼 = sup({𝑘 ∈ (0..^𝑀) ∣ (𝑄𝑘) < 𝐶}, ℝ, < )       (𝜑 → ∃𝑗 ∈ (0..^𝑀)𝐶 ∈ ((𝑄𝑗)(,)(𝑄‘(𝑗 + 1))))
 
Theoremfourierdlem26 40668* Periodic image of a point 𝑌 that's in the period that begins with the point 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → (𝐸𝑋) = 𝐵)    &   (𝜑𝑌 ∈ (𝑋(,](𝑋 + 𝑇)))       (𝜑 → (𝐸𝑌) = (𝐴 + (𝑌𝑋)))
 
Theoremfourierdlem27 40669 A partition open interval is a subset of the partitioned open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝑄:(0...𝑀)⟶(𝐴[,]𝐵))    &   (𝜑𝐼 ∈ (0..^𝑀))       (𝜑 → ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (𝐴(,)𝐵))
 
Theoremfourierdlem28 40670* Derivative of (𝐹‘(𝑋 + 𝑠)). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   𝐷 = (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))    &   (𝜑𝐷:((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)       (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐹‘(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐷‘(𝑋 + 𝑠))))
 
Theoremfourierdlem29 40671* Explicit function value for 𝐾 applied to 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))       (𝐴 ∈ (-π[,]π) → (𝐾𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))
 
Theoremfourierdlem30 40672* Sum of three small pieces is less than ε. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐼 ↦ (𝐹 · -𝐺)) ∈ 𝐿1)    &   ((𝜑𝑥𝐼) → 𝐹 ∈ ℂ)    &   ((𝜑𝑥𝐼) → 𝐺 ∈ ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   𝑋 = (abs‘𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   𝑌 = (abs‘𝐶)    &   𝑍 = (abs‘∫𝐼(𝐹 · -𝐺) d𝑥)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑅 ∈ ℝ)    &   (𝜑 → ((((𝑋 + 𝑌) + 𝑍) / 𝐸) + 1) ≤ 𝑅)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (abs‘𝐵) ≤ 1)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑 → (abs‘𝐷) ≤ 1)       (𝜑 → (abs‘(((𝐴 · -(𝐵 / 𝑅)) − (𝐶 · -(𝐷 / 𝑅))) − ∫𝐼(𝐹 · -(𝐺 / 𝑅)) d𝑥)) < 𝐸)
 
Theoremfourierdlem31 40673* If 𝐴 is finite and for any element in 𝐴 there is a number 𝑚 such that a property holds for all numbers larger than 𝑚, then there is a number 𝑛 such that the property holds for all numbers larger than 𝑛 and for all elements in 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 29-Sep-2020.)
𝑖𝜑    &   𝑟𝜑    &   𝑖𝑉    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ∀𝑖𝐴𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)    &   𝑀 = {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}    &   𝑉 = (𝑖𝐴 ↦ inf(𝑀, ℝ, < ))    &   𝑁 = sup(ran 𝑉, ℝ, < )       (𝜑 → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖𝐴 𝜒)
 
Theoremfourierdlem32 40674 Limit of a continuous function on an open subinterval. Lower bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝑅 ∈ (𝐹 lim 𝐴))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))    &   𝑌 = if(𝐶 = 𝐴, 𝑅, (𝐹𝐶))    &   𝐽 = ((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵))       (𝜑𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim 𝐶))
 
Theoremfourierdlem33 40675 Limit of a continuous function on an open subinterval. Upper bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑𝐿 ∈ (𝐹 lim 𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   (𝜑 → (𝐶(,)𝐷) ⊆ (𝐴(,)𝐵))    &   𝑌 = if(𝐷 = 𝐵, 𝐿, (𝐹𝐷))    &   𝐽 = ((TopOpen‘ℂfld) ↾t ((𝐴(,)𝐵) ∪ {𝐵}))       (𝜑𝑌 ∈ ((𝐹 ↾ (𝐶(,)𝐷)) lim 𝐷))
 
Theoremfourierdlem34 40676* A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))       (𝜑𝑄:(0...𝑀)–1-1→ℝ)
 
Theoremfourierdlem35 40677 There is a single point in (𝐴(,]𝐵) that's distant from 𝑋 a multiple integer of 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐼 ∈ ℤ)    &   (𝜑𝐽 ∈ ℤ)    &   (𝜑 → (𝑋 + (𝐼 · 𝑇)) ∈ (𝐴(,]𝐵))    &   (𝜑 → (𝑋 + (𝐽 · 𝑇)) ∈ (𝐴(,]𝐵))       (𝜑𝐼 = 𝐽)
 
