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Type | Label | Description |
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Statement | ||
Theorem | stoweidlem8 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.) |
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝐺 ⇒ ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐺‘𝑡))) ∈ 𝐴) | ||
Theorem | stoweidlem9 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.) |
⊢ (𝜑 → 𝑇 = ∅) & ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < 𝐸) | ||
Theorem | stoweidlem10 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 ∈ ℕ0 ∧ 𝐴 ≤ 1) → (1 − (𝑁 · 𝐴)) ≤ ((1 − 𝐴)↑𝑁)) | ||
Theorem | stoweidlem11 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.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑡 ∈ 𝑇) & ⊢ (𝜑 → 𝑗 ∈ (1...𝑁)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝑋‘𝑖):𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ≤ 1) & ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → ((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸)) | ||
Theorem | stoweidlem12 38905* | Lemma for stoweid 38956. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) & ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) = ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) | ||
Theorem | stoweidlem13 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.) |
⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝑋 ∈ ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝑗 ∈ ℝ) & ⊢ (𝜑 → ((𝑗 − (4 / 3)) · 𝐸) < 𝑋) & ⊢ (𝜑 → 𝑋 ≤ ((𝑗 − (1 / 3)) · 𝐸)) & ⊢ (𝜑 → ((𝑗 − (4 / 3)) · 𝐸) < 𝑌) & ⊢ (𝜑 → 𝑌 < ((𝑗 + (1 / 3)) · 𝐸)) ⇒ ⊢ (𝜑 → (abs‘(𝑌 − 𝑋)) < (2 · 𝐸)) | ||
Theorem | stoweidlem14 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} & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 < 1) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)) | ||
Theorem | stoweidlem15 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 (𝐺‘𝐼) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ (((𝜑 ∧ 𝐼 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → (((𝐺‘𝐼)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝐼)‘𝑆) ∧ ((𝐺‘𝐼)‘𝑆) ≤ 1)) | ||
Theorem | stoweidlem16 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.) |
⊢ Ⅎ𝑡𝜑 & ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} & ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → 𝐻 ∈ 𝑌) | ||
Theorem | stoweidlem17 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.) |
⊢ Ⅎ𝑡𝜑 & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑋:(0...𝑁)⟶𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝐸 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) ∈ 𝐴) | ||
Theorem | stoweidlem18 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.) |
⊢ Ⅎ𝑡𝐷 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 = ∅) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) | ||
Theorem | stoweidlem19 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝜑 & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴) | ||
Theorem | stoweidlem20 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.) |
⊢ Ⅎ𝑡𝜑 & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ (𝜑 → 𝐹 ∈ 𝐴) | ||
Theorem | stoweidlem21 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.) |
⊢ Ⅎ𝑡𝐺 & ⊢ Ⅎ𝑡𝐻 & ⊢ Ⅎ𝑡𝑆 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐻‘𝑡) + 𝑆)) & ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) & ⊢ (𝜑 → 𝑆 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → ∀𝑓 ∈ 𝐴 𝑓:𝑇⟶ℝ) & ⊢ (𝜑 → 𝐻 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (abs‘((𝐻‘𝑡) − ((𝐹‘𝑡) − 𝑆))) < 𝐸) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸) | ||
Theorem | stoweidlem22 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.) |
⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝐺 & ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) & ⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1) & ⊢ 𝐿 = (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴) | ||
Theorem | stoweidlem23 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.) |
⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑡𝐺 & ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) − (𝐺‘𝑍))) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝑇) & ⊢ (𝜑 → 𝑍 ∈ 𝑇) & ⊢ (𝜑 → 𝐺 ∈ 𝐴) & ⊢ (𝜑 → (𝐺‘𝑆) ≠ (𝐺‘𝑍)) ⇒ ⊢ (𝜑 → (𝐻 ∈ 𝐴 ∧ (𝐻‘𝑆) ≠ (𝐻‘𝑍) ∧ (𝐻‘𝑍) = 0)) | ||
Theorem | stoweidlem24 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 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} & ⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) & ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁))) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑄‘𝑡)) | ||
Theorem | stoweidlem25 38918* | This lemma proves that for n sufficiently large, qn( t ) < ε, for all 𝑡 in 𝑇 ∖ 𝑈: 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 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) & ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸) ⇒ ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑄‘𝑡) < 𝐸) | ||
Theorem | stoweidlem26 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}) & ⊢ 𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)}) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑆 ∈ ((𝐷‘𝐿) ∖ (𝐷‘(𝐿 − 1)))) & ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝑋‘𝑖):𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑋‘𝑖)‘𝑡)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁) ∧ 𝑡 ∈ (𝐵‘𝑖)) → (1 − (𝐸 / 𝑁)) < ((𝑋‘𝑖)‘𝑡)) ⇒ ⊢ (𝜑 → ((𝐿 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑆)) | ||
Theorem | stoweidlem27 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 (𝑞‘𝑖) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ 𝐺 = (𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) & ⊢ (𝜑 → 𝑄 ∈ V) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑌 Fn ran 𝐺) & ⊢ (𝜑 → ran 𝐺 ∈ V) & ⊢ ((𝜑 ∧ 𝑙 ∈ ran 𝐺) → (𝑌‘𝑙) ∈ 𝑙) & ⊢ (𝜑 → 𝐹:(1...𝑀)–1-1-onto→ran 𝐺) & ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑋) & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎℎ𝑄 ⇒ ⊢ (𝜑 → ∃𝑞(𝑀 ∈ ℕ ∧ (𝑞:(1...𝑀)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑀)0 < ((𝑞‘𝑖)‘𝑡)))) | ||
Theorem | stoweidlem28 38921* | There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on 𝑇 ∖ 𝑈. Here 𝑑 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) ⇒ ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡))) | ||
Theorem | stoweidlem29 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑇 ≠ ∅) ⇒ ⊢ (𝜑 → (inf(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ inf(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡 ∈ 𝑇 inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑡))) | ||
Theorem | stoweidlem30 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, (𝐺‘𝑖) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))) | ||
Theorem | stoweidlem31 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.) |
⊢ Ⅎℎ𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} & ⊢ 𝑉 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))} & ⊢ 𝐺 = (𝑤 ∈ 𝑅 ↦ {ℎ ∈ 𝐴 ∣ (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − (𝐸 / 𝑀)) < (ℎ‘𝑡))}) & ⊢ (𝜑 → 𝑅 ⊆ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑣:(1...𝑀)–1-1-onto→𝑅) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ⊆ (𝑇 ∖ 𝑈)) & ⊢ (𝜑 → 𝑉 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → ran 𝐺 ∈ Fin) ⇒ ⊢ (𝜑 → ∃𝑥(𝑥:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑥‘𝑖)‘𝑡)))) | ||
Theorem | stoweidlem32 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.) |
⊢ Ⅎ𝑡𝜑 & ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ (𝑌 · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)) & ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ 𝑌) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ (𝜑 → 𝑃 ∈ 𝐴) | ||
Theorem | stoweidlem33 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝐺 & ⊢ Ⅎ𝑡𝜑 & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) ⇒ ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴) | ||
