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Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstoweidlem8 38901* Lemma for stoweid 38956: two class variables replace two setvar variables, for the sum of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   𝑡𝐹    &   𝑡𝐺       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) + (𝐺𝑡))) ∈ 𝐴)
 
Theoremstoweidlem9 38902* Lemma for stoweid 38956: here the Stone Weierstrass theorem is proven for the trivial case, T is the empty set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑇 = ∅)    &   (𝜑 → (𝑡𝑇 ↦ 1) ∈ 𝐴)       (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < 𝐸)
 
Theoremstoweidlem10 38903 Lemma for stoweid 38956. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90, this lemma is an application of Bernoulli's inequality. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0𝐴 ≤ 1) → (1 − (𝑁 · 𝐴)) ≤ ((1 − 𝐴)↑𝑁))
 
Theoremstoweidlem11 38904* This lemma is used to prove that there is a function 𝑔 as in the proof of [BrosowskiDeutsh] p. 92 (at the top of page 92): this lemma proves that g(t) < ( j + 1 / 3 ) * ε. Here 𝐸 is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝑡𝑇)    &   (𝜑𝑗 ∈ (1...𝑁))    &   ((𝜑𝑖 ∈ (0...𝑁)) → (𝑋𝑖):𝑇⟶ℝ)    &   ((𝜑𝑖 ∈ (0...𝑁)) → ((𝑋𝑖)‘𝑡) ≤ 1)    &   ((𝜑𝑖 ∈ (𝑗...𝑁)) → ((𝑋𝑖)‘𝑡) < (𝐸 / 𝑁))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸))
 
Theoremstoweidlem12 38905* Lemma for stoweid 38956. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)       ((𝜑𝑡𝑇) → (𝑄𝑡) = ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))
 
Theoremstoweidlem13 38906 Lemma for stoweid 38956. This lemma is used to prove the statement abs( f(t) - g(t) ) < 2 epsilon, in the last step of the proof in [BrosowskiDeutsh] p. 92. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑋 ∈ ℝ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝑗 ∈ ℝ)    &   (𝜑 → ((𝑗 − (4 / 3)) · 𝐸) < 𝑋)    &   (𝜑𝑋 ≤ ((𝑗 − (1 / 3)) · 𝐸))    &   (𝜑 → ((𝑗 − (4 / 3)) · 𝐸) < 𝑌)    &   (𝜑𝑌 < ((𝑗 + (1 / 3)) · 𝐸))       (𝜑 → (abs‘(𝑌𝑋)) < (2 · 𝐸))
 
Theoremstoweidlem14 38907* There exists a 𝑘 as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 𝑘 is an integer and 1 < k * δ < 2. 𝐷 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐴 = {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗}    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐷 < 1)       (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))
 
Theoremstoweidlem15 38908* This lemma is used to prove the existence of a function 𝑝 as in Lemma 1 from [BrosowskiDeutsh] p. 90: 𝑝 is in the subalgebra, such that 0 ≤ p ≤ 1, p(t_0) = 0, and p > 0 on T - U. Here (𝐺𝐼) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   (𝜑𝐺:(1...𝑀)⟶𝑄)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (((𝜑𝐼 ∈ (1...𝑀)) ∧ 𝑆𝑇) → (((𝐺𝐼)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺𝐼)‘𝑆) ∧ ((𝐺𝐼)‘𝑆) ≤ 1))
 
Theoremstoweidlem16 38909* Lemma for stoweid 38956. The subset 𝑌 of functions in the algebra 𝐴, with values in [ 0 , 1 ], is closed under multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝐻 = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)       ((𝜑𝑓𝑌𝑔𝑌) → 𝐻𝑌)
 
Theoremstoweidlem17 38910* This lemma proves that the function 𝑔 (as defined in [BrosowskiDeutsh] p. 91, at the end of page 91) belongs to the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑋:(0...𝑁)⟶𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝐸 ∈ ℝ)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (𝜑 → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡))) ∈ 𝐴)
 
