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Theorem List for Metamath Proof Explorer - 44001-44100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstoweidlem50 44001* This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, contain a finite subcover of T \ U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘‘π‘ˆ    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   π‘„ = {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}    &   π‘Š = {𝑀 ∈ 𝐽 ∣ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ π‘ˆ ∈ 𝐽)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆƒπ‘’(𝑒 ∈ Fin ∧ 𝑒 βŠ† π‘Š ∧ (𝑇 βˆ– π‘ˆ) βŠ† βˆͺ 𝑒))
 
Theoremstoweidlem51 44002* There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here 𝐷 is used to represent 𝐴 in the paper, because here 𝐴 is used for the subalgebra of functions. 𝐸 is used to represent Ξ΅ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘–πœ‘    &   β„²π‘‘πœ‘    &   β„²π‘€πœ‘    &   β„²π‘€π‘‰    &   π‘Œ = {β„Ž ∈ 𝐴 ∣ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1)}    &   π‘ƒ = (𝑓 ∈ π‘Œ, 𝑔 ∈ π‘Œ ↦ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))))    &   π‘‹ = (seq1(𝑃, π‘ˆ)β€˜π‘€)    &   πΉ = (𝑑 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((π‘ˆβ€˜π‘–)β€˜π‘‘)))    &   π‘ = (𝑑 ∈ 𝑇 ↦ (seq1( Β· , (πΉβ€˜π‘‘))β€˜π‘€))    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ π‘Š:(1...𝑀)βŸΆπ‘‰)    &   (πœ‘ β†’ π‘ˆ:(1...𝑀)βŸΆπ‘Œ)    &   ((πœ‘ ∧ 𝑀 ∈ 𝑉) β†’ 𝑀 βŠ† 𝑇)    &   (πœ‘ β†’ 𝐷 βŠ† βˆͺ ran π‘Š)    &   (πœ‘ β†’ 𝐷 βŠ† 𝑇)    &   (πœ‘ β†’ 𝐡 βŠ† 𝑇)    &   ((πœ‘ ∧ 𝑖 ∈ (1...𝑀)) β†’ βˆ€π‘‘ ∈ (π‘Šβ€˜π‘–)((π‘ˆβ€˜π‘–)β€˜π‘‘) < (𝐸 / 𝑀))    &   ((πœ‘ ∧ 𝑖 ∈ (1...𝑀)) β†’ βˆ€π‘‘ ∈ 𝐡 (1 βˆ’ (𝐸 / 𝑀)) < ((π‘ˆβ€˜π‘–)β€˜π‘‘))    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝑓:π‘‡βŸΆβ„)    &   (πœ‘ β†’ 𝑇 ∈ V)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    β‡’   (πœ‘ β†’ βˆƒπ‘₯(π‘₯ ∈ 𝐴 ∧ (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘₯β€˜π‘‘) ∧ (π‘₯β€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝐷 (π‘₯β€˜π‘‘) < 𝐸 ∧ βˆ€π‘‘ ∈ 𝐡 (1 βˆ’ 𝐸) < (π‘₯β€˜π‘‘))))
 
Theoremstoweidlem52 44003* There exists a neighborhood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t0 in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘‘π‘ˆ    &   β„²π‘‘πœ‘    &   β„²π‘‘𝑃    &   πΎ = (topGenβ€˜ran (,))    &   π‘‰ = {𝑑 ∈ 𝑇 ∣ (π‘ƒβ€˜π‘‘) < (𝐷 / 2)}    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘Ž) ∈ 𝐴)    &   (πœ‘ β†’ 𝐷 ∈ ℝ+)    &   (πœ‘ β†’ 𝐷 < 1)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐽)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑃 ∈ 𝐴)    &   (πœ‘ β†’ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘ƒβ€˜π‘‘) ∧ (π‘ƒβ€˜π‘‘) ≀ 1))    &   (πœ‘ β†’ (π‘ƒβ€˜π‘) = 0)    &   (πœ‘ β†’ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)𝐷 ≀ (π‘ƒβ€˜π‘‘))    β‡’   (πœ‘ β†’ βˆƒπ‘£ ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 βŠ† π‘ˆ) ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘₯β€˜π‘‘) ∧ (π‘₯β€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝑣 (π‘₯β€˜π‘‘) < 𝑒 ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)(1 βˆ’ 𝑒) < (π‘₯β€˜π‘‘))))
 
Theoremstoweidlem53 44004* This lemma is used to prove the existence of a function 𝑝 as in Lemma 1 of [BrosowskiDeutsh] p. 90: 𝑝 is in the subalgebra, such that 0 ≀ 𝑝 ≀ 1, p_(t0) = 0, and 0 < 𝑝 on 𝑇 βˆ– π‘ˆ. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘‘π‘ˆ    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   π‘„ = {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}    &   π‘Š = {𝑀 ∈ 𝐽 ∣ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ π‘ˆ ∈ 𝐽)    &   (πœ‘ β†’ (𝑇 βˆ– π‘ˆ) β‰  βˆ…)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘β€˜π‘‘) ∧ (π‘β€˜π‘‘) ≀ 1) ∧ (π‘β€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)0 < (π‘β€˜π‘‘)))
 
Theoremstoweidlem54 44005* There exists a function π‘₯ as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here 𝐷 is used to represent 𝐴 in the paper, because here 𝐴 is used for the subalgebra of functions. 𝐸 is used to represent Ξ΅ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘–πœ‘    &   β„²π‘‘πœ‘    &   β„²π‘¦πœ‘    &   β„²π‘€πœ‘    &   π‘‡ = βˆͺ 𝐽    &   π‘Œ = {β„Ž ∈ 𝐴 ∣ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1)}    &   π‘ƒ = (𝑓 ∈ π‘Œ, 𝑔 ∈ π‘Œ ↦ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))))    &   πΉ = (𝑑 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((π‘¦β€˜π‘–)β€˜π‘‘)))    &   π‘ = (𝑑 ∈ 𝑇 ↦ (seq1( Β· , (πΉβ€˜π‘‘))β€˜π‘€))    &   π‘‰ = {𝑀 ∈ 𝐽 ∣ βˆ€π‘’ ∈ ℝ+ βˆƒβ„Ž ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝑀 (β„Žβ€˜π‘‘) < 𝑒 ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)(1 βˆ’ 𝑒) < (β„Žβ€˜π‘‘))}    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴) β†’ 𝑓:π‘‡βŸΆβ„)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ π‘Š:(1...𝑀)βŸΆπ‘‰)    &   (πœ‘ β†’ 𝐡 βŠ† 𝑇)    &   (πœ‘ β†’ 𝐷 βŠ† βˆͺ ran π‘Š)    &   (πœ‘ β†’ 𝐷 βŠ† 𝑇)    &   (πœ‘ β†’ βˆƒπ‘¦(𝑦:(1...𝑀)βŸΆπ‘Œ ∧ βˆ€π‘– ∈ (1...𝑀)(βˆ€π‘‘ ∈ (π‘Šβ€˜π‘–)((π‘¦β€˜π‘–)β€˜π‘‘) < (𝐸 / 𝑀) ∧ βˆ€π‘‘ ∈ 𝐡 (1 βˆ’ (𝐸 / 𝑀)) < ((π‘¦β€˜π‘–)β€˜π‘‘))))    &   (πœ‘ β†’ 𝑇 ∈ V)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘₯β€˜π‘‘) ∧ (π‘₯β€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝐷 (π‘₯β€˜π‘‘) < 𝐸 ∧ βˆ€π‘‘ ∈ 𝐡 (1 βˆ’ 𝐸) < (π‘₯β€˜π‘‘)))
 
Theoremstoweidlem55 44006* This lemma proves the existence of a function p as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Here Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘‘π‘ˆ    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ π‘ˆ ∈ 𝐽)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    &   π‘„ = {β„Ž ∈ 𝐴 ∣ ((β„Žβ€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1))}    &   π‘Š = {𝑀 ∈ 𝐽 ∣ βˆƒβ„Ž ∈ 𝑄 𝑀 = {𝑑 ∈ 𝑇 ∣ 0 < (β„Žβ€˜π‘‘)}}    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘β€˜π‘‘) ∧ (π‘β€˜π‘‘) ≀ 1) ∧ (π‘β€˜π‘) = 0 ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)0 < (π‘β€˜π‘‘)))
 
Theoremstoweidlem56 44007* This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here 𝑍 is used to represent t0 in the paper, 𝑣 is used to represent 𝑉 in the paper, and 𝑒 is used to represent Ξ΅. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
β„²π‘‘π‘ˆ    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ π‘ˆ ∈ 𝐽)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ βˆƒπ‘£ ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 βŠ† π‘ˆ) ∧ βˆ€π‘’ ∈ ℝ+ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘₯β€˜π‘‘) ∧ (π‘₯β€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝑣 (π‘₯β€˜π‘‘) < 𝑒 ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)(1 βˆ’ 𝑒) < (π‘₯β€˜π‘‘))))
 