Theoremfourierdlem36 40678* 𝐹 is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐹 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴))    &   𝑁 = ((#‘𝐴) − 1)       (𝜑𝐹 Isom < , < ((0...𝑁), 𝐴))
 
Theoremfourierdlem37 40679* 𝐼 is a function that maps any real point to the point that in the partition that immediately precedes the corresponding periodic point in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   𝑇 = (𝐵𝐴)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   𝐿 = (𝑦 ∈ (𝐴(,]𝐵) ↦ if(𝑦 = 𝐵, 𝐴, 𝑦))    &   𝐼 = (𝑥 ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))}, ℝ, < ))       (𝜑 → (𝐼:ℝ⟶(0..^𝑀) ∧ (𝑥 ∈ ℝ → sup({𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))}, ℝ, < ) ∈ {𝑖 ∈ (0..^𝑀) ∣ (𝑄𝑖) ≤ (𝐿‘(𝐸𝑥))})))
 
Theoremfourierdlem38 40680* The function 𝐹 is continuous on every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (dom 𝐹cn→ℂ))    &   𝑃 = (𝑛 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑛)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑛) = π) ∧ ∀𝑖 ∈ (0..^𝑛)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   𝐻 = (𝐴 ∪ ((-π[,]π) ∖ dom 𝐹))    &   (𝜑 → ran 𝑄 = 𝐻)       ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))
 
Theoremfourierdlem39 40681* Integration by parts of ∫(𝐴(,)𝐵)((𝐹𝑥) · (sin‘(𝑅 · 𝑥))) d𝑥 (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   𝐺 = (ℝ D 𝐹)    &   (𝜑𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐺𝑥)) ≤ 𝑦)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → ∫(𝐴(,)𝐵)((𝐹𝑥) · (sin‘(𝑅 · 𝑥))) d𝑥 = ((((𝐹𝐵) · -((cos‘(𝑅 · 𝐵)) / 𝑅)) − ((𝐹𝐴) · -((cos‘(𝑅 · 𝐴)) / 𝑅))) − ∫(𝐴(,)𝐵)((𝐺𝑥) · -((cos‘(𝑅 · 𝑥)) / 𝑅)) d𝑥))
 
Theoremfourierdlem40 40682* 𝐻 is a continuous function on any partition interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝐴 ∈ (-π[,]π))    &   (𝜑𝐵 ∈ (-π[,]π))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))    &   (𝜑 → (𝐹 ↾ ((𝐴 + 𝑋)(,)(𝐵 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐵 + 𝑋))–cn→ℂ))    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))       (𝜑 → (𝐻 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
 
Theoremfourierdlem41 40683* Lemma used to prove that every real is a limit point for the domain of the derivative of the periodic function to be approximated. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)    &   (𝜑𝑋 ∈ ℝ)    &   𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍𝑥)))    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))) ⊆ 𝐷)       (𝜑 → (∃𝑦 ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) ⊆ 𝐷) ∧ ∃𝑦 ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) ⊆ 𝐷)))
 
Theoremfourierdlem42 40684* The set of points in a moved partition are finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 29-Sep-2020.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐵 < 𝐶)    &   𝑇 = (𝐶𝐵)    &   (𝜑𝐴 ⊆ (𝐵[,]𝐶))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)    &   𝐷 = (abs ∘ − )    &   𝐼 = ((𝐴 × 𝐴) ∖ I )    &   𝑅 = ran (𝐷𝐼)    &   𝐸 = inf(𝑅, ℝ, < )    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   𝐽 = (topGen‘ran (,))    &   𝐾 = (𝐽t (𝑋[,]𝑌))    &   𝐻 = {𝑥 ∈ (𝑋[,]𝑌) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ 𝐴}    &   (𝜓 ↔ ((𝜑 ∧ (𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ 𝑎 < 𝑏)) ∧ ∃𝑗 ∈ ℤ ∃𝑘 ∈ ℤ ((𝑎 + (𝑗 · 𝑇)) ∈ 𝐴 ∧ (𝑏 + (𝑘 · 𝑇)) ∈ 𝐴)))       (𝜑𝐻 ∈ Fin)
 