Theoremstoweidlem18 38911* This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 92 when A is empty, the trivial case. Here D is used to denote the set A of Lemma 2, because the variable A is used for the subalgebra. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐷    &   𝑡𝜑    &   𝐹 = (𝑡𝑇 ↦ 1)    &   𝑇 = 𝐽    &   ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)    &   (𝜑𝐵 ∈ (Clsd‘𝐽))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐷 = ∅)       (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
 
Theoremstoweidlem19 38912* If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝐹𝐴)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡)↑𝑁)) ∈ 𝐴)
 
Theoremstoweidlem20 38913* If a set A of real functions from a common domain T is closed under the sum of two functions, then it is closed under the sum of a finite number of functions, indexed by G. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝐹 = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑀)⟶𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (𝜑𝐹𝐴)
 
Theoremstoweidlem21 38914* Once the Stone Weierstrass theorem has been proven for approximating nonnegative functions, then this lemma is used to extend the result to functions with (possibly) negative values. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐺    &   𝑡𝐻    &   𝑡𝑆    &   𝑡𝜑    &   𝐺 = (𝑡𝑇 ↦ ((𝐻𝑡) + 𝑆))    &   (𝜑𝐹:𝑇⟶ℝ)    &   (𝜑𝑆 ∈ ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑 → ∀𝑓𝐴 𝑓:𝑇⟶ℝ)    &   (𝜑𝐻𝐴)    &   (𝜑 → ∀𝑡𝑇 (abs‘((𝐻𝑡) − ((𝐹𝑡) − 𝑆))) < 𝐸)       (𝜑 → ∃𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸)
 
Theoremstoweidlem22 38915* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑡𝐹    &   𝑡𝐺    &   𝐻 = (𝑡𝑇 ↦ ((𝐹𝑡) − (𝐺𝑡)))    &   𝐼 = (𝑡𝑇 ↦ -1)    &   𝐿 = (𝑡𝑇 ↦ ((𝐼𝑡) · (𝐺𝑡)))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝐺𝑡))) ∈ 𝐴)
 
Theoremstoweidlem23 38916* This lemma is used to prove the existence of a function pt as in the beginning of Lemma 1 [BrosowskiDeutsh] p. 90: for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑡𝐺    &   𝐻 = (𝑡𝑇 ↦ ((𝐺𝑡) − (𝐺𝑍)))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝑆𝑇)    &   (𝜑𝑍𝑇)    &   (𝜑𝐺𝐴)    &   (𝜑 → (𝐺𝑆) ≠ (𝐺𝑍))       (𝜑 → (𝐻𝐴 ∧ (𝐻𝑆) ≠ (𝐻𝑍) ∧ (𝐻𝑍) = 0))
 
Theoremstoweidlem24 38917* This lemma proves that for 𝑛 sufficiently large, qn( t ) > ( 1 - epsilon ), for all 𝑡 in 𝑉: see Lemma 1 [BrosowskiDeutsh] p. 90, (at the bottom of page 90). 𝑄 is used to represent qn in the paper, 𝑁 to represent 𝑛 in the paper, 𝐾 to represent 𝑘, 𝐷 to represent δ, and 𝐸 to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}    &   𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁)))    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))       ((𝜑𝑡𝑉) → (1 − 𝐸) < (𝑄𝑡))
 
Theoremstoweidlem25 38918* This lemma proves that for n sufficiently large, qn( t ) < ε, for all 𝑡 in 𝑇𝑈: see Lemma 1 [BrosowskiDeutsh] p. 91 (at the top of page 91). 𝑄 is used to represent qn in the paper, 𝑁 to represent n in the paper, 𝐾 to represent k, 𝐷 to represent δ, 𝑃 to represent p, and 𝐸 to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸)       ((𝜑𝑡 ∈ (𝑇𝑈)) → (𝑄𝑡) < 𝐸)
 