Theoremstoweidlem57 44008* There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. In this theorem, it is proven the non-trivial case (the closed set D is nonempty). Here D is used to represent A in the paper, because the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑑𝐷    &   β„²π‘‘π‘ˆ    &   β„²π‘‘πœ‘    &   π‘Œ = {β„Ž ∈ 𝐴 ∣ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1)}    &   π‘‰ = {𝑀 ∈ 𝐽 ∣ βˆ€π‘’ ∈ ℝ+ βˆƒβ„Ž ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (β„Žβ€˜π‘‘) ∧ (β„Žβ€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝑀 (β„Žβ€˜π‘‘) < 𝑒 ∧ βˆ€π‘‘ ∈ (𝑇 βˆ– π‘ˆ)(1 βˆ’ 𝑒) < (β„Žβ€˜π‘‘))}    &   πΎ = (topGenβ€˜ran (,))    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   π‘ˆ = (𝑇 βˆ– 𝐡)    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘Ž) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ 𝐡 ∈ (Clsdβ€˜π½))    &   (πœ‘ β†’ 𝐷 ∈ (Clsdβ€˜π½))    &   (πœ‘ β†’ (𝐡 ∩ 𝐷) = βˆ…)    &   (πœ‘ β†’ 𝐷 β‰  βˆ…)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘₯β€˜π‘‘) ∧ (π‘₯β€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝐷 (π‘₯β€˜π‘‘) < 𝐸 ∧ βˆ€π‘‘ ∈ 𝐡 (1 βˆ’ 𝐸) < (π‘₯β€˜π‘‘)))
 
Theoremstoweidlem58 44009* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑑𝐷    &   β„²π‘‘π‘ˆ    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘Ž) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ 𝐡 ∈ (Clsdβ€˜π½))    &   (πœ‘ β†’ 𝐷 ∈ (Clsdβ€˜π½))    &   (πœ‘ β†’ (𝐡 ∩ 𝐷) = βˆ…)    &   π‘ˆ = (𝑇 βˆ– 𝐡)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝐴 (βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘₯β€˜π‘‘) ∧ (π‘₯β€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ 𝐷 (π‘₯β€˜π‘‘) < 𝐸 ∧ βˆ€π‘‘ ∈ 𝐡 (1 βˆ’ 𝐸) < (π‘₯β€˜π‘‘)))
 
Theoremstoweidlem59 44010* This lemma proves that there exists a function π‘₯ as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: xj is in the subalgebra, 0 <= xj <= 1, xj < Ξ΅ / n on Aj (meaning A in the paper), xj > 1 - \epsilon / n on Bj. Here 𝐷 is used to represent A in the paper (because A is used for the subalgebra of functions), 𝐸 is used to represent Ξ΅. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑑𝐹    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   π· = (𝑗 ∈ (0...𝑁) ↦ {𝑑 ∈ 𝑇 ∣ (πΉβ€˜π‘‘) ≀ ((𝑗 βˆ’ (1 / 3)) Β· 𝐸)})    &   π΅ = (𝑗 ∈ (0...𝑁) ↦ {𝑑 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) Β· 𝐸) ≀ (πΉβ€˜π‘‘)})    &   π‘Œ = {𝑦 ∈ 𝐴 ∣ βˆ€π‘‘ ∈ 𝑇 (0 ≀ (π‘¦β€˜π‘‘) ∧ (π‘¦β€˜π‘‘) ≀ 1)}    &   π» = (𝑗 ∈ (0...𝑁) ↦ {𝑦 ∈ π‘Œ ∣ (βˆ€π‘‘ ∈ (π·β€˜π‘—)(π‘¦β€˜π‘‘) < (𝐸 / 𝑁) ∧ βˆ€π‘‘ ∈ (π΅β€˜π‘—)(1 βˆ’ (𝐸 / 𝑁)) < (π‘¦β€˜π‘‘))})    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ 𝐹 ∈ 𝐢)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ βˆƒπ‘₯(π‘₯:(0...𝑁)⟢𝐴 ∧ βˆ€π‘— ∈ (0...𝑁)(βˆ€π‘‘ ∈ 𝑇 (0 ≀ ((π‘₯β€˜π‘—)β€˜π‘‘) ∧ ((π‘₯β€˜π‘—)β€˜π‘‘) ≀ 1) ∧ βˆ€π‘‘ ∈ (π·β€˜π‘—)((π‘₯β€˜π‘—)β€˜π‘‘) < (𝐸 / 𝑁) ∧ βˆ€π‘‘ ∈ (π΅β€˜π‘—)(1 βˆ’ (𝐸 / 𝑁)) < ((π‘₯β€˜π‘—)β€˜π‘‘))))
 
Theoremstoweidlem60 44011* This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all 𝑑 in 𝑇, there is a 𝑗 such that (j-4/3)*Ξ΅ < f(t) <= (j-1/3)*Ξ΅ and (j-4/3)*Ξ΅ < g(t) < (j+1/3)*Ξ΅. Here 𝐹 is used to represent f in the paper, and 𝐸 is used to represent Ξ΅. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑑𝐹    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   π· = (𝑗 ∈ (0...𝑛) ↦ {𝑑 ∈ 𝑇 ∣ (πΉβ€˜π‘‘) ≀ ((𝑗 βˆ’ (1 / 3)) Β· 𝐸)})    &   π΅ = (𝑗 ∈ (0...𝑛) ↦ {𝑑 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) Β· 𝐸) ≀ (πΉβ€˜π‘‘)})    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   (πœ‘ β†’ 𝑇 β‰  βˆ…)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑦 ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ 𝐹 ∈ 𝐢)    &   (πœ‘ β†’ βˆ€π‘‘ ∈ 𝑇 0 ≀ (πΉβ€˜π‘‘))    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    β‡’   (πœ‘ β†’ βˆƒπ‘” ∈ 𝐴 βˆ€π‘‘ ∈ 𝑇 βˆƒπ‘— ∈ ℝ ((((𝑗 βˆ’ (4 / 3)) Β· 𝐸) < (πΉβ€˜π‘‘) ∧ (πΉβ€˜π‘‘) ≀ ((𝑗 βˆ’ (1 / 3)) Β· 𝐸)) ∧ ((π‘”β€˜π‘‘) < ((𝑗 + (1 / 3)) Β· 𝐸) ∧ ((𝑗 βˆ’ (4 / 3)) Β· 𝐸) < (π‘”β€˜π‘‘))))
 
Theoremstoweidlem61 44012* This lemma proves that there exists a function 𝑔 as in the proof in [BrosowskiDeutsh] p. 92: 𝑔 is in the subalgebra, and for all 𝑑 in 𝑇, abs( f(t) - g(t) ) < 2*Ξ΅. Here 𝐹 is used to represent f in the paper, and 𝐸 is used to represent Ξ΅. For this lemma there's the further assumption that the function 𝐹 to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑑𝐹    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   π‘‡ = βˆͺ 𝐽    &   (πœ‘ β†’ 𝑇 β‰  βˆ…)    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ 𝐹 ∈ 𝐢)    &   (πœ‘ β†’ βˆ€π‘‘ ∈ 𝑇 0 ≀ (πΉβ€˜π‘‘))    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    β‡’   (πœ‘ β†’ βˆƒπ‘” ∈ 𝐴 βˆ€π‘‘ ∈ 𝑇 (absβ€˜((π‘”β€˜π‘‘) βˆ’ (πΉβ€˜π‘‘))) < (2 Β· 𝐸))
 
Theoremstoweidlem62 44013* This theorem proves the Stone Weierstrass theorem for the non-trivial case in which T is nonempty. The proof follows [BrosowskiDeutsh] p. 89 (through page 92). (Contributed by Glauco Siliprandi, 20-Apr-2017.) (Revised by AV, 13-Sep-2020.)
Ⅎ𝑑𝐹    &   β„²π‘“πœ‘    &   β„²π‘‘πœ‘    &   π» = (𝑑 ∈ 𝑇 ↦ ((πΉβ€˜π‘‘) βˆ’ inf(ran 𝐹, ℝ, < )))    &   πΎ = (topGenβ€˜ran (,))    &   π‘‡ = βˆͺ 𝐽    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒπ‘ž ∈ 𝐴 (π‘žβ€˜π‘Ÿ) β‰  (π‘žβ€˜π‘‘))    &   (πœ‘ β†’ 𝐹 ∈ 𝐢)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝑇 β‰  βˆ…)    &   (πœ‘ β†’ 𝐸 < (1 / 3))    β‡’   (πœ‘ β†’ βˆƒπ‘“ ∈ 𝐴 βˆ€π‘‘ ∈ 𝑇 (absβ€˜((π‘“β€˜π‘‘) βˆ’ (πΉβ€˜π‘‘))) < 𝐸)
 
Theoremstoweid 44014* This theorem proves the Stone-Weierstrass theorem for real-valued functions: let 𝐽 be a compact topology on 𝑇, and 𝐢 be the set of real continuous functions on 𝑇. Assume that 𝐴 is a subalgebra of 𝐢 (closed under addition and multiplication of functions) containing constant functions and discriminating points (if π‘Ÿ and 𝑑 are distinct points in 𝑇, then there exists a function β„Ž in 𝐴 such that h(r) is distinct from h(t) ). Then, for any continuous function 𝐹 and for any positive real 𝐸, there exists a function 𝑓 in the subalgebra 𝐴, such that 𝑓 approximates 𝐹 up to 𝐸 (𝐸 represents the usual Ξ΅ value). As a classical example, given any a, b reals, the closed interval 𝑇 = [π‘Ž, 𝑏] could be taken, along with the subalgebra 𝐴 of real polynomials on 𝑇, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on [π‘Ž, 𝑏]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Ⅎ𝑑𝐹    &   β„²π‘‘πœ‘    &   πΎ = (topGenβ€˜ran (,))    &   (πœ‘ β†’ 𝐽 ∈ Comp)    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐢)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((πœ‘ ∧ (π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑)) β†’ βˆƒβ„Ž ∈ 𝐴 (β„Žβ€˜π‘Ÿ) β‰  (β„Žβ€˜π‘‘))    &   (πœ‘ β†’ 𝐹 ∈ 𝐢)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    β‡’   (πœ‘ β†’ βˆƒπ‘“ ∈ 𝐴 βˆ€π‘‘ ∈ 𝑇 (absβ€˜((π‘“β€˜π‘‘) βˆ’ (πΉβ€˜π‘‘))) < 𝐸)
 
