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Theorem List for Metamath Proof Explorer - 41001-41100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfvvolioof 41001 The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ* × ℝ*))    &   (𝜑𝑋𝐴)       (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))

Theoremvolioore 41002 The measure of an open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) = if(𝐴𝐵, (𝐵𝐴), 0))

Theoremfvvolicof 41003 The function value of the Lebesgue measure of a left-closed right-open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ* × ℝ*))    &   (𝜑𝑋𝐴)       (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))

Theoremvoliooico 41004 An open interval and a left-closed, right-open interval with the same real bounds, have the same Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))

Theoremismbl4 41005* The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 23693, but here +𝑒 is used, and the precondition (vol*‘𝑥) ∈ ℝ can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ(vol*‘𝑥) = ((vol*‘(𝑥𝐴)) +𝑒 (vol*‘(𝑥𝐴)))))

Theoremvolioofmpt 41006* ((vol ∘ (,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))       (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥𝐴 ↦ (vol‘((1st ‘(𝐹𝑥))(,)(2nd ‘(𝐹𝑥))))))

Theoremvolicoff 41007 ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ × ℝ*))       (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞))

Theoremvoliooicof 41008 The Lebesgue measure of open intervals is the same as the Lebesgue measure of left-closed right-open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝐹:𝐴⟶(ℝ × ℝ))       (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹))

Theoremvolicofmpt 41009* ((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝑥𝐹    &   (𝜑𝐹:𝐴⟶(ℝ × ℝ*))       (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑥𝐴 ↦ (vol‘((1st ‘(𝐹𝑥))[,)(2nd ‘(𝐹𝑥))))))

Theoremvolicc 41010 The Lebesgue measure of a closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴𝐵) → (vol‘(𝐴[,]𝐵)) = (𝐵𝐴))

Theoremvoliccico 41011 A closed interval and a left-closed, right-open interval with the same real bounds, have the same Lebesgue measure. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (vol‘(𝐴[,]𝐵)) = (vol‘(𝐴[,)𝐵)))

Theoremmbfdmssre 41012 The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ)

20.33.12  Stone Weierstrass theorem - real version

Theoremstoweidlem1 41013 Lemma for stoweid 41075. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 13285. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝐷 ∈ ℝ+)    &   (𝜑𝐴 ∈ ℝ+)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 ≤ 1)    &   (𝜑𝐷𝐴)       (𝜑 → ((1 − (𝐴𝑁))↑(𝐾𝑁)) ≤ (1 / ((𝐾 · 𝐷)↑𝑁)))

Theoremstoweidlem2 41014* lemma for stoweid 41075: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)    &   ((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)    &   ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)    &   (𝜑𝐸 ∈ ℝ)    &   (𝜑𝐹𝐴)       (𝜑 → (𝑡𝑇 ↦ (𝐸 · (𝐹𝑡))) ∈ 𝐴)

Theoremstoweidlem3 41015* Lemma for stoweid 41075: if 𝐴 is positive and all 𝑀 terms of a finite product are larger than 𝐴, then the finite product is larger than A^M. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑖𝐹    &   𝑖𝜑    &   𝑋 = seq1( · , 𝐹)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝐹:(1...𝑀)⟶ℝ)    &   ((𝜑𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹𝑖))    &   (𝜑𝐴 ∈ ℝ+)       (𝜑 → (𝐴𝑀) < (𝑋𝑀))

Theoremstoweidlem4 41016* Lemma for stoweid 41075: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
((𝜑𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)       ((𝜑𝐵 ∈ ℝ) → (𝑡𝑇𝐵) ∈ 𝐴)

Theoremstoweidlem5 41017* There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on 𝑇𝑈. Here 𝐷 is used to represent δ in the paper and 𝑄 to represent 𝑇𝑈 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡𝜑    &   𝐷 = if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2))    &   (𝜑𝑃:𝑇⟶ℝ)    &   (𝜑𝑄𝑇)    &   (𝜑𝐶 ∈ ℝ+)    &   (𝜑 → ∀𝑡𝑄 𝐶 ≤ (𝑃𝑡))       (𝜑 → ∃𝑑(𝑑 ∈ ℝ+𝑑 < 1 ∧ ∀𝑡𝑄 𝑑 ≤ (𝑃𝑡)))

Theoremstoweidlem6 41018* Lemma for stoweid 41075: two class variables replace two setvar variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝑡 𝑓 = 𝐹    &   𝑡 𝑔 = 𝐺    &   ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)       ((𝜑𝐹𝐴𝐺𝐴) → (𝑡𝑇 ↦ ((𝐹𝑡) · (𝐺𝑡))) ∈ 𝐴)

Theoremstoweidlem7 41019* This lemma is used to prove that qn as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91, (at the top of page 91), is such that qn < ε on 𝑇𝑈, and qn > 1 - ε on 𝑉. Here it is proven that, for 𝑛 large enough, 1-(k*δ/2)^n > 1 - ε , and 1/(k*δ)^n < ε. The variable 𝐴 is used to represent (k*δ) in the paper, and 𝐵 is used to represent (k*δ/2). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
𝐹 = (𝑖 ∈ ℕ0 ↦ ((1 / 𝐴)↑𝑖))    &   𝐺 = (𝑖 ∈ ℕ0 ↦ (𝐵𝑖))    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐵 < 1)    &   (𝜑𝐸 ∈ ℝ+)       (𝜑 → ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (𝐵𝑛)) ∧ (1 / (𝐴𝑛)) < 𝐸))

Theoremstoweidlem8 41020* Lemma for stoweid 41075: 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 41021* Lemma for stoweid 41075: 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 41022 Lemma for stoweid 41075. 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 41023* 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 41024* Lemma for stoweid 41075. 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 41025 Lemma for stoweid 41075. 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 41026* 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 41027* 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 41028* Lemma for stoweid 41075. 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 41029* 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 41030* 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 41031* 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 41032* 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 41033* 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 41034* 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 41035* 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 41036* 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 41037* 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 41038* 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 41039* 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 41040* 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 41041* 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 41042* 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 41043* 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 41044* 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 41045* 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 41046* 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 41047* 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 41048* 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 41049* 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 41050* 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 41051* 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 41052* 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 41053* 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 41054* 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 41055* 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 41056* 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 41057* 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 41058* 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 41059* 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 41060* 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 41061* 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 41062* 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 41063* 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 41064* 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 41065* 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 41066* 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 41067* 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 41068* 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 41069* 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 41070* 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 41071* 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 41072* 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 41073* 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 41074* 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 41075* 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 41076* 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 41075: often times it will be better to use stoweid 41075 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

20.33.13  Wallis' product for π

Theoremwallispilem1 41077* 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 41078* 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 41079* 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 41080* 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 41081* 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 41082* 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 41083 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 41084 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 41085 An alternative version of Wallis' formula for π ; this second formula uses factorials and it is later used to prove Stirling's approximation formula. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
𝑉 = (𝑛 ∈ ℕ ↦ ((((2↑(4 · 𝑛)) · ((!‘𝑛)↑4)) / ((!‘(2 · 𝑛))↑2)) / ((2 · 𝑛) + 1)))       𝑉 ⇝ (π / 2)

20.33.14  Stirling's approximation formula for ` n ` factorial

Theoremstirlinglem1 41086 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 41087 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 41088 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 41089* 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 41090* 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 41091* 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 41092* 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 41093 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 41094* ((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 41095* 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 41096* 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 41097* 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 41098* 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 41099* 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 41100* The Stirling's formula is proven using a number of local definitions. The main theorem stirling 41101 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)

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