Theorem | stoweidlem34 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}) & ⊢ 𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)}) & ⊢ 𝐽 = (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑁) ∣ 𝑡 ∈ (𝐷‘𝑗)}) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡)) & ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < ((𝑁 − 1) · 𝐸)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑋‘𝑗):𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑋‘𝑗)‘𝑡)) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑡 ∈ 𝑇) → ((𝑋‘𝑗)‘𝑡) ≤ 1) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ((𝑋‘𝑗)‘𝑡) < (𝐸 / 𝑁)) & ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝑡 ∈ (𝐵‘𝑗)) → (1 − (𝐸 / 𝑁)) < ((𝑋‘𝑗)‘𝑡)) ⇒ ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡)))) | ||
Theorem | stoweidlem35 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 (𝑞‘𝑖) is used to represent p(t_i) in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎℎ𝜑 & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} & ⊢ 𝐺 = (𝑤 ∈ 𝑋 ↦ {ℎ ∈ 𝑄 ∣ 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝑋 ∈ Fin) & ⊢ (𝜑 → 𝑋 ⊆ 𝑊) & ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑋) & ⊢ (𝜑 → (𝑇 ∖ 𝑈) ≠ ∅) ⇒ ⊢ (𝜑 → ∃𝑚∃𝑞(𝑚 ∈ ℕ ∧ (𝑞:(1...𝑚)⟶𝑄 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑖 ∈ (1...𝑚)0 < ((𝑞‘𝑖)‘𝑡)))) | ||
Theorem | stoweidlem36 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.) |
⊢ Ⅎℎ𝑄 & ⊢ Ⅎ𝑡𝐻 & ⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝐺 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐹‘𝑡))) & ⊢ 𝑁 = sup(ran 𝐺, ℝ, < ) & ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐺‘𝑡) / 𝑁)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝑆 ∈ 𝑇) & ⊢ (𝜑 → 𝑍 ∈ 𝑇) & ⊢ (𝜑 → 𝐹 ∈ 𝐴) & ⊢ (𝜑 → (𝐹‘𝑆) ≠ (𝐹‘𝑍)) & ⊢ (𝜑 → (𝐹‘𝑍) = 0) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))) | ||
Theorem | stoweidlem37 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, (𝐺‘𝑖) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ (𝜑 → 𝑍 ∈ 𝑇) ⇒ ⊢ (𝜑 → (𝑃‘𝑍) = 0) | ||
Theorem | stoweidlem38 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, (𝐺‘𝑖) is used for p(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ⇒ ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (0 ≤ (𝑃‘𝑆) ∧ (𝑃‘𝑆) ≤ 1)) | ||
Theorem | stoweidlem39 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.) |
⊢ Ⅎℎ𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ 𝑈 = (𝑇 ∖ 𝐵) & ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} & ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))} & ⊢ (𝜑 → 𝑟 ∈ (𝒫 𝑊 ∩ Fin)) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑟) & ⊢ (𝜑 → 𝐷 ≠ ∅) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ⊆ 𝑇) & ⊢ (𝜑 → 𝑊 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) ⇒ ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑊 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑥(𝑥:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑥‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑥‘𝑖)‘𝑡))))) | ||
Theorem | stoweidlem40 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.) |
⊢ Ⅎ𝑡𝑃 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑𝑀)) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) & ⊢ 𝐺 = (𝑡 ∈ 𝑇 ↦ 1) & ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) ⇒ ⊢ (𝜑 → 𝑄 ∈ 𝐴) | ||
Theorem | stoweidlem41 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.) |
⊢ Ⅎ𝑡𝜑 & ⊢ 𝑋 = (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡))) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1) & ⊢ 𝑉 ⊆ 𝑇 & ⊢ (𝜑 → 𝑦 ∈ 𝐴) & ⊢ (𝜑 → 𝑦:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑤) ∈ 𝐴) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡)) & ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡))) | ||
Theorem | stoweidlem42 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.) |
⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑡𝑌 & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) & ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝐵 ⊆ 𝑇) ⇒ ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)) | ||
Theorem | stoweidlem43 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.) |