Theoremstowei 44015* This theorem proves the Stone-Weierstrass theorem for real-valued functions: let 𝐽 be a compact topology on 𝑇, and 𝐢 be the set of real continuous functions on 𝑇. Assume that 𝐴 is a subalgebra of 𝐢 (closed under addition and multiplication of functions) containing constant functions and discriminating points (if π‘Ÿ and 𝑑 are distinct points in 𝑇, then there exists a function β„Ž in 𝐴 such that h(r) is distinct from h(t) ). Then, for any continuous function 𝐹 and for any positive real 𝐸, there exists a function 𝑓 in the subalgebra 𝐴, such that 𝑓 approximates 𝐹 up to 𝐸 (𝐸 represents the usual Ξ΅ value). As a classical example, given any a, b reals, the closed interval 𝑇 = [π‘Ž, 𝑏] could be taken, along with the subalgebra 𝐴 of real polynomials on 𝑇, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on [π‘Ž, 𝑏]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. The deduction version of this theorem is stoweid 44014: often times it will be better to use stoweid 44014 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐾 = (topGenβ€˜ran (,))    &   π½ ∈ Comp    &   π‘‡ = βˆͺ 𝐽    &   πΆ = (𝐽 Cn 𝐾)    &   π΄ βŠ† 𝐢    &   ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) + (π‘”β€˜π‘‘))) ∈ 𝐴)    &   ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) β†’ (𝑑 ∈ 𝑇 ↦ ((π‘“β€˜π‘‘) Β· (π‘”β€˜π‘‘))) ∈ 𝐴)    &   (π‘₯ ∈ ℝ β†’ (𝑑 ∈ 𝑇 ↦ π‘₯) ∈ 𝐴)    &   ((π‘Ÿ ∈ 𝑇 ∧ 𝑑 ∈ 𝑇 ∧ π‘Ÿ β‰  𝑑) β†’ βˆƒβ„Ž ∈ 𝐴 (β„Žβ€˜π‘Ÿ) β‰  (β„Žβ€˜π‘‘))    &   πΉ ∈ 𝐢    &   πΈ ∈ ℝ+    β‡’   βˆƒπ‘“ ∈ 𝐴 βˆ€π‘‘ ∈ 𝑇 (absβ€˜((π‘“β€˜π‘‘) βˆ’ (πΉβ€˜π‘‘))) < 𝐸
 
21.38.13  Wallis' product for Ο€
 
Theoremwallispilem1 44016* 𝐼 is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐼 = (𝑛 ∈ β„•0 ↦ ∫(0(,)Ο€)((sinβ€˜π‘₯)↑𝑛) dπ‘₯)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (πΌβ€˜(𝑁 + 1)) ≀ (πΌβ€˜π‘))
 
Theoremwallispilem2 44017* 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 ∈ β„•0 ↦ ∫(0(,)Ο€)((sinβ€˜π‘₯)↑𝑛) dπ‘₯)    β‡’   ((πΌβ€˜0) = Ο€ ∧ (πΌβ€˜1) = 2 ∧ (𝑁 ∈ (β„€β‰₯β€˜2) β†’ (πΌβ€˜π‘) = (((𝑁 βˆ’ 1) / 𝑁) Β· (πΌβ€˜(𝑁 βˆ’ 2)))))
 
Theoremwallispilem3 44018* I maps to real values. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐼 = (𝑛 ∈ β„•0 ↦ ∫(0(,)Ο€)((sinβ€˜π‘₯)↑𝑛) dπ‘₯)    β‡’   (𝑁 ∈ β„•0 β†’ (πΌβ€˜π‘) ∈ ℝ+)
 
Theoremwallispilem4 44019* 𝐹 maps to explicit expression for the ratio of two consecutive values of 𝐼. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
𝐹 = (π‘˜ ∈ β„• ↦ (((2 Β· π‘˜) / ((2 Β· π‘˜) βˆ’ 1)) Β· ((2 Β· π‘˜) / ((2 Β· π‘˜) + 1))))    &   πΌ = (𝑛 ∈ β„•0 ↦ ∫(0(,)Ο€)((sinβ€˜π‘§)↑𝑛) d𝑧)    &   πΊ = (𝑛 ∈ β„• ↦ ((πΌβ€˜(2 Β· 𝑛)) / (πΌβ€˜((2 Β· 𝑛) + 1))))    &   π» = (𝑛 ∈ β„• ↦ ((Ο€ / 2) Β· (1 / (seq1( Β· , 𝐹)β€˜π‘›))))    β‡’   πΊ = 𝐻
 
Theoremwallispilem5 44020* The sequence 𝐻 converges to 1. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
𝐹 = (π‘˜ ∈ β„• ↦ (((2 Β· π‘˜) / ((2 Β· π‘˜) βˆ’ 1)) Β· ((2 Β· π‘˜) / ((2 Β· π‘˜) + 1))))    &   πΌ = (𝑛 ∈ β„•0 ↦ ∫(0(,)Ο€)((sinβ€˜π‘₯)↑𝑛) dπ‘₯)    &   πΊ = (𝑛 ∈ β„• ↦ ((πΌβ€˜(2 Β· 𝑛)) / (πΌβ€˜((2 Β· 𝑛) + 1))))    &   π» = (𝑛 ∈ β„• ↦ ((Ο€ / 2) Β· (1 / (seq1( Β· , 𝐹)β€˜π‘›))))    &   πΏ = (𝑛 ∈ β„• ↦ (((2 Β· 𝑛) + 1) / (2 Β· 𝑛)))    β‡’   π» ⇝ 1
 
Theoremwallispi 44021* Wallis' formula for Ο€ : Wallis' product converges to Ο€ / 2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐹 = (π‘˜ ∈ β„• ↦ (((2 Β· π‘˜) / ((2 Β· π‘˜) βˆ’ 1)) Β· ((2 Β· π‘˜) / ((2 Β· π‘˜) + 1))))    &   π‘Š = (𝑛 ∈ β„• ↦ (seq1( Β· , 𝐹)β€˜π‘›))    β‡’   π‘Š ⇝ (Ο€ / 2)
 
Theoremwallispi2lem1 44022 An intermediate step between the first version of the Wallis' formula for Ο€ 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 ∈ β„• β†’ (seq1( Β· , (π‘˜ ∈ β„• ↦ (((2 Β· π‘˜) / ((2 Β· π‘˜) βˆ’ 1)) Β· ((2 Β· π‘˜) / ((2 Β· π‘˜) + 1)))))β€˜π‘) = ((1 / ((2 Β· 𝑁) + 1)) Β· (seq1( Β· , (π‘˜ ∈ β„• ↦ (((2 Β· π‘˜)↑4) / (((2 Β· π‘˜) Β· ((2 Β· π‘˜) βˆ’ 1))↑2))))β€˜π‘)))
 
Theoremwallispi2lem2 44023 Two expressions are proven to be equal, and this is used to complete the proof of the second version of Wallis' formula for Ο€ . (Contributed by Glauco Siliprandi, 30-Jun-2017.)
(𝑁 ∈ β„• β†’ (seq1( Β· , (π‘˜ ∈ β„• ↦ (((2 Β· π‘˜)↑4) / (((2 Β· π‘˜) Β· ((2 Β· π‘˜) βˆ’ 1))↑2))))β€˜π‘) = (((2↑(4 Β· 𝑁)) Β· ((!β€˜π‘)↑4)) / ((!β€˜(2 Β· 𝑁))↑2)))
 
Theoremwallispi2 44024 An alternative version of Wallis' formula for Ο€ ; 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 ∈ β„• ↦ ((((2↑(4 Β· 𝑛)) Β· ((!β€˜π‘›)↑4)) / ((!β€˜(2 Β· 𝑛))↑2)) / ((2 Β· 𝑛) + 1)))    β‡’   π‘‰ ⇝ (Ο€ / 2)
 
21.38.14  Stirling's approximation formula for ` n ` factorial
 
Theoremstirlinglem1 44025 A simple limit of fractions is computed. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
𝐻 = (𝑛 ∈ β„• ↦ ((𝑛↑2) / (𝑛 Β· ((2 Β· 𝑛) + 1))))    &   πΉ = (𝑛 ∈ β„• ↦ (1 βˆ’ (1 / ((2 Β· 𝑛) + 1))))    &   πΊ = (𝑛 ∈ β„• ↦ (1 / ((2 Β· 𝑛) + 1)))    &   πΏ = (𝑛 ∈ β„• ↦ (1 / 𝑛))    β‡’   π» ⇝ (1 / 2)
 
Theoremstirlinglem2 44026 𝐴 maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    β‡’   (𝑁 ∈ β„• β†’ (π΄β€˜π‘) ∈ ℝ+)
 