Theoremstoweidlem26 38919* This lemma is used to prove that there is a function 𝑔 as in the proof of [BrosowskiDeutsh] p. 92: this lemma proves that g(t) > ( j - 4 / 3 ) * ε. Here 𝐿 is used to represnt j in the paper, 𝐷 is used to represent A in the paper, 𝑆 is used to represent t, and 𝐸 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑗𝜑    &   𝑡𝜑    &   𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})    &   𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇 ∈ V)    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝑆 ∈ ((𝐷𝐿) ∖ (𝐷‘(𝐿 − 1))))    &   (𝜑𝐹:𝑇⟶ℝ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))    &   ((𝜑𝑖 ∈ (0...𝑁)) → (𝑋𝑖):𝑇⟶ℝ)    &   ((𝜑𝑖 ∈ (0...𝑁) ∧ 𝑡𝑇) → 0 ≤ ((𝑋𝑖)‘𝑡))    &   ((𝜑𝑖 ∈ (0...𝑁) ∧ 𝑡 ∈ (𝐵𝑖)) → (1 − (𝐸 / 𝑁)) < ((𝑋𝑖)‘𝑡))       (𝜑 → ((𝐿 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑆))
 
Theoremstoweidlem27 38920* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Here (𝑞𝑖) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐺 = (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})    &   (𝜑𝑄 ∈ V)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑌 Fn ran 𝐺)    &   (𝜑 → ran 𝐺 ∈ V)    &   ((𝜑𝑙 ∈ ran 𝐺) → (𝑌𝑙) ∈ 𝑙)    &   (𝜑𝐹:(1...𝑀)–1-1-onto→ran 𝐺)    &   (𝜑 → (𝑇𝑈) ⊆ 𝑋)    &   𝑡𝜑    &   𝑤𝜑    &   𝑄       (𝜑 → ∃𝑞(𝑀 ∈ ℕ ∧ (𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞𝑖)‘𝑡))))
 
Theoremstoweidlem28 38921* There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on 𝑇𝑈. Here 𝑑 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑈    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑃 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)0 < (𝑃𝑡))    &   (𝜑𝑈𝐽)       (𝜑 → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇𝑈)𝑑 ≤ (𝑃𝑡)))
 
Theoremstoweidlem29 38922* When the hypothesis for the extreme value theorem hold, then the inf of the range of the function belongs to the range, it is real and it a lower bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.) (Revised by AV, 13-Sep-2020.)
𝑡𝐹    &   𝑡𝜑    &   𝑇 = 𝐽    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑𝑇 ≠ ∅)       (𝜑 → (inf(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ inf(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡𝑇 inf(ran 𝐹, ℝ, < ) ≤ (𝐹𝑡)))
 
Theoremstoweidlem30 38923* This lemma is used to prove the existence of a function p as in Lemma 1 [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺𝑖) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑃 = (𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑀)⟶𝑄)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       ((𝜑𝑆𝑇) → (𝑃𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑆)))
 
Theoremstoweidlem31 38924* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that 𝑅 is a finite subset of 𝑉, 𝑥 indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all 𝑖 ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on 𝐵. Here M is used to represent m in the paper, 𝐸 is used to represent ε in the paper, vi is used to represent V(ti). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝜑    &   𝑡𝜑    &   𝑤𝜑    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}    &   𝐺 = (𝑤𝑅 ↦ {𝐴 ∣ (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − (𝐸 / 𝑀)) < (𝑡))})    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑣:(1...𝑀)–1-1-onto𝑅)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐵 ⊆ (𝑇𝑈))    &   (𝜑𝑉 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑 → ran 𝐺 ∈ Fin)       (𝜑 → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥𝑖)‘𝑡))))
 
Theoremstoweidlem32 38925* If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑃 = (𝑡𝑇 ↦ (𝑌 · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))    &   𝐹 = (𝑡𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡))    &   𝐻 = (𝑡𝑇𝑌)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑌 ∈ ℝ)    &   (𝜑𝐺:(1...𝑀)⟶𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       (𝜑𝑃𝐴)
 
Theoremstoweidlem33 38926* If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝐺    &   𝑡𝜑    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) − (𝐺𝑡))) ∈ 𝐴)
 