⊢ Ⅎ𝑔𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎℎ𝑄 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑇 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑙 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑙‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑙 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑙‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑔 ∈ 𝐴 (𝑔‘𝑟) ≠ (𝑔‘𝑡)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝑆 ∈ (𝑇 ∖ 𝑈)) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ ∈ 𝑄 ∧ 0 < (ℎ‘𝑆))) | ||
Theorem | stoweidlem44 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.) |
⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄) & ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)∃𝑗 ∈ (1...𝑀)0 < ((𝐺‘𝑗)‘𝑡)) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝑍 ∈ 𝑇) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))) | ||
Theorem | stoweidlem45 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.) |
⊢ Ⅎ𝑡𝑃 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} & ⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ ℕ) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 < 1) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) & ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁))) & ⊢ (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)) | ||
Theorem | stoweidlem46 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.) |
⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎℎ𝑄 & ⊢ Ⅎ𝑞𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} & ⊢ 𝑇 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝑇 ∈ V) ⇒ ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ ∪ 𝑊) | ||
Theorem | stoweidlem47 38940* | Subtracting a constant from a real continuous function gives another continuous function. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝑆 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐺 = (𝑇 × {-𝑆}) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Top) & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐹 ∈ 𝐶) & ⊢ (𝜑 → 𝑆 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − 𝑆)) ∈ 𝐶) | ||
Theorem | stoweidlem48 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.) |
⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) & ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊) & ⊢ (𝜑 → 𝐷 ⊆ 𝑇) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < 𝐸) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸) | ||
Theorem | stoweidlem49 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 𝑇 ∖ 𝑈, 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.) |
⊢ Ⅎ𝑡𝑃 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 < 1) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) & ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)) | ||
Theorem | stoweidlem50 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.) |
⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑢(𝑢 ∈ Fin ∧ 𝑢 ⊆ 𝑊 ∧ (𝑇 ∖ 𝑈) ⊆ ∪ 𝑢)) | ||
Theorem | stoweidlem51 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.) |
⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑤𝑉 & ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))) & ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀)) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉) & ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌) & ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ⊆ 𝑇) & ⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊) & ⊢ (𝜑 → 𝐷 ⊆ 𝑇) & ⊢ (𝜑 → 𝐵 ⊆ 𝑇) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀)) & ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡)) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))) | ||
Theorem | stoweidlem52 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.) |
⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑡𝑃 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ (𝜑 → 𝐷 < 1) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1)) & ⊢ (𝜑 → (𝑃‘𝑍) = 0) & ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡)) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))) | ||
Theorem | stoweidlem53 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.) |
⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → (𝑇 ∖ 𝑈) ≠ ∅) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))) | ||
Theorem | stoweidlem54 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.) |