Theoremstirlinglem3 44027 Long but simple algebraic transformations are applied to show that 𝑉, the Wallis formula for Ο€ , 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 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π· = (𝑛 ∈ β„• ↦ (π΄β€˜(2 Β· 𝑛)))    &   πΈ = (𝑛 ∈ β„• ↦ ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛)))    &   π‘‰ = (𝑛 ∈ β„• ↦ ((((2↑(4 Β· 𝑛)) Β· ((!β€˜π‘›)↑4)) / ((!β€˜(2 Β· 𝑛))↑2)) / ((2 Β· 𝑛) + 1)))    β‡’   π‘‰ = (𝑛 ∈ β„• ↦ ((((π΄β€˜π‘›)↑4) / ((π·β€˜π‘›)↑2)) Β· ((𝑛↑2) / (𝑛 Β· ((2 Β· 𝑛) + 1)))))
 
Theoremstirlinglem4 44028* Algebraic manipulation of ((𝐡 n ) - ( B (𝑛 + 1))). It will be used in other theorems to show that 𝐡 is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π΅ = (𝑛 ∈ β„• ↦ (logβ€˜(π΄β€˜π‘›)))    &   π½ = (𝑛 ∈ β„• ↦ ((((1 + (2 Β· 𝑛)) / 2) Β· (logβ€˜((𝑛 + 1) / 𝑛))) βˆ’ 1))    β‡’   (𝑁 ∈ β„• β†’ ((π΅β€˜π‘) βˆ’ (π΅β€˜(𝑁 + 1))) = (π½β€˜π‘))
 
Theoremstirlinglem5 44029* 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 ∈ β„• ↦ ((-1↑(𝑗 βˆ’ 1)) Β· ((𝑇↑𝑗) / 𝑗)))    &   πΈ = (𝑗 ∈ β„• ↦ ((𝑇↑𝑗) / 𝑗))    &   πΉ = (𝑗 ∈ β„• ↦ (((-1↑(𝑗 βˆ’ 1)) Β· ((𝑇↑𝑗) / 𝑗)) + ((𝑇↑𝑗) / 𝑗)))    &   π» = (𝑗 ∈ β„•0 ↦ (2 Β· ((1 / ((2 Β· 𝑗) + 1)) Β· (𝑇↑((2 Β· 𝑗) + 1)))))    &   πΊ = (𝑗 ∈ β„•0 ↦ ((2 Β· 𝑗) + 1))    &   (πœ‘ β†’ 𝑇 ∈ ℝ+)    &   (πœ‘ β†’ (absβ€˜π‘‡) < 1)    β‡’   (πœ‘ β†’ seq0( + , 𝐻) ⇝ (logβ€˜((1 + 𝑇) / (1 βˆ’ 𝑇))))
 
Theoremstirlinglem6 44030* A series that converges to log((𝑁 + 1) / 𝑁). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐻 = (𝑗 ∈ β„•0 ↦ (2 Β· ((1 / ((2 Β· 𝑗) + 1)) Β· ((1 / ((2 Β· 𝑁) + 1))↑((2 Β· 𝑗) + 1)))))    β‡’   (𝑁 ∈ β„• β†’ seq0( + , 𝐻) ⇝ (logβ€˜((𝑁 + 1) / 𝑁)))
 
Theoremstirlinglem7 44031* Algebraic manipulation of the formula for J(n). (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐽 = (𝑛 ∈ β„• ↦ ((((1 + (2 Β· 𝑛)) / 2) Β· (logβ€˜((𝑛 + 1) / 𝑛))) βˆ’ 1))    &   πΎ = (π‘˜ ∈ β„• ↦ ((1 / ((2 Β· π‘˜) + 1)) Β· ((1 / ((2 Β· 𝑁) + 1))↑(2 Β· π‘˜))))    &   π» = (π‘˜ ∈ β„•0 ↦ (2 Β· ((1 / ((2 Β· π‘˜) + 1)) Β· ((1 / ((2 Β· 𝑁) + 1))↑((2 Β· π‘˜) + 1)))))    β‡’   (𝑁 ∈ β„• β†’ seq1( + , 𝐾) ⇝ (π½β€˜π‘))
 
Theoremstirlinglem8 44032 If 𝐴 converges to 𝐢, then 𝐹 converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.)
β„²π‘›πœ‘    &   β„²π‘›π΄    &   β„²π‘›π·    &   π· = (𝑛 ∈ β„• ↦ (π΄β€˜(2 Β· 𝑛)))    &   (πœ‘ β†’ 𝐴:β„•βŸΆβ„+)    &   πΉ = (𝑛 ∈ β„• ↦ (((π΄β€˜π‘›)↑4) / ((π·β€˜π‘›)↑2)))    &   πΏ = (𝑛 ∈ β„• ↦ ((π΄β€˜π‘›)↑4))    &   π‘€ = (𝑛 ∈ β„• ↦ ((π·β€˜π‘›)↑2))    &   ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π·β€˜π‘›) ∈ ℝ+)    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ 𝐴 ⇝ 𝐢)    β‡’   (πœ‘ β†’ 𝐹 ⇝ (𝐢↑2))
 
Theoremstirlinglem9 44033* ((π΅β€˜π‘) βˆ’ (π΅β€˜(𝑁 + 1))) is expressed as a limit of a series. This result will be used both to prove that 𝐡 is decreasing and to prove that 𝐡 is bounded (below). It will follow that 𝐡 converges in the reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π΅ = (𝑛 ∈ β„• ↦ (logβ€˜(π΄β€˜π‘›)))    &   π½ = (𝑛 ∈ β„• ↦ ((((1 + (2 Β· 𝑛)) / 2) Β· (logβ€˜((𝑛 + 1) / 𝑛))) βˆ’ 1))    &   πΎ = (π‘˜ ∈ β„• ↦ ((1 / ((2 Β· π‘˜) + 1)) Β· ((1 / ((2 Β· 𝑁) + 1))↑(2 Β· π‘˜))))    β‡’   (𝑁 ∈ β„• β†’ seq1( + , 𝐾) ⇝ ((π΅β€˜π‘) βˆ’ (π΅β€˜(𝑁 + 1))))
 
Theoremstirlinglem10 44034* A bound for any B(N)-B(N + 1) that will allow to find a lower bound for the whole 𝐡 sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π΅ = (𝑛 ∈ β„• ↦ (logβ€˜(π΄β€˜π‘›)))    &   πΎ = (π‘˜ ∈ β„• ↦ ((1 / ((2 Β· π‘˜) + 1)) Β· ((1 / ((2 Β· 𝑁) + 1))↑(2 Β· π‘˜))))    &   πΏ = (π‘˜ ∈ β„• ↦ ((1 / (((2 Β· 𝑁) + 1)↑2))β†‘π‘˜))    β‡’   (𝑁 ∈ β„• β†’ ((π΅β€˜π‘) βˆ’ (π΅β€˜(𝑁 + 1))) ≀ ((1 / 4) Β· (1 / (𝑁 Β· (𝑁 + 1)))))
 
Theoremstirlinglem11 44035* 𝐡 is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π΅ = (𝑛 ∈ β„• ↦ (logβ€˜(π΄β€˜π‘›)))    &   πΎ = (π‘˜ ∈ β„• ↦ ((1 / ((2 Β· π‘˜) + 1)) Β· ((1 / ((2 Β· 𝑁) + 1))↑(2 Β· π‘˜))))    β‡’   (𝑁 ∈ β„• β†’ (π΅β€˜(𝑁 + 1)) < (π΅β€˜π‘))
 
Theoremstirlinglem12 44036* The sequence 𝐡 is bounded below. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π΅ = (𝑛 ∈ β„• ↦ (logβ€˜(π΄β€˜π‘›)))    &   πΉ = (𝑛 ∈ β„• ↦ (1 / (𝑛 Β· (𝑛 + 1))))    β‡’   (𝑁 ∈ β„• β†’ ((π΅β€˜1) βˆ’ (1 / 4)) ≀ (π΅β€˜π‘))
 
Theoremstirlinglem13 44037* 𝐡 is decreasing and has a lower bound, then it converges. Since 𝐡 is log𝐴, in another theorem it is proven that 𝐴 converges as well. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π΅ = (𝑛 ∈ β„• ↦ (logβ€˜(π΄β€˜π‘›)))    β‡’   βˆƒπ‘‘ ∈ ℝ 𝐡 ⇝ 𝑑
 
Theoremstirlinglem14 44038* 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 Ο€& , such constant is exactly determined, thus proving the Stirling's formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐴 = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π΅ = (𝑛 ∈ β„• ↦ (logβ€˜(π΄β€˜π‘›)))    β‡’   βˆƒπ‘ ∈ ℝ+ 𝐴 ⇝ 𝑐
 
