P. 436 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43501-43600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ditgeqiooicc 43501*	A function 𝐹 on an open interval, has the same directed integral as its extension 𝐺 on the closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐵](𝐹‘𝑥) d𝑥 = ⨜[𝐴 → 𝐵](𝐺‘𝑥) d𝑥)
Theorem	volge0 43502	The volume of a set is always nonnegative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ dom vol → 0 ≤ (vol‘𝐴))
Theorem	cnbdibl 43503*	A continuous bounded function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ dom vol)    &   ⊢ (𝜑 → (vol‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)
Theorem	snmbl 43504	A singleton is measurable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℝ → {𝐴} ∈ dom vol)
Theorem	ditgeq3d 43505*	Equality theorem for the directed integral. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐷 = 𝐸)    ⇒   ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐷 d𝑥 = ⨜[𝐴 → 𝐵]𝐸 d𝑥)
Theorem	iblempty 43506	The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ∅ ∈ 𝐿1
Theorem	iblsplit 43507*	The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿1)
Theorem	volsn 43508	A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℝ → (vol‘{𝐴}) = 0)
Theorem	itgvol0 43509*	If the domani is negligible, the function is integrable and the integral is 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ∧ ∫𝐴𝐵 d𝑥 = 0))
Theorem	itgcoscmulx 43510*	Exercise: the integral of 𝑥 ↦ cos𝑎𝑥 on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → ∫(𝐵(,)𝐶)(cos‘(𝐴 · 𝑥)) d𝑥 = (((sin‘(𝐴 · 𝐶)) − (sin‘(𝐴 · 𝐵))) / 𝐴))
Theorem	iblsplitf 43511*	A version of iblsplit 43507 using bound-variable hypotheses instead of distinct variable conditions". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿1)
Theorem	ibliooicc 43512*	If a function is integrable on an open interval, it is integrable on the corresponding closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐶) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐶) ∈ 𝐿1)
Theorem	volioc 43513	The measure of a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,]𝐵)) = (𝐵 − 𝐴))
Theorem	iblspltprt 43514*	If a function is integrable on any interval of a partition, then it is integrable on the whole interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑡𝜑    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀...𝑁)) → (𝑃‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))    &   ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑃‘𝑀)[,](𝑃‘𝑁))) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑡 ∈ ((𝑃‘𝑖)[,](𝑃‘(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (𝑡 ∈ ((𝑃‘𝑀)[,](𝑃‘𝑁)) ↦ 𝐴) ∈ 𝐿1)
Theorem	itgsincmulx 43515*	Exercise: the integral of 𝑥 ↦ sin𝑎𝑥 on an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ∫(𝐵(,)𝐶)(sin‘(𝐴 · 𝑥)) d𝑥 = (((cos‘(𝐴 · 𝐵)) − (cos‘(𝐴 · 𝐶))) / 𝐴))
Theorem	itgsubsticclem 43516*	lemma for itgsubsticc 43517. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶)    &   ⊢ 𝐺 = (𝑢 ∈ ℝ ↦ if(𝑢 ∈ (𝐾[,]𝐿), (𝐹‘𝑢), if(𝑢 < 𝐾, (𝐹‘𝐾), (𝐹‘𝐿))))    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐾[,]𝐿)–cn→ℂ))    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    &   ⊢ (𝜑 → 𝐿 ∈ ℝ)    &   ⊢ (𝜑 → 𝐾 ≤ 𝐿)    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)    &   ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)    ⇒   ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
Theorem	itgsubsticc 43517*	Integration by u-substitution. The main difference with respect to itgsubst 25213 is that here we consider the range of 𝐴(𝑥) to be in the closed interval (𝐾[,]𝐿). If 𝐴(𝑥) is a continuous, differentiable function from [𝑋, 𝑌] to (𝑍, 𝑊), whose derivative is continuous and integrable, and 𝐶(𝑢) is a continuous function on (𝑍, 𝑊), then the integral of 𝐶(𝑢) from 𝐾 = 𝐴(𝑋) to 𝐿 = 𝐴(𝑌) is equal to the integral of 𝐶(𝐴(𝑥)) D 𝐴(𝑥) from 𝑋 to 𝑌. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴) ∈ ((𝑋[,]𝑌)–cn→(𝐾[,]𝐿)))    &   ⊢ (𝜑 → (𝑢 ∈ (𝐾[,]𝐿) ↦ 𝐶) ∈ ((𝐾[,]𝐿)–cn→ℂ))    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ (((𝑋(,)𝑌)–cn→ℂ) ∩ 𝐿1))    &   ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑋[,]𝑌) ↦ 𝐴)) = (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵))    &   ⊢ (𝑢 = 𝐴 → 𝐶 = 𝐸)    &   ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐾)    &   ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐿)    &   ⊢ (𝜑 → 𝐾 ∈ ℝ)    &   ⊢ (𝜑 → 𝐿 ∈ ℝ)    ⇒   ⊢ (𝜑 → ⨜[𝐾 → 𝐿]𝐶 d𝑢 = ⨜[𝑋 → 𝑌](𝐸 · 𝐵) d𝑥)