Theoremstoweidlem34 38927* This lemma proves that for all 𝑡 in 𝑇 there is a 𝑗 as in the proof of [BrosowskiDeutsh] p. 91 (at the bottom of page 91 and at the top of page 92): (j-4/3) * ε < f(t) <= (j-1/3) * ε , g(t) < (j+1/3) * ε, and g(t) > (j-4/3) * ε. Here 𝐸 is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑗𝜑    &   𝑡𝜑    &   𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})    &   𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})    &   𝐽 = (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑁) ∣ 𝑡 ∈ (𝐷𝑗)})    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑇 ∈ V)    &   (𝜑𝐹:𝑇⟶ℝ)    &   ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))    &   ((𝜑𝑡𝑇) → (𝐹𝑡) < ((𝑁 − 1) · 𝐸))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))    &   ((𝜑𝑗 ∈ (0...𝑁)) → (𝑋𝑗):𝑇⟶ℝ)    &   ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝑡𝑇) → 0 ≤ ((𝑋𝑗)‘𝑡))    &   ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝑡𝑇) → ((𝑋𝑗)‘𝑡) ≤ 1)    &   ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝑡 ∈ (𝐷𝑗)) → ((𝑋𝑗)‘𝑡) < (𝐸 / 𝑁))    &   ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝑡 ∈ (𝐵𝑗)) → (1 − (𝐸 / 𝑁)) < ((𝑋𝑗)‘𝑡))       (𝜑 → ∀𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋𝑖)‘𝑡)))‘𝑡))))
 
Theoremstoweidlem35 38928* 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(t_0) = 0, and p > 0 on T - U. Here (𝑞𝑖) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑤𝜑    &   𝜑    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}    &   𝐺 = (𝑤𝑋 ↦ {𝑄𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}})    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑋 ∈ Fin)    &   (𝜑𝑋𝑊)    &   (𝜑 → (𝑇𝑈) ⊆ 𝑋)    &   (𝜑 → (𝑇𝑈) ≠ ∅)       (𝜑 → ∃𝑚𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞𝑖)‘𝑡))))
 
Theoremstoweidlem36 38929* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function p in the subalgebra, such that pt ( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Z is used for t0 , S is used for t e. T - U , h is used for pt . G is used for (ht)^2 and the final h is a normalized version of G ( divided by its norm, see the variable N ). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄    &   𝑡𝐻    &   𝑡𝐹    &   𝑡𝐺    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑇 = 𝐽    &   𝐺 = (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐹𝑡)))    &   𝑁 = sup(ran 𝐺, ℝ, < )    &   𝐻 = (𝑡𝑇 ↦ ((𝐺𝑡) / 𝑁))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝑆𝑇)    &   (𝜑𝑍𝑇)    &   (𝜑𝐹𝐴)    &   (𝜑 → (𝐹𝑆) ≠ (𝐹𝑍))    &   (𝜑 → (𝐹𝑍) = 0)       (𝜑 → ∃(𝑄 ∧ 0 < (𝑆)))
 
Theoremstoweidlem37 38930* 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(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺𝑖) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑃 = (𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑀)⟶𝑄)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   (𝜑𝑍𝑇)       (𝜑 → (𝑃𝑍) = 0)
 
Theoremstoweidlem38 38931* 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(t_0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺𝑖) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑃 = (𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑀)⟶𝑄)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)       ((𝜑𝑆𝑇) → (0 ≤ (𝑃𝑆) ∧ (𝑃𝑆) ≤ 1))
 
Theoremstoweidlem39 38932* This lemma is used to prove that there exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91: assuming that 𝑟 is a finite subset of 𝑊, 𝑥 indexes a finite set of functions in the subalgebra (of the Stone Weierstrass theorem), such that for all i ranging in the finite indexing set, 0 ≤ xi ≤ 1, xi < ε / m on V(ti), and xi > 1 - ε / m on 𝐵. Here 𝐷 is used to represent A in the paper's Lemma 2 (because 𝐴 is used for the subalgebra), 𝑀 is used to represent m in the paper, 𝐸 is used to represent ε, and vi is used to represent V(ti). 𝑊 is just a local definition, used to shorten statements. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝜑    &   𝑡𝜑    &   𝑤𝜑    &   𝑈 = (𝑇𝐵)    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝑊 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}    &   (𝜑𝑟 ∈ (𝒫 𝑊 ∩ Fin))    &   (𝜑𝐷 𝑟)    &   (𝜑𝐷 ≠ ∅)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐵𝑇)    &   (𝜑𝑊 ∈ V)    &   (𝜑𝐴 ∈ V)       (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊𝐷 ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣𝑖)((𝑥𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥𝑖)‘𝑡)))))
 