Theoremstirlinglem15 44039* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 44040 will use this final lemma, but it will not expose the local definitions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
β„²π‘›πœ‘    &   π‘† = (𝑛 ∈ β„•0 ↦ ((βˆšβ€˜((2 Β· Ο€) Β· 𝑛)) Β· ((𝑛 / e)↑𝑛)))    &   π΄ = (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛))))    &   π· = (𝑛 ∈ β„• ↦ (π΄β€˜(2 Β· 𝑛)))    &   πΈ = (𝑛 ∈ β„• ↦ ((βˆšβ€˜(2 Β· 𝑛)) Β· ((𝑛 / e)↑𝑛)))    &   π‘‰ = (𝑛 ∈ β„• ↦ ((((2↑(4 Β· 𝑛)) Β· ((!β€˜π‘›)↑4)) / ((!β€˜(2 Β· 𝑛))↑2)) / ((2 Β· 𝑛) + 1)))    &   πΉ = (𝑛 ∈ β„• ↦ (((π΄β€˜π‘›)↑4) / ((π·β€˜π‘›)↑2)))    &   π» = (𝑛 ∈ β„• ↦ ((𝑛↑2) / (𝑛 Β· ((2 Β· 𝑛) + 1))))    &   (πœ‘ β†’ 𝐢 ∈ ℝ+)    &   (πœ‘ β†’ 𝐴 ⇝ 𝐢)    β‡’   (πœ‘ β†’ (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / (π‘†β€˜π‘›))) ⇝ 1)
 
Theoremstirling 44040 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 Ο€ it is proven that the unknown constant is the square root of Ο€ and then the exact Stirling's formula is established. This is Metamath 100 proof #90. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑆 = (𝑛 ∈ β„•0 ↦ ((βˆšβ€˜((2 Β· Ο€) Β· 𝑛)) Β· ((𝑛 / e)↑𝑛)))    β‡’   (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / (π‘†β€˜π‘›))) ⇝ 1
 
Theoremstirlingr 44041 Stirling's approximation formula for 𝑛 factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 44040 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 ∈ β„•0 ↦ ((βˆšβ€˜((2 Β· Ο€) Β· 𝑛)) Β· ((𝑛 / e)↑𝑛)))    &   π‘… = (β‡π‘‘β€˜(topGenβ€˜ran (,)))    β‡’   (𝑛 ∈ β„• ↦ ((!β€˜π‘›) / (π‘†β€˜π‘›)))𝑅1
 
21.38.15  Dirichlet kernel
 
Theoremdirkerval 44042* The Nth Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))    β‡’   (𝑁 ∈ β„• β†’ (π·β€˜π‘) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
 
Theoremdirker2re 44043 The Dirichlet Kernel value is a real if the argument is not a multiple of Ο€ . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(((𝑁 ∈ β„• ∧ 𝑆 ∈ ℝ) ∧ Β¬ (𝑆 mod (2 Β· Ο€)) = 0) β†’ ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑆)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑆 / 2)))) ∈ ℝ)
 
Theoremdirkerdenne0 44044 The Dirichlet Kernel denominator is never 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
((𝑆 ∈ ℝ ∧ Β¬ (𝑆 mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(𝑆 / 2))) β‰  0)
 
Theoremdirkerval2 44045* The Nth Dirichlet Kernel evaluated at a specific point 𝑆. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))    β‡’   ((𝑁 ∈ β„• ∧ 𝑆 ∈ ℝ) β†’ ((π·β€˜π‘)β€˜π‘†) = if((𝑆 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑆)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑆 / 2))))))
 
Theoremdirkerre 44046* The Dirichlet Kernel at any point evaluates to a real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))    β‡’   ((𝑁 ∈ β„• ∧ 𝑆 ∈ ℝ) β†’ ((π·β€˜π‘)β€˜π‘†) ∈ ℝ)
 
Theoremdirkerper 44047* the Dirichlet Kernel has period 2Ο€. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))    &   π‘‡ = (2 Β· Ο€)    β‡’   ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((π·β€˜π‘)β€˜(π‘₯ + 𝑇)) = ((π·β€˜π‘)β€˜π‘₯))
 
Theoremdirkerf 44048* For any natural number 𝑁, the Dirichlet Kernel (π·β€˜π‘) is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))    β‡’   (𝑁 ∈ β„• β†’ (π·β€˜π‘):β„βŸΆβ„)
 
Theoremdirkertrigeqlem1 44049* Sum of an even number of alternating cos values. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐾 ∈ β„• β†’ Σ𝑛 ∈ (1...(2 Β· 𝐾))(cosβ€˜(𝑛 Β· Ο€)) = 0)
 
Theoremdirkertrigeqlem2 44050* Trigonomic equality lemma for the Dirichlet Kernel trigonomic equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ (sinβ€˜π΄) β‰  0)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (((1 / 2) + Σ𝑛 ∈ (1...𝑁)(cosβ€˜(𝑛 Β· 𝐴))) / Ο€) = ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝐴)) / ((2 Β· Ο€) Β· (sinβ€˜(𝐴 / 2)))))
 
Theoremdirkertrigeqlem3 44051* Trigonometric equality lemma for the Dirichlet Kernel trigonometric equality. Here we handle the case for an angle that's an odd multiple of Ο€. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐾 ∈ β„€)    &   π΄ = (((2 Β· 𝐾) + 1) Β· Ο€)    β‡’   (πœ‘ β†’ (((1 / 2) + Σ𝑛 ∈ (1...𝑁)(cosβ€˜(𝑛 Β· 𝐴))) / Ο€) = ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝐴)) / ((2 Β· Ο€) Β· (sinβ€˜(𝐴 / 2)))))
 
Theoremdirkertrigeq 44052* Trigonometric equality for the Dirichlet kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   πΉ = (π·β€˜π‘)    &   π» = (𝑠 ∈ ℝ ↦ (((1 / 2) + Ξ£π‘˜ ∈ (1...𝑁)(cosβ€˜(π‘˜ Β· 𝑠))) / Ο€))    β‡’   (πœ‘ β†’ 𝐹 = 𝐻)
 
Theoremdirkeritg 44053* The definite integral of the Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (π‘₯ ∈ ℝ ↦ if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   πΉ = (π·β€˜π‘)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   πΊ = (π‘₯ ∈ (𝐴[,]𝐡) ↦ (((π‘₯ / 2) + Ξ£π‘˜ ∈ (1...𝑁)((sinβ€˜(π‘˜ Β· π‘₯)) / π‘˜)) / Ο€))    β‡’   (πœ‘ β†’ ∫(𝐴(,)𝐡)(πΉβ€˜π‘₯) dπ‘₯ = ((πΊβ€˜π΅) βˆ’ (πΊβ€˜π΄)))
 
Theoremdirkercncflem1 44054* If π‘Œ is a multiple of Ο€ then it belongs to an open inerval (𝐴(,)𝐡) such that for any other point 𝑦 in the interval, cos y/2 and sin y/2 are nonzero. Such an interval is needed to apply De L'Hopital theorem. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐴 = (π‘Œ βˆ’ Ο€)    &   π΅ = (π‘Œ + Ο€)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ (π‘Œ mod (2 Β· Ο€)) = 0)    β‡’   (πœ‘ β†’ (π‘Œ ∈ (𝐴(,)𝐡) ∧ βˆ€π‘¦ ∈ ((𝐴(,)𝐡) βˆ– {π‘Œ})((sinβ€˜(𝑦 / 2)) β‰  0 ∧ (cosβ€˜(𝑦 / 2)) β‰  0)))
 
Theoremdirkercncflem2 44055* Lemma used to prove that the Dirichlet Kernel is continuous at π‘Œ points that are multiples of (2 Β· Ο€). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))    &   πΉ = (𝑦 ∈ ((𝐴(,)𝐡) βˆ– {π‘Œ}) ↦ (sinβ€˜((𝑁 + (1 / 2)) Β· 𝑦)))    &   πΊ = (𝑦 ∈ ((𝐴(,)𝐡) βˆ– {π‘Œ}) ↦ ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2))))    &   ((πœ‘ ∧ 𝑦 ∈ ((𝐴(,)𝐡) βˆ– {π‘Œ})) β†’ (sinβ€˜(𝑦 / 2)) β‰  0)    &   π» = (𝑦 ∈ ((𝐴(,)𝐡) βˆ– {π‘Œ}) ↦ ((𝑁 + (1 / 2)) Β· (cosβ€˜((𝑁 + (1 / 2)) Β· 𝑦))))    &   πΌ = (𝑦 ∈ ((𝐴(,)𝐡) βˆ– {π‘Œ}) ↦ (Ο€ Β· (cosβ€˜(𝑦 / 2))))    &   πΏ = (𝑀 ∈ (𝐴(,)𝐡) ↦ (((𝑁 + (1 / 2)) Β· (cosβ€˜((𝑁 + (1 / 2)) Β· 𝑀))) / (Ο€ Β· (cosβ€˜(𝑀 / 2)))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ π‘Œ ∈ (𝐴(,)𝐡))    &   (πœ‘ β†’ (π‘Œ mod (2 Β· Ο€)) = 0)    &   ((πœ‘ ∧ 𝑦 ∈ ((𝐴(,)𝐡) βˆ– {π‘Œ})) β†’ (cosβ€˜(𝑦 / 2)) β‰  0)    β‡’   (πœ‘ β†’ ((π·β€˜π‘)β€˜π‘Œ) ∈ (((π·β€˜π‘) β†Ύ ((𝐴(,)𝐡) βˆ– {π‘Œ})) limβ„‚ π‘Œ))
 
Theoremdirkercncflem3 44056* The Dirichlet Kernel is continuous at π‘Œ points that are multiples of (2 Β· Ο€). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))    &   π΄ = (π‘Œ βˆ’ Ο€)    &   π΅ = (π‘Œ + Ο€)    &   πΉ = (𝑦 ∈ (𝐴(,)𝐡) ↦ ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))    &   πΊ = (𝑦 ∈ (𝐴(,)𝐡) ↦ ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ (π‘Œ mod (2 Β· Ο€)) = 0)    β‡’   (πœ‘ β†’ ((π·β€˜π‘)β€˜π‘Œ) ∈ ((π·β€˜π‘) limβ„‚ π‘Œ))
 