Theorem	itgioocnicc 43518*	The integral of a piecewise continuous function 𝐹 on an open interval is equal to the integral of the continuous function 𝐺, in the corresponding closed interval. 𝐺 is equal to 𝐹 on the open interval, but it is continuous at the two boundaries, also. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴(,)𝐵)) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ dom 𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴))    &   ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵))    &   ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ if(𝑥 = 𝐴, 𝑅, if(𝑥 = 𝐵, 𝐿, (𝐹‘𝑥))))    ⇒   ⊢ (𝜑 → (𝐺 ∈ 𝐿1 ∧ ∫(𝐴[,]𝐵)(𝐺‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥))
Theorem	iblcncfioo 43519	A continuous function 𝐹 on an open interval (𝐴(,)𝐵) with a finite right limit 𝑅 in 𝐴 and a finite left limit 𝐿 in 𝐵 is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ (𝐹 limℂ 𝐴))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)
Theorem	itgspltprt 43520*	The ∫ integral splits on a given partition 𝑃. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1)))    &   ⊢ (𝜑 → 𝑃:(𝑀...𝑁)⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))    &   ⊢ ((𝜑 ∧ 𝑡 ∈ ((𝑃‘𝑀)[,](𝑃‘𝑁))) → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑀..^𝑁)) → (𝑡 ∈ ((𝑃‘𝑖)[,](𝑃‘(𝑖 + 1))) ↦ 𝐴) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → ∫((𝑃‘𝑀)[,](𝑃‘𝑁))𝐴 d𝑡 = Σ𝑖 ∈ (𝑀..^𝑁)∫((𝑃‘𝑖)[,](𝑃‘(𝑖 + 1)))𝐴 d𝑡)
Theorem	itgiccshift 43521*	The integral of a function, 𝐹 stays the same if its closed interval domain is shifted by 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝑇 ∈ ℝ+)    &   ⊢ 𝐺 = (𝑥 ∈ ((𝐴 + 𝑇)[,](𝐵 + 𝑇)) ↦ (𝐹‘(𝑥 − 𝑇)))    ⇒   ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐺‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
Theorem	itgperiod 43522*	The integral of a periodic function, with period 𝑇 stays the same if the domain of integration is shifted. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐹:ℝ⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘(𝑥 + 𝑇)) = (𝐹‘𝑥))    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴[,]𝐵)) ∈ ((𝐴[,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → ∫((𝐴 + 𝑇)[,](𝐵 + 𝑇))(𝐹‘𝑥) d𝑥 = ∫(𝐴[,]𝐵)(𝐹‘𝑥) d𝑥)
Theorem	itgsbtaddcnst 43523*	Integral substitution, adding a constant to the function's argument. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → ⨜[(𝐴 − 𝑋) → (𝐵 − 𝑋)](𝐹‘(𝑋 + 𝑠)) d𝑠 = ⨜[𝐴 → 𝐵](𝐹‘𝑡) d𝑡)
Theorem	volico 43524	The measure of left-closed, right-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 < 𝐵, (𝐵 − 𝐴), 0))
Theorem	sublevolico 43525	The Lebesgue measure of a left-closed, right-open interval is greater than or equal to the difference of the two bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol‘(𝐴[,)𝐵)))
Theorem	dmvolss 43526	Lebesgue measurable sets are subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ dom vol ⊆ 𝒫 ℝ
Theorem	ismbl3 43527*	The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl2 24691, 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 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴))) ≤ (vol*‘𝑥)))
Theorem	volioof 43528	The function that assigns the Lebesgue measure to open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (vol ∘ (,)):(ℝ* × ℝ*)⟶(0[,]+∞)
Theorem	ovolsplit 43529	The Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts, using addition for extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵))))
Theorem	fvvolioof 43530	The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))
Theorem	volioore 43531	The measure of an open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴(,)𝐵)) = if(𝐴 ≤ 𝐵, (𝐵 − 𝐴), 0))
Theorem	fvvolicof 43532	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.)
⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))
Theorem	voliooico 43533	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol‘(𝐴(,)𝐵)) = (vol‘(𝐴[,)𝐵)))
Theorem	ismbl4 43534*	The predicate "𝐴 is Lebesgue-measurable". Similar to ismbl 24690, 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*‘(𝑥 ∩ 𝐴)) +𝑒 (vol*‘(𝑥 ∖ 𝐴)))))
Theorem	volioofmpt 43535*	((vol ∘ (,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*))    ⇒   ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (vol‘((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))))))
Theorem	volicoff 43536	((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶(ℝ × ℝ*))    ⇒   ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹):𝐴⟶(0[,]+∞))
Theorem	voliooicof 43537	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.)