Theoremstoweidlem40 38933* This lemma proves that qn is in the subalgebra, as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent qn in the paper, N is used to represent n in the paper, and M is used to represent k^n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑃    &   𝑡𝜑    &   𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑𝑀))    &   𝐹 = (𝑡𝑇 ↦ (1 − ((𝑃𝑡)↑𝑁)))    &   𝐺 = (𝑡𝑇 ↦ 1)    &   𝐻 = (𝑡𝑇 ↦ ((𝑃𝑡)↑𝑁))    &   (𝜑𝑃𝐴)    &   (𝜑𝑃:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ)       (𝜑𝑄𝐴)
 
Theoremstoweidlem41 38934* This lemma is used to prove that there exists x as in Lemma 1 of [BrosowskiDeutsh] p. 90: 0 <= x(t) <= 1 for all t in T, x(t) < epsilon for all t in V, x(t) > 1 - epsilon for all t in T \ U. Here we prove the very last step of the proof of Lemma 1: "The result follows from taking x = 1 - qn";. Here 𝐸 is used to represent ε in the paper, and 𝑦 to represent qn in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝑋 = (𝑡𝑇 ↦ (1 − (𝑦𝑡)))    &   𝐹 = (𝑡𝑇 ↦ 1)    &   𝑉𝑇    &   (𝜑𝑦𝐴)    &   (𝜑𝑦:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑤 ∈ ℝ) → (𝑡𝑇𝑤) ∈ 𝐴)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1))    &   (𝜑 → ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡))    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸)       (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑉 (𝑥𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝐸) < (𝑥𝑡)))
 
Theoremstoweidlem42 38935* This lemma is used to prove that 𝑥 built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x > 1 - ε on B. Here 𝑋 is used to represent 𝑥 in the paper, and E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝜑    &   𝑡𝜑    &   𝑡𝑌    &   𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))    &   𝑋 = (seq1(𝑃, 𝑈)‘𝑀)    &   𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))    &   𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑈:(1...𝑀)⟶𝑌)    &   ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈𝑖)‘𝑡))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))    &   ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)    &   (𝜑𝑇 ∈ V)    &   (𝜑𝐵𝑇)       (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))
 
Theoremstoweidlem43 38936* This lemma is used to prove the existence of a function pt as in Lemma 1 of [BrosowskiDeutsh] p. 90 (at the beginning of Lemma 1): for all t in T - U, there exists a function pt in the subalgebra, such that pt( t0 ) = 0 , pt ( t ) > 0, and 0 <= pt <= 1. Hera Z is used for t0 , S is used for t e. T - U , h is used for pt. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑔𝜑    &   𝑡𝜑    &   𝑄    &   𝐾 = (topGen‘ran (,))    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑇 = 𝐽    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))    &   ((𝜑𝑓𝐴𝑙𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑙𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑙𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑙𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑔𝐴 (𝑔𝑟) ≠ (𝑔𝑡))    &   (𝜑𝑈𝐽)    &   (𝜑𝑍𝑈)    &   (𝜑𝑆 ∈ (𝑇𝑈))       (𝜑 → ∃(𝑄 ∧ 0 < (𝑆)))
 
Theoremstoweidlem44 38937* 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(t_0) = 0, and p > 0 on T - U. Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑗𝜑    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑃 = (𝑡𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺𝑖)‘𝑡)))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐺:(1...𝑀)⟶𝑄)    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺𝑗)‘𝑡))    &   𝑇 = 𝐽    &   (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝑍𝑇)       (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
 
Theoremstoweidlem45 38938* This lemma proves that, given an appropriate 𝐾 (in another theorem we prove such a 𝐾 exists), there exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 ( at the top of page 91): 0 <= qn <= 1 , qn < ε on T \ U, and qn > 1 - ε on 𝑉. We use y to represent the final qn in the paper (the one with n large enough), 𝑁 to represent 𝑛 in the paper, 𝐾 to represent 𝑘, 𝐷 to represent δ, 𝐸 to represent ε, and 𝑃 to represent 𝑝. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑃    &   𝑡𝜑    &   𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}    &   𝑄 = (𝑡𝑇 ↦ ((1 − ((𝑃𝑡)↑𝑁))↑(𝐾𝑁)))    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐷 < 1)    &   (𝜑𝑃𝐴)    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁)))    &   (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸)       (𝜑 → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸))
 