Theoremdirkercncflem4 44057* The Dirichlet Kernel is continuos at points that are not multiple of 2 Ο€ . This is the easier condition, for the proof of the continuity of the Dirichlet kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ (π‘Œ mod (2 Β· Ο€)) β‰  0)    &   π΄ = (βŒŠβ€˜(π‘Œ / (2 Β· Ο€)))    &   π΅ = (𝐴 + 1)    &   πΆ = (𝐴 Β· (2 Β· Ο€))    &   πΈ = (𝐡 Β· (2 Β· Ο€))    β‡’   (πœ‘ β†’ (π·β€˜π‘) ∈ (((topGenβ€˜ran (,)) CnP (topGenβ€˜ran (,)))β€˜π‘Œ))
 
Theoremdirkercncf 44058* For any natural number 𝑁, the Dirichlet Kernel (π·β€˜π‘) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))    β‡’   (𝑁 ∈ β„• β†’ (π·β€˜π‘) ∈ (ℝ–cn→ℝ))
 
21.38.16  Fourier Series
 
Theoremfourierdlem1 44059 A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))    &   (πœ‘ β†’ 𝐼 ∈ (0..^𝑀))    &   (πœ‘ β†’ 𝑋 ∈ ((π‘„β€˜πΌ)[,](π‘„β€˜(𝐼 + 1))))    β‡’   (πœ‘ β†’ 𝑋 ∈ (𝐴[,]𝐡))
 
Theoremfourierdlem2 44060* Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    β‡’   (𝑀 ∈ β„• β†’ (𝑄 ∈ (π‘ƒβ€˜π‘€) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))))
 
Theoremfourierdlem3 44061* Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ ((-Ο€[,]Ο€) ↑m (0...π‘š)) ∣ (((π‘β€˜0) = -Ο€ ∧ (π‘β€˜π‘š) = Ο€) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    β‡’   (𝑀 ∈ β„• β†’ (𝑄 ∈ (π‘ƒβ€˜π‘€) ↔ (𝑄 ∈ ((-Ο€[,]Ο€) ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = -Ο€ ∧ (π‘„β€˜π‘€) = Ο€) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))))
 
Theoremfourierdlem4 44062* 𝐸 is a function that maps any point to a periodic corresponding point in (𝐴, 𝐡]. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   πΈ = (π‘₯ ∈ ℝ ↦ (π‘₯ + ((βŒŠβ€˜((𝐡 βˆ’ π‘₯) / 𝑇)) Β· 𝑇)))    β‡’   (πœ‘ β†’ 𝐸:β„βŸΆ(𝐴(,]𝐡))
 
Theoremfourierdlem5 44063* 𝑆 is a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑆 = (π‘₯ ∈ (-Ο€[,]Ο€) ↦ (sinβ€˜((𝑋 + (1 / 2)) Β· π‘₯)))    β‡’   (𝑋 ∈ ℝ β†’ 𝑆:(-Ο€[,]Ο€)βŸΆβ„)
 
Theoremfourierdlem6 44064 𝑋 is in the periodic partition, when the considered interval is centered at 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ 𝐼 ∈ β„€)    &   (πœ‘ β†’ 𝐽 ∈ β„€)    &   (πœ‘ β†’ 𝐼 < 𝐽)    &   (πœ‘ β†’ (𝑋 + (𝐼 Β· 𝑇)) ∈ (𝐴[,]𝐡))    &   (πœ‘ β†’ (𝑋 + (𝐽 Β· 𝑇)) ∈ (𝐴[,]𝐡))    β‡’   (πœ‘ β†’ 𝐽 = (𝐼 + 1))
 
Theoremfourierdlem7 44065* The difference between the periodic sawtooth function and the identity function is decreasing. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   πΈ = (π‘₯ ∈ ℝ ↦ (π‘₯ + ((βŒŠβ€˜((𝐡 βˆ’ π‘₯) / 𝑇)) Β· 𝑇)))    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ≀ π‘Œ)    β‡’   (πœ‘ β†’ ((πΈβ€˜π‘Œ) βˆ’ π‘Œ) ≀ ((πΈβ€˜π‘‹) βˆ’ 𝑋))
 
Theoremfourierdlem8 44066 A partition interval is a subset of the partitioned interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))    &   (πœ‘ β†’ 𝐼 ∈ (0..^𝑀))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΌ)[,](π‘„β€˜(𝐼 + 1))) βŠ† (𝐴[,]𝐡))
 
Theoremfourierdlem9 44067* 𝐻 is a complex function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ π‘Š ∈ ℝ)    &   π» = (𝑠 ∈ (-Ο€[,]Ο€) ↦ if(𝑠 = 0, 0, (((πΉβ€˜(𝑋 + 𝑠)) βˆ’ if(0 < 𝑠, π‘Œ, π‘Š)) / 𝑠)))    β‡’   (πœ‘ β†’ 𝐻:(-Ο€[,]Ο€)βŸΆβ„)
 
Theoremfourierdlem10 44068 Condition on the bounds of a nonempty subinterval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐷 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 < 𝐷)    &   (πœ‘ β†’ (𝐢(,)𝐷) βŠ† (𝐴(,)𝐡))    β‡’   (πœ‘ β†’ (𝐴 ≀ 𝐢 ∧ 𝐷 ≀ 𝐡))
 
Theoremfourierdlem11 44069* If there is a partition, than the lower bound is strictly less than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))    β‡’   (πœ‘ β†’ (𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐴 < 𝐡))
 
Theoremfourierdlem12 44070* 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 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))    &   (πœ‘ β†’ 𝑋 ∈ ran 𝑄)    β‡’   ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) β†’ Β¬ 𝑋 ∈ ((π‘„β€˜π‘–)(,)(π‘„β€˜(𝑖 + 1))))
 
Theoremfourierdlem13 44071* Value of 𝑉 in terms of value of 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   π‘ƒ = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = (𝐴 + 𝑋) ∧ (π‘β€˜π‘š) = (𝐡 + 𝑋)) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑉 ∈ (π‘ƒβ€˜π‘€))    &   (πœ‘ β†’ 𝐼 ∈ (0...𝑀))    &   π‘„ = (𝑖 ∈ (0...𝑀) ↦ ((π‘‰β€˜π‘–) βˆ’ 𝑋))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΌ) = ((π‘‰β€˜πΌ) βˆ’ 𝑋) ∧ (π‘‰β€˜πΌ) = (𝑋 + (π‘„β€˜πΌ))))
 
Theoremfourierdlem14 44072* Given the partition 𝑉, 𝑄 is the partition shifted to the left by 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   π‘ƒ = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = (𝐴 + 𝑋) ∧ (π‘β€˜π‘š) = (𝐡 + 𝑋)) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   π‘‚ = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑉 ∈ (π‘ƒβ€˜π‘€))    &   π‘„ = (𝑖 ∈ (0...𝑀) ↦ ((π‘‰β€˜π‘–) βˆ’ 𝑋))    β‡’   (πœ‘ β†’ 𝑄 ∈ (π‘‚β€˜π‘€))
 
Theoremfourierdlem15 44073* The range of the partition is between its starting point and its ending point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))    β‡’   (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))
 
Theoremfourierdlem16 44074* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   πΆ = (-Ο€(,)Ο€)    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐢) ∈ 𝐿1)    &   π΄ = (𝑛 ∈ β„•0 ↦ (∫𝐢((πΉβ€˜π‘₯) Β· (cosβ€˜(𝑛 Β· π‘₯))) dπ‘₯ / Ο€))    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (((π΄β€˜π‘) ∈ ℝ ∧ (π‘₯ ∈ 𝐢 ↦ (πΉβ€˜π‘₯)) ∈ 𝐿1) ∧ ∫𝐢((πΉβ€˜π‘₯) Β· (cosβ€˜(𝑁 Β· π‘₯))) dπ‘₯ ∈ ℝ))
 
Theoremfourierdlem17 44075* The defined 𝐿 is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   πΏ = (π‘₯ ∈ (𝐴(,]𝐡) ↦ if(π‘₯ = 𝐡, 𝐴, π‘₯))    β‡’   (πœ‘ β†’ 𝐿:(𝐴(,]𝐡)⟢(𝐴[,]𝐡))
 
Theoremfourierdlem18 44076* The function 𝑆 is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑁 ∈ ℝ)    &   π‘† = (𝑠 ∈ (-Ο€[,]Ο€) ↦ (sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)))    β‡’   (πœ‘ β†’ 𝑆 ∈ ((-Ο€[,]Ο€)–cn→ℝ))
 
Theoremfourierdlem19 44077* If two elements of 𝐷 have the same periodic image in (𝐴(,]𝐡) then they are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   π· = {𝑦 ∈ ((𝐴 + 𝑋)(,](𝐡 + 𝑋)) ∣ βˆƒπ‘˜ ∈ β„€ (𝑦 + (π‘˜ Β· 𝑇)) ∈ 𝐢}    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   πΈ = (π‘₯ ∈ ℝ ↦ (π‘₯ + ((βŒŠβ€˜((𝐡 βˆ’ π‘₯) / 𝑇)) Β· 𝑇)))    &   (πœ‘ β†’ π‘Š ∈ 𝐷)    &   (πœ‘ β†’ 𝑍 ∈ 𝐷)    &   (πœ‘ β†’ (πΈβ€˜π‘) = (πΈβ€˜π‘Š))    β‡’   (πœ‘ β†’ Β¬ π‘Š < 𝑍)
 