Theoremfourierdlem43 40685 𝐾 is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))       𝐾:(-π[,]π)⟶ℝ
 
Theoremfourierdlem44 40686 A condition for having (sin‘(𝐴 / 2)) non zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝐴 ∈ (-π[,]π) ∧ 𝐴 ≠ 0) → (sin‘(𝐴 / 2)) ≠ 0)
 
Theoremfourierdlem46 40687* The function 𝐹 has a limit at the bounds of every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹 ∈ (dom 𝐹cn→ℂ))    &   ((𝜑𝑥 ∈ ((-π[,)π) ∖ dom 𝐹)) → ((𝐹 ↾ (𝑥(,)+∞)) lim 𝑥) ≠ ∅)    &   ((𝜑𝑥 ∈ ((-π(,]π) ∖ dom 𝐹)) → ((𝐹 ↾ (-∞(,)𝑥)) lim 𝑥) ≠ ∅)    &   (𝜑𝑄 Isom < , < ((0...𝑀), 𝐻))    &   (𝜑𝑄:(0...𝑀)⟶𝐻)    &   (𝜑𝐼 ∈ (0..^𝑀))    &   (𝜑 → (𝑄𝐼) < (𝑄‘(𝐼 + 1)))    &   (𝜑 → ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1))) ⊆ (-π(,)π))    &   (𝜑𝐶 ∈ ℝ)    &   𝐻 = ({-π, π, 𝐶} ∪ ((-π[,]π) ∖ dom 𝐹))    &   (𝜑 → ran 𝑄 = 𝐻)       (𝜑 → (((𝐹 ↾ ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1)))) lim (𝑄𝐼)) ≠ ∅ ∧ ((𝐹 ↾ ((𝑄𝐼)(,)(𝑄‘(𝐼 + 1)))) lim (𝑄‘(𝐼 + 1))) ≠ ∅))
 
Theoremfourierdlem47 40688* For 𝑟 large enough, the final expression is less than the given positive 𝐸. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → (𝑥𝐼𝐹) ∈ 𝐿1)    &   ((𝜑𝑟 ∈ ℝ) → (𝑥𝐼 ↦ (𝐹 · -𝐺)) ∈ 𝐿1)    &   ((𝜑𝑥𝐼) → 𝐹 ∈ ℂ)    &   (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℂ) → 𝐺 ∈ ℂ)    &   (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ) → (abs‘𝐺) ≤ 1)    &   (𝜑𝐴 ∈ ℂ)    &   𝑋 = (abs‘𝐴)    &   (𝜑𝐶 ∈ ℂ)    &   𝑌 = (abs‘𝐶)    &   𝑍 = ∫𝐼(abs‘𝐹) d𝑥    &   (𝜑𝐸 ∈ ℝ+)    &   ((𝜑𝑟 ∈ ℂ) → 𝐵 ∈ ℂ)    &   ((𝜑𝑟 ∈ ℝ) → (abs‘𝐵) ≤ 1)    &   ((𝜑𝑟 ∈ ℂ) → 𝐷 ∈ ℂ)    &   ((𝜑𝑟 ∈ ℝ) → (abs‘𝐷) ≤ 1)    &   𝑀 = ((⌊‘((((𝑋 + 𝑌) + 𝑍) / 𝐸) + 1)) + 1)       (𝜑 → ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)(abs‘(((𝐴 · -(𝐵 / 𝑟)) − (𝐶 · -(𝐷 / 𝑟))) − ∫𝐼(𝐹 · -(𝐺 / 𝑟)) d𝑥)) < 𝐸)
 
Theoremfourierdlem48 40689* The given periodic function 𝐹 has a right limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑅 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄𝑖)))    &   (𝜑𝑋 ∈ ℝ)    &   𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍𝑥)))    &   (𝜒 ↔ ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑦 ∈ ((𝑄𝑖)[,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ ℤ) ∧ 𝑦 = (𝑋 + (𝑘 · 𝑇))))       (𝜑 → ((𝐹 ↾ (𝑋(,)+∞)) lim 𝑋) ≠ ∅)
 