⊢ Ⅎ𝑖𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} & ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) & ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑦‘𝑖)‘𝑡))) & ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀)) & ⊢ 𝑉 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))} & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉) & ⊢ (𝜑 → 𝐵 ⊆ 𝑇) & ⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊) & ⊢ (𝜑 → 𝐷 ⊆ 𝑇) & ⊢ (𝜑 → ∃𝑦(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) & ⊢ (𝜑 → 𝑇 ∈ V) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) | ||
Theorem | stoweidlem55 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.) |
⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} & ⊢ 𝑊 = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ 𝑄 𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))) | ||
Theorem | stoweidlem56 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.) |
⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝑈 ∈ 𝐽) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))) | ||
Theorem | stoweidlem57 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.) |
⊢ Ⅎ𝑡𝐷 & ⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} & ⊢ 𝑉 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))} & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ 𝑈 = (𝑇 ∖ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) & ⊢ (𝜑 → 𝐷 ≠ ∅) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) | ||
Theorem | stoweidlem58 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.) |
⊢ Ⅎ𝑡𝐷 & ⊢ Ⅎ𝑡𝑈 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽)) & ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅) & ⊢ 𝑈 = (𝑇 ∖ 𝐵) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) | ||
Theorem | stoweidlem59 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ 𝐷 = (𝑗 ∈ (0...𝑁) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}) & ⊢ 𝐵 = (𝑗 ∈ (0...𝑁) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)}) & ⊢ 𝑌 = {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} & ⊢ 𝐻 = (𝑗 ∈ (0...𝑁) ↦ {𝑦 ∈ 𝑌 ∣ (∀𝑡 ∈ (𝐷‘𝑗)(𝑦‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑁)) < (𝑦‘𝑡))}) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝐹 ∈ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → ∃𝑥(𝑥:(0...𝑁)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑁)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑁) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑁)) < ((𝑥‘𝑗)‘𝑡)))) | ||
Theorem | stoweidlem60 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ 𝐷 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)}) & ⊢ 𝐵 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)}) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝑇 ≠ ∅) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝐹 ∈ 𝐶) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐹‘𝑡)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))) | ||
Theorem | stoweidlem61 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ (𝜑 → 𝑇 ≠ ∅) & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝐹 ∈ 𝐶) & ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐹‘𝑡)) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑔‘𝑡) − (𝐹‘𝑡))) < (2 · 𝐸)) | ||
Theorem | stoweidlem62 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑓𝜑 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) & ⊢ (𝜑 → 𝐹 ∈ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) & ⊢ (𝜑 → 𝑇 ≠ ∅) & ⊢ (𝜑 → 𝐸 < (1 / 3)) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸) | ||
Theorem | stoweid 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 ℎ 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.) |
⊢ Ⅎ𝑡𝐹 & ⊢ Ⅎ𝑡𝜑 & ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾) & ⊢ (𝜑 → 𝐴 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃ℎ ∈ 𝐴 (ℎ‘𝑟) ≠ (ℎ‘𝑡)) & ⊢ (𝜑 → 𝐹 ∈ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸) | ||
Theorem | stowei 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 ℎ 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) & ⊢ 𝐴 ⊆ 𝐶 & ⊢ ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) & ⊢ ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) & ⊢ (𝑥 ∈ ℝ → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) & ⊢ ((𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡) → ∃ℎ ∈ 𝐴 (ℎ‘𝑟) ≠ (ℎ‘𝑡)) & ⊢ 𝐹 ∈ 𝐶 & ⊢ 𝐸 ∈ ℝ+ ⇒ ⊢ ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 | ||
Theorem | wallispilem1 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) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐼‘(𝑁 + 1)) ≤ (𝐼‘𝑁)) | ||
Theorem | wallispilem2 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) → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))) | ||
Theorem | wallispilem3 38960* | I maps to real values. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ 𝐼 = (𝑛 ∈ ℕ0 ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐼‘𝑁) ∈ ℝ+) | ||
Theorem | wallispilem4 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 | ||
Theorem | wallispilem5 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 | ||
Theorem | wallispi 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) | ||
Theorem | wallispi2lem1 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))) | ||
Theorem | wallispi2lem2 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))) | ||
Theorem | wallispi2 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) | ||
Theorem | stirlinglem1 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) | ||
Theorem | stirlinglem2 38968 | 𝐴 maps to positive reals. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝐴‘𝑁) ∈ ℝ+) | ||
Theorem | stirlinglem3 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 ↦ ((((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2)) · ((𝑛↑2) / (𝑛 · ((2 · 𝑛) + 1))))) | ||
Theorem | stirlinglem4 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‘(𝐴‘𝑛))) & ⊢ 𝐽 = (𝑛 ∈ ℕ ↦ ((((1 + (2 · 𝑛)) / 2) · (log‘((𝑛 + 1) / 𝑛))) − 1)) ⇒ ⊢ (𝑁 ∈ ℕ → ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1))) = (𝐽‘𝑁)) | ||
Theorem | stirlinglem5 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)) · ((𝑇↑𝑗) / 𝑗))) & ⊢ 𝐸 = (𝑗 ∈ ℕ ↦ ((𝑇↑𝑗) / 𝑗)) & ⊢ 𝐹 = (𝑗 ∈ ℕ ↦ (((-1↑(𝑗 − 1)) · ((𝑇↑𝑗) / 𝑗)) + ((𝑇↑𝑗) / 𝑗))) & ⊢ 𝐻 = (𝑗 ∈ ℕ0 ↦ (2 · ((1 / ((2 · 𝑗) + 1)) · (𝑇↑((2 · 𝑗) + 1))))) & ⊢ 𝐺 = (𝑗 ∈ ℕ0 ↦ ((2 · 𝑗) + 1)) & ⊢ (𝜑 → 𝑇 ∈ ℝ+) & ⊢ (𝜑 → (abs‘𝑇) < 1) ⇒ ⊢ (𝜑 → seq0( + , 𝐻) ⇝ (log‘((1 + 𝑇) / (1 − 𝑇)))) | ||
Theorem | stirlinglem6 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))) | ||
Theorem | stirlinglem7 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) ⇝ (𝐽‘𝑁)) | ||
Theorem | stirlinglem8 38974 | If 𝐴 converges to 𝐶, then 𝐹 converges to C^2 . (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑛𝐴 & ⊢ Ⅎ𝑛𝐷 & ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝐴‘(2 · 𝑛))) & ⊢ (𝜑 → 𝐴:ℕ⟶ℝ+) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2))) & ⊢ 𝐿 = (𝑛 ∈ ℕ ↦ ((𝐴‘𝑛)↑4)) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ((𝐷‘𝑛)↑2)) & ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ ℝ+) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 ⇝ 𝐶) ⇒ ⊢ (𝜑 → 𝐹 ⇝ (𝐶↑2)) | ||
Theorem | stirlinglem9 38975* | ((𝐵‘𝑁) − (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‘(𝐴‘𝑛))) & ⊢ 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) ⇝ ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1)))) | ||
Theorem | stirlinglem10 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‘(𝐴‘𝑛))) & ⊢ 𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘)))) & ⊢ 𝐿 = (𝑘 ∈ ℕ ↦ ((1 / (((2 · 𝑁) + 1)↑2))↑𝑘)) ⇒ ⊢ (𝑁 ∈ ℕ → ((𝐵‘𝑁) − (𝐵‘(𝑁 + 1))) ≤ ((1 / 4) · (1 / (𝑁 · (𝑁 + 1))))) | ||
Theorem | stirlinglem11 38977* | 𝐵 is decreasing. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
⊢ 𝐴 = (𝑛 ∈ ℕ ↦ ((!‘𝑛) / ((√‘(2 · 𝑛)) · ((𝑛 / e)↑𝑛)))) & ⊢ 𝐵 = (𝑛 ∈ ℕ ↦ (log‘(𝐴‘𝑛))) & ⊢ 𝐾 = (𝑘 ∈ ℕ ↦ ((1 / ((2 · 𝑘) + 1)) · ((1 / ((2 · 𝑁) + 1))↑(2 · 𝑘)))) ⇒ ⊢ (𝑁 ∈ ℕ → (𝐵‘(𝑁 + 1)) < (𝐵‘𝑁)) | ||
Theorem | stirlinglem12 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‘(𝐴‘𝑛))) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1)))) ⇒ ⊢ (𝑁 ∈ ℕ → ((𝐵‘1) − (1 / 4)) ≤ (𝐵‘𝑁)) | ||
Theorem | stirlinglem13 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‘(𝐴‘𝑛))) ⇒ ⊢ ∃𝑑 ∈ ℝ 𝐵 ⇝ 𝑑 | ||