Theoremfourierdlem20 44078* Every interval in the partition 𝑆 is included in an interval of the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝑄:(0...𝑀)βŸΆβ„)    &   (πœ‘ β†’ (π‘„β€˜0) ≀ 𝐴)    &   (πœ‘ β†’ 𝐡 ≀ (π‘„β€˜π‘€))    &   (πœ‘ β†’ 𝐽 ∈ (0..^𝑁))    &   π‘‡ = ({𝐴, 𝐡} βˆͺ (ran 𝑄 ∩ (𝐴(,)𝐡)))    &   (πœ‘ β†’ 𝑆 Isom < , < ((0...𝑁), 𝑇))    &   πΌ = sup({π‘˜ ∈ (0..^𝑀) ∣ (π‘„β€˜π‘˜) ≀ (π‘†β€˜π½)}, ℝ, < )    β‡’   (πœ‘ β†’ βˆƒπ‘– ∈ (0..^𝑀)((π‘†β€˜π½)(,)(π‘†β€˜(𝐽 + 1))) βŠ† ((π‘„β€˜π‘–)(,)(π‘„β€˜(𝑖 + 1))))
 
Theoremfourierdlem21 44079* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   πΆ = (-Ο€(,)Ο€)    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐢) ∈ 𝐿1)    &   π΅ = (𝑛 ∈ β„• ↦ (∫𝐢((πΉβ€˜π‘₯) Β· (sinβ€˜(𝑛 Β· π‘₯))) dπ‘₯ / Ο€))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (((π΅β€˜π‘) ∈ ℝ ∧ (π‘₯ ∈ 𝐢 ↦ ((πΉβ€˜π‘₯) Β· (sinβ€˜(𝑁 Β· π‘₯)))) ∈ 𝐿1) ∧ ∫𝐢((πΉβ€˜π‘₯) Β· (sinβ€˜(𝑁 Β· π‘₯))) dπ‘₯ ∈ ℝ))
 
Theoremfourierdlem22 44080* The coefficients of the fourier series are integrable and reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   πΆ = (-Ο€(,)Ο€)    &   (πœ‘ β†’ (𝐹 β†Ύ 𝐢) ∈ 𝐿1)    &   π΄ = (𝑛 ∈ β„•0 ↦ (∫𝐢((πΉβ€˜π‘₯) Β· (cosβ€˜(𝑛 Β· π‘₯))) dπ‘₯ / Ο€))    &   π΅ = (𝑛 ∈ β„• ↦ (∫𝐢((πΉβ€˜π‘₯) Β· (sinβ€˜(𝑛 Β· π‘₯))) dπ‘₯ / Ο€))    β‡’   (πœ‘ β†’ ((𝑛 ∈ β„•0 β†’ (π΄β€˜π‘›) ∈ ℝ) ∧ (𝑛 ∈ β„• β†’ (π΅β€˜π‘›) ∈ ℝ)))
 
Theoremfourierdlem23 44081* If 𝐹 is continuous and 𝑋 is constant, then (πΉβ€˜(𝑋 + 𝑠)) is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 βŠ† β„‚)    &   (πœ‘ β†’ 𝐹 ∈ (𝐴–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐡 βŠ† β„‚)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   ((πœ‘ ∧ 𝑠 ∈ 𝐡) β†’ (𝑋 + 𝑠) ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝑠 ∈ 𝐡 ↦ (πΉβ€˜(𝑋 + 𝑠))) ∈ (𝐡–cnβ†’β„‚))
 
Theoremfourierdlem24 44082 A sufficient condition for module being nonzero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝐴 ∈ ((-Ο€[,]Ο€) βˆ– {0}) β†’ (𝐴 mod (2 Β· Ο€)) β‰  0)
 
Theoremfourierdlem25 44083* If 𝐢 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.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄:(0...𝑀)βŸΆβ„)    &   (πœ‘ β†’ 𝐢 ∈ ((π‘„β€˜0)[,](π‘„β€˜π‘€)))    &   (πœ‘ β†’ Β¬ 𝐢 ∈ ran 𝑄)    &   πΌ = sup({π‘˜ ∈ (0..^𝑀) ∣ (π‘„β€˜π‘˜) < 𝐢}, ℝ, < )    β‡’   (πœ‘ β†’ βˆƒπ‘— ∈ (0..^𝑀)𝐢 ∈ ((π‘„β€˜π‘—)(,)(π‘„β€˜(𝑗 + 1))))
 
Theoremfourierdlem26 44084* Periodic image of a point π‘Œ that's in the period that begins with the point 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   πΈ = (π‘₯ ∈ ℝ ↦ (π‘₯ + ((βŒŠβ€˜((𝐡 βˆ’ π‘₯) / 𝑇)) Β· 𝑇)))    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ (πΈβ€˜π‘‹) = 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ (𝑋(,](𝑋 + 𝑇)))    β‡’   (πœ‘ β†’ (πΈβ€˜π‘Œ) = (𝐴 + (π‘Œ βˆ’ 𝑋)))
 
Theoremfourierdlem27 44085 A partition open interval is a subset of the partitioned open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    &   (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))    &   (πœ‘ β†’ 𝐼 ∈ (0..^𝑀))    β‡’   (πœ‘ β†’ ((π‘„β€˜πΌ)(,)(π‘„β€˜(𝐼 + 1))) βŠ† (𝐴(,)𝐡))
 
Theoremfourierdlem28 44086* Derivative of (πΉβ€˜(𝑋 + 𝑠)). (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   π· = (ℝ D (𝐹 β†Ύ ((𝑋 + 𝐴)(,)(𝑋 + 𝐡))))    &   (πœ‘ β†’ 𝐷:((𝑋 + 𝐴)(,)(𝑋 + 𝐡))βŸΆβ„)    β‡’   (πœ‘ β†’ (ℝ D (𝑠 ∈ (𝐴(,)𝐡) ↦ (πΉβ€˜(𝑋 + 𝑠)))) = (𝑠 ∈ (𝐴(,)𝐡) ↦ (π·β€˜(𝑋 + 𝑠))))
 
Theoremfourierdlem29 44087* Explicit function value for 𝐾 applied to 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝐾 = (𝑠 ∈ (-Ο€[,]Ο€) ↦ if(𝑠 = 0, 1, (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2))))))    β‡’   (𝐴 ∈ (-Ο€[,]Ο€) β†’ (πΎβ€˜π΄) = if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))))
 
Theoremfourierdlem30 44088* Sum of three small pieces is less than Ξ΅. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ (π‘₯ ∈ 𝐼 ↦ (𝐹 Β· -𝐺)) ∈ 𝐿1)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝐹 ∈ β„‚)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝐺 ∈ β„‚)    &   (πœ‘ β†’ 𝐴 ∈ β„‚)    &   π‘‹ = (absβ€˜π΄)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   π‘Œ = (absβ€˜πΆ)    &   π‘ = (absβ€˜βˆ«πΌ(𝐹 Β· -𝐺) dπ‘₯)    &   (πœ‘ β†’ 𝐸 ∈ ℝ+)    &   (πœ‘ β†’ 𝑅 ∈ ℝ)    &   (πœ‘ β†’ ((((𝑋 + π‘Œ) + 𝑍) / 𝐸) + 1) ≀ 𝑅)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π΅) ≀ 1)    &   (πœ‘ β†’ 𝐷 ∈ β„‚)    &   (πœ‘ β†’ (absβ€˜π·) ≀ 1)    β‡’   (πœ‘ β†’ (absβ€˜(((𝐴 Β· -(𝐡 / 𝑅)) βˆ’ (𝐢 Β· -(𝐷 / 𝑅))) βˆ’ ∫𝐼(𝐹 Β· -(𝐺 / 𝑅)) dπ‘₯)) < 𝐸)
 
Theoremfourierdlem31 44089* If 𝐴 is finite and for any element in 𝐴 there is a number π‘š such that a property holds for all numbers larger than π‘š, 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.)
β„²π‘–πœ‘    &   β„²π‘Ÿπœ‘    &   β„²π‘–𝑉    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ βˆ€π‘– ∈ 𝐴 βˆƒπ‘š ∈ β„• βˆ€π‘Ÿ ∈ (π‘š(,)+∞)πœ’)    &   π‘€ = {π‘š ∈ β„• ∣ βˆ€π‘Ÿ ∈ (π‘š(,)+∞)πœ’}    &   π‘‰ = (𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < ))    &   π‘ = sup(ran 𝑉, ℝ, < )    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ β„• βˆ€π‘Ÿ ∈ (𝑛(,)+∞)βˆ€π‘– ∈ 𝐴 πœ’)
 
Theoremfourierdlem32 44090 Limit of a continuous function on an open subinterval. Lower bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝑅 ∈ (𝐹 limβ„‚ 𝐴))    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐷 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 < 𝐷)    &   (πœ‘ β†’ (𝐢(,)𝐷) βŠ† (𝐴(,)𝐡))    &   π‘Œ = if(𝐢 = 𝐴, 𝑅, (πΉβ€˜πΆ))    &   π½ = ((TopOpenβ€˜β„‚fld) β†Ύt (𝐴[,)𝐡))    β‡’   (πœ‘ β†’ π‘Œ ∈ ((𝐹 β†Ύ (𝐢(,)𝐷)) limβ„‚ 𝐢))
 