Theoremfourierdlem49 40690* The given periodic function 𝐹 has a left limit at every point in the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝑇 = (𝐵𝐴)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐷 ⊆ ℝ)    &   (𝜑𝐹:𝐷⟶ℝ)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝑥 + (𝑘 · 𝑇)) ∈ 𝐷)    &   ((𝜑𝑥𝐷𝑘 ∈ ℤ) → (𝐹‘(𝑥 + (𝑘 · 𝑇))) = (𝐹𝑥))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → (𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∈ (((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))–cn→ℂ))    &   ((𝜑𝑖 ∈ (0..^𝑀)) → 𝐿 ∈ ((𝐹 ↾ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) lim (𝑄‘(𝑖 + 1))))    &   (𝜑𝑋 ∈ ℝ)    &   𝑍 = (𝑥 ∈ ℝ ↦ ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇))    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + (𝑍𝑥)))       (𝜑 → ((𝐹 ↾ (-∞(,)𝑋)) lim 𝑋) ≠ ∅)
 
Theoremfourierdlem50 40691* Continuity of 𝑂 and its limits with respect to the 𝑆 partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑋 ∈ ℝ)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (-π + 𝑋) ∧ (𝑝𝑚) = (π + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑉 ∈ (𝑃𝑀))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 𝐵)    &   (𝜑 → (𝐴[,]𝐵) ⊆ (-π[,]π))    &   𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))    &   𝑇 = ({𝐴, 𝐵} ∪ (ran 𝑄 ∩ (𝐴(,)𝐵)))    &   𝑁 = ((#‘𝑇) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   (𝜑𝐽 ∈ (0..^𝑁))    &   𝑈 = (𝑖 ∈ (0..^𝑀)((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1))))    &   (𝜒 ↔ ((((𝜑𝑖 ∈ (0..^𝑀)) ∧ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑖)(,)(𝑄‘(𝑖 + 1)))) ∧ 𝑘 ∈ (0..^𝑀)) ∧ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑘)(,)(𝑄‘(𝑘 + 1)))))       (𝜑 → (𝑈 ∈ (0..^𝑀) ∧ ((𝑆𝐽)(,)(𝑆‘(𝐽 + 1))) ⊆ ((𝑄𝑈)(,)(𝑄‘(𝑈 + 1)))))
 
Theoremfourierdlem51 40692* 𝑋 is in the periodic partition, when the considered interval is centered at 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐴 < 0)    &   (𝜑 → 0 < 𝐵)    &   𝑇 = (𝐵𝐴)    &   (𝜑𝐶 ∈ Fin)    &   (𝜑𝐶 ⊆ (𝐴[,]𝐵))    &   (𝜑𝐵𝐶)    &   𝐸 = (𝑥 ∈ ℝ ↦ (𝑥 + ((⌊‘((𝐵𝑥) / 𝑇)) · 𝑇)))    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑 → (𝐸𝑋) ∈ 𝐶)    &   𝐷 = ({(𝐴 + 𝑋), (𝐵 + 𝑋)} ∪ {𝑦 ∈ ((𝐴 + 𝑋)[,](𝐵 + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ 𝐶})    &   𝐹 = (℩𝑓𝑓 Isom < , < ((0...((#‘𝐷) − 1)), 𝐷))    &   𝐻 = {𝑦 ∈ ((𝐴 + 𝑋)(,](𝐵 + 𝑋)) ∣ ∃𝑘 ∈ ℤ (𝑦 + (𝑘 · 𝑇)) ∈ 𝐶}       (𝜑𝑋 ∈ ran 𝐹)
 
Theoremfourierdlem52 40693* d16:d17,d18:jca |- ( ph -> ( ( S 0) ≤ 𝐴𝐴 ≤ (𝑆 0 ) ) ) . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝑇 ∈ Fin)    &   𝑁 = ((#‘𝑇) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝑇 ⊆ (𝐴[,]𝐵))    &   (𝜑𝐴𝑇)    &   (𝜑𝐵𝑇)       (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆𝑁) = 𝐵))
 
Theoremfourierdlem53 40694* The limit of 𝐹(𝑠) at (𝑋 + 𝐷) is the limit of 𝐹(𝑋 + 𝑠) at 𝐷. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ⊆ ℝ)    &   𝐺 = (𝑠𝐴 ↦ (𝐹‘(𝑋 + 𝑠)))    &   ((𝜑𝑠𝐴) → (𝑋 + 𝑠) ∈ 𝐵)    &   (𝜑𝐵 ⊆ ℝ)    &   ((𝜑𝑠𝐴) → 𝑠𝐷)    &   (𝜑𝐶 ∈ ((𝐹𝐵) lim (𝑋 + 𝐷)))    &   (𝜑𝐷 ∈ ℂ)       (𝜑𝐶 ∈ (𝐺 lim 𝐷))
 