Theoremstoweidlem46 38939* This lemma proves that sets U(t) as defined in Lemma 1 of [BrosowskiDeutsh] p. 90, are a cover of T \ U. Using this lemma, in a later theorem we will prove that a finite subcover exists. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑈    &   𝑄    &   𝑞𝜑    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}    &   𝑇 = 𝐽    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴 ⊆ (𝐽 Cn 𝐾))    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝑈𝐽)    &   (𝜑𝑍𝑈)    &   (𝜑𝑇 ∈ V)       (𝜑 → (𝑇𝑈) ⊆ 𝑊)
 
Theoremstoweidlem47 38940* Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝑆    &   𝑡𝜑    &   𝑇 = 𝐽    &   𝐺 = (𝑇 × {-𝑆})    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Top)    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐹𝐶)    &   (𝜑𝑆 ∈ ℝ)       (𝜑 → (𝑡𝑇 ↦ ((𝐹𝑡) − 𝑆)) ∈ 𝐶)
 
Theoremstoweidlem48 38941* This lemma is used to prove that 𝑥 built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x < ε on 𝐴. Here 𝑋 is used to represent 𝑥 in the paper, 𝐸 is used to represent ε in the paper, and 𝐷 is used to represent 𝐴 in the paper (because 𝐴 is always used to represent the subalgebra). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝜑    &   𝑡𝜑    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))    &   𝑋 = (seq1(𝑃, 𝑈)‘𝑀)    &   𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))    &   𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑊:(1...𝑀)⟶𝑉)    &   (𝜑𝑈:(1...𝑀)⟶𝑌)    &   (𝜑𝐷 ran 𝑊)    &   (𝜑𝐷𝑇)    &   ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < 𝐸)    &   (𝜑𝑇 ∈ V)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∀𝑡𝐷 (𝑋𝑡) < 𝐸)
 
Theoremstoweidlem49 38942* There exists a function qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 (at the top of page 91): 0 <= qn <= 1 , qn < ε on 𝑇𝑈, and qn > 1 - ε on 𝑉. Here y is used to represent the final qn in the paper (the one with n large enough), 𝑁 represents 𝑛 in the paper, 𝐾 represents 𝑘, 𝐷 represents δ, 𝐸 represents ε, and 𝑃 represents 𝑝. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑃    &   𝑡𝜑    &   𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐷 < 1)    &   (𝜑𝑃𝐴)    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑦𝐴 (∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1) ∧ ∀𝑡𝑉 (1 − 𝐸) < (𝑦𝑡) ∧ ∀𝑡 ∈ (𝑇𝑈)(𝑦𝑡) < 𝐸))
 
Theoremstoweidlem50 38943* 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.)
𝑡𝑈    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝑈𝐽)    &   (𝜑𝑍𝑈)       (𝜑 → ∃𝑢(𝑢 ∈ Fin ∧ 𝑢𝑊 ∧ (𝑇𝑈) ⊆ 𝑢))
 
Theoremstoweidlem51 38944* 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 ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝜑    &   𝑡𝜑    &   𝑤𝜑    &   𝑤𝑉    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))    &   𝑋 = (seq1(𝑃, 𝑈)‘𝑀)    &   𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))    &   𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑊:(1...𝑀)⟶𝑉)    &   (𝜑𝑈:(1...𝑀)⟶𝑌)    &   ((𝜑𝑤𝑉) → 𝑤𝑇)    &   (𝜑𝐷 ran 𝑊)    &   (𝜑𝐷𝑇)    &   (𝜑𝐵𝑇)    &   ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))    &   ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈𝑖)‘𝑡))    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   (𝜑𝑇 ∈ V)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))))
 