Theorem | stirlinglem14 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‘(𝐴‘𝑛))) ⇒ ⊢ ∃𝑐 ∈ ℝ+ 𝐴 ⇝ 𝑐 | ||
Theorem | stirlinglem15 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.) |
⊢ Ⅎ𝑛𝜑 & ⊢ 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 ↦ (((𝐴‘𝑛)↑4) / ((𝐷‘𝑛)↑2))) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝑛↑2) / (𝑛 · ((2 · 𝑛) + 1)))) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) & ⊢ (𝜑 → 𝐴 ⇝ 𝐶) ⇒ ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((!‘𝑛) / (𝑆‘𝑛))) ⇝ 1) | ||
Theorem | stirling 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) / (𝑆‘𝑛))) ⇝ 1 | ||
Theorem | stirlingr 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) / (𝑆‘𝑛)))𝑅1 | ||
Theorem | dirkerval 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 → (𝐷‘𝑁) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))) | ||
Theorem | dirker2re 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) | ||
Theorem | dirkerdenne0 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) | ||
Theorem | dirkerval2 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) → ((𝐷‘𝑁)‘𝑆) = if((𝑆 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2)))))) | ||
Theorem | dirkerre 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) → ((𝐷‘𝑁)‘𝑆) ∈ ℝ) | ||
Theorem | dirkerper 38989* | the Dirichlet Kernel has period 2π. (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) → ((𝐷‘𝑁)‘(𝑥 + 𝑇)) = ((𝐷‘𝑁)‘𝑥)) | ||
Theorem | dirkerf 38990* | For any natural number 𝑁, the Dirichlet Kernel (𝐷‘𝑁) 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 → (𝐷‘𝑁):ℝ⟶ℝ) | ||
Theorem | dirkertrigeqlem1 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) | ||
Theorem | dirkertrigeqlem2 38992* | Trigonomic equality lemma for the Dirichlet Kernel trigonomic equality. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (sin‘𝐴) ≠ 0) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → (((1 / 2) + Σ𝑛 ∈ (1...𝑁)(cos‘(𝑛 · 𝐴))) / π) = ((sin‘((𝑁 + (1 / 2)) · 𝐴)) / ((2 · π) · (sin‘(𝐴 / 2))))) | ||
Theorem | dirkertrigeqlem3 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.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ 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))))) | ||
Theorem | dirkertrigeq 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))))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐹 = (𝐷‘𝑁) & ⊢ 𝐻 = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐻) | ||
Theorem | dirkeritg 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))))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐹 = (𝐷‘𝑁) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) / π)) ⇒ ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ((𝐺‘𝐵) − (𝐺‘𝐴))) | ||
Theorem | dirkercncflem1 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) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → (𝑌 mod (2 · π)) = 0) ⇒ ⊢ (𝜑 → (𝑌 ∈ (𝐴(,)𝐵) ∧ ∀𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})((sin‘(𝑦 / 2)) ≠ 0 ∧ (cos‘(𝑦 / 2)) ≠ 0))) | ||
Theorem | dirkercncflem2 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)))) & ⊢ ((𝜑 ∧ 𝑦 ∈ ((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))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑌 ∈ (𝐴(,)𝐵)) & ⊢ (𝜑 → (𝑌 mod (2 · π)) = 0) & ⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐴(,)𝐵) ∖ {𝑌})) → (cos‘(𝑦 / 2)) ≠ 0) ⇒ ⊢ (𝜑 → ((𝐷‘𝑁)‘𝑌) ∈ (((𝐷‘𝑁) ↾ ((𝐴(,)𝐵) ∖ {𝑌})) limℂ 𝑌)) | ||
Theorem | dirkercncflem3 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)))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → (𝑌 mod (2 · π)) = 0) ⇒ ⊢ (𝜑 → ((𝐷‘𝑁)‘𝑌) ∈ ((𝐷‘𝑁) limℂ 𝑌)) | ||
Theorem | dirkercncflem4 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))))))) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → (𝑌 mod (2 · π)) ≠ 0) & ⊢ 𝐴 = (⌊‘(𝑌 / (2 · π))) & ⊢ 𝐵 = (𝐴 + 1) & ⊢ 𝐶 = (𝐴 · (2 · π)) & ⊢ 𝐸 = (𝐵 · (2 · π)) ⇒ ⊢ (𝜑 → (𝐷‘𝑁) ∈ (((topGen‘ran (,)) CnP (topGen‘ran (,)))‘𝑌)) | ||
Theorem | dirkercncf 39000* | For any natural number 𝑁, the Dirichlet Kernel (𝐷‘𝑁) 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 → (𝐷‘𝑁) ∈ (ℝ–cn→ℝ)) |
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