Theoremfourierdlem54 40695* Given a partition 𝑄 and an arbitrary interval [𝐶, 𝐷], a partition 𝑆 on [𝐶, 𝐷] is built such that it preserves any periodic function piecewise continuous on 𝑄 will be piecewise continuous on 𝑆, with the same limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑇 = (𝐵𝐴)    &   𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑄 ∈ (𝑃𝑀))    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐷 ∈ ℝ)    &   (𝜑𝐶 < 𝐷)    &   𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐶 ∧ (𝑝𝑚) = 𝐷) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})    &   𝐻 = ({𝐶, 𝐷} ∪ {𝑥 ∈ (𝐶[,]𝐷) ∣ ∃𝑘 ∈ ℤ (𝑥 + (𝑘 · 𝑇)) ∈ ran 𝑄})    &   𝑁 = ((#‘𝐻) − 1)    &   𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐻))       (𝜑 → ((𝑁 ∈ ℕ ∧ 𝑆 ∈ (𝑂𝑁)) ∧ 𝑆 Isom < , < ((0...𝑁), 𝐻)))
 
Theoremfourierdlem55 40696* 𝑈 is a real function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑊 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 0, (((𝐹‘(𝑋 + 𝑠)) − if(0 < 𝑠, 𝑌, 𝑊)) / 𝑠)))    &   𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   𝑈 = (𝑠 ∈ (-π[,]π) ↦ ((𝐻𝑠) · (𝐾𝑠)))       (𝜑𝑈:(-π[,]π)⟶ℝ)
 
Theoremfourierdlem56 40697* Derivative of the 𝐾 function on an interval non containing ' 0 '. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))    &   (𝜑 → (𝐴(,)𝐵) ⊆ ((-π[,]π) ∖ {0}))    &   ((𝜑𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ≠ 0)       (𝜑 → (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (𝐾𝑠))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ ((((sin‘(𝑠 / 2)) − (((cos‘(𝑠 / 2)) / 2) · 𝑠)) / ((sin‘(𝑠 / 2))↑2)) / 2)))
 
Theoremfourierdlem57 40698* The derivative of 𝑂. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))):((𝑋 + 𝐴)(,)(𝑋 + 𝐵))⟶ℝ)    &   (𝜑 → (𝐴(,)𝐵) ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))    &   (𝜑𝐶 ∈ ℝ)    &   𝑂 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / (2 · (sin‘(𝑠 / 2)))))       ((𝜑 → ((ℝ D 𝑂):(𝐴(,)𝐵)⟶ℝ ∧ (ℝ D 𝑂) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((((ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵))))‘(𝑋 + 𝑠)) · (2 · (sin‘(𝑠 / 2)))) − ((cos‘(𝑠 / 2)) · ((𝐹‘(𝑋 + 𝑠)) − 𝐶))) / ((2 · (sin‘(𝑠 / 2)))↑2))))) ∧ (ℝ D (𝑠 ∈ (𝐴(,)𝐵) ↦ (2 · (sin‘(𝑠 / 2))))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑠 / 2))))
 
Theoremfourierdlem58 40699* The derivative of 𝐾 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠𝐴 ↦ (𝑠 / (2 · (sin‘(𝑠 / 2)))))    &   (𝜑𝐴 ⊆ (-π[,]π))    &   (𝜑 → ¬ 0 ∈ 𝐴)    &   (𝜑𝐴 ∈ (topGen‘ran (,)))       (𝜑 → (ℝ D 𝐾) ∈ (𝐴cn→ℝ))
 
Theoremfourierdlem59 40700* The derivative of 𝐻 is continuous on the given interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝐹:ℝ⟶ℝ)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → ¬ 0 ∈ (𝐴(,)𝐵))    &   (𝜑 → (ℝ D (𝐹 ↾ ((𝑋 + 𝐴)(,)(𝑋 + 𝐵)))) ∈ (((𝑋 + 𝐴)(,)(𝑋 + 𝐵))–cn→ℝ))    &   (𝜑𝐶 ∈ ℝ)    &   𝐻 = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((𝐹‘(𝑋 + 𝑠)) − 𝐶) / 𝑠))       (𝜑 → (ℝ D 𝐻) ∈ ((𝐴(,)𝐵)–cn→ℝ))
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