Theoremstoweidlem52 38945* There exists a neighborood 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.)
𝑡𝑈    &   𝑡𝜑    &   𝑡𝑃    &   𝐾 = (topGen‘ran (,))    &   𝑉 = {𝑡𝑇 ∣ (𝑃𝑡) < (𝐷 / 2)}    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐷 < 1)    &   (𝜑𝑈𝐽)    &   (𝜑𝑍𝑈)    &   (𝜑𝑃𝐴)    &   (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑃𝑡) ∧ (𝑃𝑡) ≤ 1))    &   (𝜑 → (𝑃𝑍) = 0)    &   (𝜑 → ∀𝑡 ∈ (𝑇𝑈)𝐷 ≤ (𝑃𝑡))       (𝜑 → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
 
Theoremstoweidlem53 38946* 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(t_0) = 0, and p > 0 on T - U. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑈    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝑈𝐽)    &   (𝜑 → (𝑇𝑈) ≠ ∅)    &   (𝜑𝑍𝑈)       (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
 
Theoremstoweidlem54 38947* 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 ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝜑    &   𝑡𝜑    &   𝑦𝜑    &   𝑤𝜑    &   𝑇 = 𝐽    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))    &   𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑦𝑖)‘𝑡)))    &   𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))    &   𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑊:(1...𝑀)⟶𝑉)    &   (𝜑𝐵𝑇)    &   (𝜑𝐷 ran 𝑊)    &   (𝜑𝐷𝑇)    &   (𝜑 → ∃𝑦(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))))    &   (𝜑𝑇 ∈ V)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
 
Theoremstoweidlem55 38948* 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(t_0) = 0, and p > 0 on T - U. Here Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑈    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝑈𝐽)    &   (𝜑𝑍𝑈)    &   𝑄 = {𝐴 ∣ ((𝑍) = 0 ∧ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1))}    &   𝑊 = {𝑤𝐽 ∣ ∃𝑄 𝑤 = {𝑡𝑇 ∣ 0 < (𝑡)}}       (𝜑 → ∃𝑝𝐴 (∀𝑡𝑇 (0 ≤ (𝑝𝑡) ∧ (𝑝𝑡) ≤ 1) ∧ (𝑝𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇𝑈)0 < (𝑝𝑡)))
 
Theoremstoweidlem56 38949* 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 ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝑈    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝑈𝐽)    &   (𝜑𝑍𝑈)       (𝜑 → ∃𝑣𝐽 ((𝑍𝑣𝑣𝑈) ∧ ∀𝑒 ∈ ℝ+𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝑣 (𝑥𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑥𝑡))))
 
Theoremstoweidlem57 38950* 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.)
𝑡𝐷    &   𝑡𝑈    &   𝑡𝜑    &   𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}    &   𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   𝑈 = (𝑇𝐵)    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝐵 ∈ (Clsd‘𝐽))    &   (𝜑𝐷 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝐵𝐷) = ∅)    &   (𝜑𝐷 ≠ ∅)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
 
Theoremstoweidlem58 38951* 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.)
𝑡𝐷    &   𝑡𝑈    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑎 ∈ ℝ) → (𝑡𝑇𝑎) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝐵 ∈ (Clsd‘𝐽))    &   (𝜑𝐷 ∈ (Clsd‘𝐽))    &   (𝜑 → (𝐵𝐷) = ∅)    &   𝑈 = (𝑇𝐵)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
 
Theoremstoweidlem59 38952* This lemma proves that there exists a function 𝑥 as in the proof in [BrosowskiDeutsh] p. 91, after Lemma 2: xj is in the subalgebra, 0 <= xj <= 1, xj < ε / n on Aj (meaning A in the paper), xj > 1 - \epslon / 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 ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})    &   𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})    &   𝑌 = {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}    &   𝐻 = (𝑗 ∈ (0...𝑁) ↦ {𝑦𝑌 ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < (𝑦𝑡))})    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝐹𝐶)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → ∃𝑥(𝑥:(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥𝑗)‘𝑡))))
 
Theoremstoweidlem60 38953* 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 𝑡 in 𝑇, there is a 𝑗 such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here 𝐹 is used to represent f in the paper, and 𝐸 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   𝐷 = (𝑗 ∈ (0...𝑛) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})    &   𝐵 = (𝑗 ∈ (0...𝑛) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝑇 ≠ ∅)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝐹𝐶)    &   (𝜑 → ∀𝑡𝑇 0 ≤ (𝐹𝑡))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))))
 