⊢ (𝜑 → 𝐹:𝐴⟶(ℝ × ℝ))    ⇒   ⊢ (𝜑 → ((vol ∘ (,)) ∘ 𝐹) = ((vol ∘ [,)) ∘ 𝐹))
Theorem	volicofmpt 43538*	((vol ∘ [,)) ∘ 𝐹) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ × ℝ*))    ⇒   ⊢ (𝜑 → ((vol ∘ [,)) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ (vol‘((1st ‘(𝐹‘𝑥))[,)(2nd ‘(𝐹‘𝑥))))))
Theorem	volicc 43539	The Lebesgue measure of a closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴[,]𝐵)) = (𝐵 − 𝐴))
Theorem	voliccico 43540	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.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) = (vol‘(𝐴[,)𝐵)))
Theorem	mbfdmssre 43541	The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ)
20.37.12  Stone Weierstrass theorem - real version
Theorem	stoweidlem1 43542	Lemma for stoweid 43604. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90; the key step uses Bernoulli's inequality bernneq 13944. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≤ 1)    &   ⊢ (𝜑 → 𝐷 ≤ 𝐴)    ⇒   ⊢ (𝜑 → ((1 − (𝐴↑𝑁))↑(𝐾↑𝑁)) ≤ (1 / ((𝐾 · 𝐷)↑𝑁)))
Theorem	stoweidlem2 43543*	lemma for stoweid 43604: here we prove that the subalgebra of continuous functions, which contains constant functions, is closed under scaling. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡𝜑    &   ⊢ ((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 → 𝐸 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ (𝐸 · (𝐹‘𝑡))) ∈ 𝐴)
Theorem	stoweidlem3 43544*	Lemma for stoweid 43604: 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.)
⊢ Ⅎ𝑖𝐹    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ 𝑋 = seq1( · , 𝐹)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹:(1...𝑀)⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐴 < (𝐹‘𝑖))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (𝐴↑𝑀) < (𝑋‘𝑀))
Theorem	stoweidlem4 43545*	Lemma for stoweid 43604: a class variable replaces a setvar variable, for constant functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐵 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝐵) ∈ 𝐴)
Theorem	stoweidlem5 43546*	There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on 𝑇 ∖ 𝑈. Here 𝐷 is used to represent δ in the paper and 𝑄 to represent 𝑇 ∖ 𝑈 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡𝜑    &   ⊢ 𝐷 = if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2))    &   ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)    &   ⊢ (𝜑 → 𝑄 ⊆ 𝑇)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐶 ≤ (𝑃‘𝑡))    ⇒   ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡)))
Theorem	stoweidlem6 43547*	Lemma for stoweid 43604: two class variables replace two setvar variables, for multiplication of two functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡 𝑓 = 𝐹    &   ⊢ Ⅎ𝑡 𝑔 = 𝐺    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)    ⇒   ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
Theorem	stoweidlem7 43548*	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 𝑇 ∖ 𝑈, 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 ↦ (𝐵↑𝑖))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 1 < 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 < 1)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ ((1 − 𝐸) < (1 − (𝐵↑𝑛)) ∧ (1 / (𝐴↑𝑛)) < 𝐸))
Theorem	stoweidlem8 43549*	Lemma for stoweid 43604: 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 43550*	Lemma for stoweid 43604: 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 43551	Lemma for stoweid 43604. 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 43552*	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 43553*	Lemma for stoweid 43604. 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 43554	Lemma for stoweid 43604. 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 43555*	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 43556*	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_(t0) = 0, and p > 0 on T - U. Here (𝐺‘𝐼) is used to represent p_(ti) 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 43557*	Lemma for stoweid 43604. 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 43558*	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 43559*	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 43560*	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 43561*	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 43562*	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 43563*	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 43564*	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 43565*	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 43566*	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 43567*	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 43568*	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_(t0) = 0, and p > 0 on T - U. Here (𝑞‘𝑖) is used to represent p_(ti) 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 43569*	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 43570*	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 43571*	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_(t0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺‘𝑖) is used for p_(ti). (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 43572*	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 43573*	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 43574*	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 43575*	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 43576*	This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Here (𝑞‘𝑖) is used to represent p_(ti) 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 43577*	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 43578*	This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺‘𝑖) is used for p_(ti). (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 43579*	This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Z is used for t0, P is used for p, (𝐺‘𝑖) is used for p_(ti). (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 43580*	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 43581*	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 43582*	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 43583*	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 43584*	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 43585*	This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and 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 43586*	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 43587*	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 43588*	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 43589*	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 43590*	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 43591*	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 43592*	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 43593*	There exists a neighborhood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t0 in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ 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 43594*	This lemma is used to prove the existence of a function 𝑝 as in Lemma 1 of [BrosowskiDeutsh] p. 90: 𝑝 is in the subalgebra, such that 0 ≤ 𝑝 ≤ 1, p_(t0) = 0, and 0 < 𝑝 on 𝑇 ∖ 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 43595*	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 43596*	This lemma proves the existence of a function p as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t0) = 0, and p > 0 on T - U. Here Z is used to represent t0 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ 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 43597*	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 43598*	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 43599*	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 43600*	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 - \epsilon / n on Bj. Here 𝐷 is used to represent A in the paper (because A is used for the subalgebra of functions), 𝐸 is used to represent 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))))