Theoremfourierdlem33 44091 Limit of a continuous function on an open subinterval. Upper bound version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ 𝐿 ∈ (𝐹 limβ„‚ 𝐡))    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐷 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 < 𝐷)    &   (πœ‘ β†’ (𝐢(,)𝐷) βŠ† (𝐴(,)𝐡))    &   π‘Œ = if(𝐷 = 𝐡, 𝐿, (πΉβ€˜π·))    &   π½ = ((TopOpenβ€˜β„‚fld) β†Ύt ((𝐴(,)𝐡) βˆͺ {𝐡}))    β‡’   (πœ‘ β†’ π‘Œ ∈ ((𝐹 β†Ύ (𝐢(,)𝐷)) limβ„‚ 𝐷))
 
Theoremfourierdlem34 44092* A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))    β‡’   (πœ‘ β†’ 𝑄:(0...𝑀)–1-1→ℝ)
 
Theoremfourierdlem35 44093 There is a single point in (𝐴(,]𝐡) that's distant from 𝑋 a multiple integer of 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ 𝐼 ∈ β„€)    &   (πœ‘ β†’ 𝐽 ∈ β„€)    &   (πœ‘ β†’ (𝑋 + (𝐼 Β· 𝑇)) ∈ (𝐴(,]𝐡))    &   (πœ‘ β†’ (𝑋 + (𝐽 Β· 𝑇)) ∈ (𝐴(,]𝐡))    β‡’   (πœ‘ β†’ 𝐼 = 𝐽)
 
Theoremfourierdlem36 44094* 𝐹 is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   πΉ = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝐴))    &   π‘ = ((β™―β€˜π΄) βˆ’ 1)    β‡’   (πœ‘ β†’ 𝐹 Isom < , < ((0...𝑁), 𝐴))
 
Theoremfourierdlem37 44095* 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 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   πΈ = (π‘₯ ∈ ℝ ↦ (π‘₯ + ((βŒŠβ€˜((𝐡 βˆ’ π‘₯) / 𝑇)) Β· 𝑇)))    &   πΏ = (𝑦 ∈ (𝐴(,]𝐡) ↦ if(𝑦 = 𝐡, 𝐴, 𝑦))    &   πΌ = (π‘₯ ∈ ℝ ↦ sup({𝑖 ∈ (0..^𝑀) ∣ (π‘„β€˜π‘–) ≀ (πΏβ€˜(πΈβ€˜π‘₯))}, ℝ, < ))    β‡’   (πœ‘ β†’ (𝐼:β„βŸΆ(0..^𝑀) ∧ (π‘₯ ∈ ℝ β†’ sup({𝑖 ∈ (0..^𝑀) ∣ (π‘„β€˜π‘–) ≀ (πΏβ€˜(πΈβ€˜π‘₯))}, ℝ, < ) ∈ {𝑖 ∈ (0..^𝑀) ∣ (π‘„β€˜π‘–) ≀ (πΏβ€˜(πΈβ€˜π‘₯))})))
 
Theoremfourierdlem38 44096* The function 𝐹 is continuous on every interval induced by the partition 𝑄. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹 ∈ (dom 𝐹–cnβ†’β„‚))    &   π‘ƒ = (𝑛 ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...𝑛)) ∣ (((π‘β€˜0) = -Ο€ ∧ (π‘β€˜π‘›) = Ο€) ∧ βˆ€π‘– ∈ (0..^𝑛)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))    &   π» = (𝐴 βˆͺ ((-Ο€[,]Ο€) βˆ– dom 𝐹))    &   (πœ‘ β†’ ran 𝑄 = 𝐻)    β‡’   ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) β†’ (𝐹 β†Ύ ((π‘„β€˜π‘–)(,)(π‘„β€˜(𝑖 + 1)))) ∈ (((π‘„β€˜π‘–)(,)(π‘„β€˜(𝑖 + 1)))–cnβ†’β„‚))
 
Theoremfourierdlem39 44097* Integration by parts of ∫(𝐴(,)𝐡)((πΉβ€˜π‘₯) Β· (sinβ€˜(𝑅 Β· π‘₯))) dπ‘₯ (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 ≀ 𝐡)    &   (πœ‘ β†’ 𝐹 ∈ ((𝐴[,]𝐡)–cnβ†’β„‚))    &   πΊ = (ℝ D 𝐹)    &   (πœ‘ β†’ 𝐺 ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘₯ ∈ (𝐴(,)𝐡)(absβ€˜(πΊβ€˜π‘₯)) ≀ 𝑦)    &   (πœ‘ β†’ 𝑅 ∈ ℝ+)    β‡’   (πœ‘ β†’ ∫(𝐴(,)𝐡)((πΉβ€˜π‘₯) Β· (sinβ€˜(𝑅 Β· π‘₯))) dπ‘₯ = ((((πΉβ€˜π΅) Β· -((cosβ€˜(𝑅 Β· 𝐡)) / 𝑅)) βˆ’ ((πΉβ€˜π΄) Β· -((cosβ€˜(𝑅 Β· 𝐴)) / 𝑅))) βˆ’ ∫(𝐴(,)𝐡)((πΊβ€˜π‘₯) Β· -((cosβ€˜(𝑅 Β· π‘₯)) / 𝑅)) dπ‘₯))
 
Theoremfourierdlem40 44098* 𝐻 is a continuous function on any partition interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(πœ‘ β†’ 𝐹:β„βŸΆβ„)    &   (πœ‘ β†’ 𝐴 ∈ (-Ο€[,]Ο€))    &   (πœ‘ β†’ 𝐡 ∈ (-Ο€[,]Ο€))    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ Β¬ 0 ∈ (𝐴(,)𝐡))    &   (πœ‘ β†’ (𝐹 β†Ύ ((𝐴 + 𝑋)(,)(𝐡 + 𝑋))) ∈ (((𝐴 + 𝑋)(,)(𝐡 + 𝑋))–cnβ†’β„‚))    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   (πœ‘ β†’ π‘Š ∈ ℝ)    &   π» = (𝑠 ∈ (-Ο€[,]Ο€) ↦ if(𝑠 = 0, 0, (((πΉβ€˜(𝑋 + 𝑠)) βˆ’ if(0 < 𝑠, π‘Œ, π‘Š)) / 𝑠)))    β‡’   (πœ‘ β†’ (𝐻 β†Ύ (𝐴(,)𝐡)) ∈ ((𝐴(,)𝐡)–cnβ†’β„‚))
 
Theoremfourierdlem41 44099* 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐴 < 𝐡)    &   π‘‡ = (𝐡 βˆ’ 𝐴)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐷 ∧ π‘˜ ∈ β„€) β†’ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ 𝐷)    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   π‘ = (π‘₯ ∈ ℝ ↦ ((βŒŠβ€˜((𝐡 βˆ’ π‘₯) / 𝑇)) Β· 𝑇))    &   πΈ = (π‘₯ ∈ ℝ ↦ (π‘₯ + (π‘β€˜π‘₯)))    &   π‘ƒ = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))    &   ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) β†’ ((π‘„β€˜π‘–)(,)(π‘„β€˜(𝑖 + 1))) βŠ† 𝐷)    β‡’   (πœ‘ β†’ (βˆƒπ‘¦ ∈ ℝ (𝑦 < 𝑋 ∧ (𝑦(,)𝑋) βŠ† 𝐷) ∧ βˆƒπ‘¦ ∈ ℝ (𝑋 < 𝑦 ∧ (𝑋(,)𝑦) βŠ† 𝐷)))
 
Theoremfourierdlem42 44100* The set of points in a moved partition are finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 29-Sep-2020.)
(πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 < 𝐢)    &   π‘‡ = (𝐢 βˆ’ 𝐡)    &   (πœ‘ β†’ 𝐴 βŠ† (𝐡[,]𝐢))    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   (πœ‘ β†’ 𝐡 ∈ 𝐴)    &   (πœ‘ β†’ 𝐢 ∈ 𝐴)    &   π· = (abs ∘ βˆ’ )    &   πΌ = ((𝐴 Γ— 𝐴) βˆ– I )    &   π‘… = ran (𝐷 β†Ύ 𝐼)    &   πΈ = inf(𝑅, ℝ, < )    &   (πœ‘ β†’ 𝑋 ∈ ℝ)    &   (πœ‘ β†’ π‘Œ ∈ ℝ)    &   π½ = (topGenβ€˜ran (,))    &   πΎ = (𝐽 β†Ύt (𝑋[,]π‘Œ))    &   π» = {π‘₯ ∈ (𝑋[,]π‘Œ) ∣ βˆƒπ‘˜ ∈ β„€ (π‘₯ + (π‘˜ Β· 𝑇)) ∈ 𝐴}    &   (πœ“ ↔ ((πœ‘ ∧ (π‘Ž ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ π‘Ž < 𝑏)) ∧ βˆƒπ‘— ∈ β„€ βˆƒπ‘˜ ∈ β„€ ((π‘Ž + (𝑗 Β· 𝑇)) ∈ 𝐴 ∧ (𝑏 + (π‘˜ Β· 𝑇)) ∈ 𝐴)))    β‡’   (πœ‘ β†’ 𝐻 ∈ Fin)
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