Theoremstoweidlem61 38954* 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 𝑡 in 𝑇, abs( f(t) - g(t) ) < 2*ε. Here 𝐹 is used to represent f in the paper, and 𝐸 is used to represent ε. 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.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   (𝜑𝑇 ≠ ∅)    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝐹𝐶)    &   (𝜑 → ∀𝑡𝑇 0 ≤ (𝐹𝑡))    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ∃𝑔𝐴𝑡𝑇 (abs‘((𝑔𝑡) − (𝐹𝑡))) < (2 · 𝐸))
 
Theoremstoweidlem62 38955* 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.)
𝑡𝐹    &   𝑓𝜑    &   𝑡𝜑    &   𝐻 = (𝑡𝑇 ↦ ((𝐹𝑡) − inf(ran 𝐹, ℝ, < )))    &   𝐾 = (topGen‘ran (,))    &   𝑇 = 𝐽    &   (𝜑𝐽 ∈ Comp)    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))    &   (𝜑𝐹𝐶)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑𝑇 ≠ ∅)    &   (𝜑𝐸 < (1 / 3))       (𝜑 → ∃𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸)
 
Theoremstoweid 38956* This theorem proves the Stone-Weierstrass theorem for real valued functions: let 𝐽 be a compact topology on 𝑇, and 𝐶 be the set of real continuous functions on 𝑇. Assume that 𝐴 is a subalgebra of 𝐶 (closed under addition and multiplication of functions) containing constant functions and discriminating points (if 𝑟 and 𝑡 are distinct points in 𝑇, then there exists a function 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 ε value). As a classical example, given any a, b reals, the closed interval 𝑇 = [𝑎, 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 [𝑎, 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.)
𝑡𝐹    &   𝑡𝜑    &   𝐾 = (topGen‘ran (,))    &   (𝜑𝐽 ∈ Comp)    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   (𝜑𝐴𝐶)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝐴 (𝑟) ≠ (𝑡))    &   (𝜑𝐹𝐶)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸)
 
Theoremstowei 38957* This theorem proves the Stone-Weierstrass theorem for real valued functions: let 𝐽 be a compact topology on 𝑇, and 𝐶 be the set of real continuous functions on 𝑇. Assume that 𝐴 is a subalgebra of 𝐶 (closed under addition and multiplication of functions) containing constant functions and discriminating points (if 𝑟 and 𝑡 are distinct points in 𝑇, then there exists a function 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 ε value). As a classical example, given any a, b reals, the closed interval 𝑇 = [𝑎, 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 [𝑎, 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 38956: often times it will be better to use stoweid 38956 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐾 = (topGen‘ran (,))    &   𝐽 ∈ Comp    &   𝑇 = 𝐽    &   𝐶 = (𝐽 Cn 𝐾)    &   𝐴𝐶    &   ((𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)    &   ((𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   (𝑥 ∈ ℝ → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝑟𝑇𝑡𝑇𝑟𝑡) → ∃𝐴 (𝑟) ≠ (𝑡))    &   𝐹𝐶    &   𝐸 ∈ ℝ+       𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸
 
21.31.13  Wallis' product for π
 
Theoremwallispilem1 38958* 𝐼 is monotone: increasing the exponent, the integral decreases. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼𝑁))
 
Theoremwallispilem2 38959* 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 38960* 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 38961* 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 38962* 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 38963* 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 38964 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 38965 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 38966 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)
 
21.31.14  Stirling's approximation formula for ` n ` factorial
 
Theoremstirlinglem1 38967 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 38968 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 38969 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 38970* 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 38971* 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 38972* 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 38973* 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 38974 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 38975* ((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 38976* 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 38977* 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 38978* 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 38979* 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 38980* 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 38981* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 38982 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 38982 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 38983 Stirling's approximation formula for 𝑛 factorial: here convergence is expressed with respect to the standard topology on the reals. The main theorem stirling 38982 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
 
21.31.15  Dirichlet kernel
 
Theoremdirkerval 38984* 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 38985 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 38986 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 38987* 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 38988* 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 38989* 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 38990* 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 38991* 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 38992* 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 38993* 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 38994* 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 38995* 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 38996* 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 38997* 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 38998* 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 38999* 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 39000* 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))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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