P. 409 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 40801-40900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hoidmvval0 40801*	The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑗𝜑    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → ∃𝑗 ∈ 𝑋 (𝐵‘𝑗) ≤ (𝐴‘𝑗))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = 0)
Theorem	hoiprodp1 40802*	The dimensional volume of a half-open interval with dimension 𝑛 + 1. Used in the first equality of step (e) of Lemma 115B of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌)    &   ⊢ 𝑋 = (𝑌 ∪ {𝑍})    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐺 = ∏𝑘 ∈ 𝑌 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = (𝐺 · (vol‘((𝐴‘𝑍)[,)(𝐵‘𝑍)))))
Theorem	sge0hsphoire 40803*	If the generalized sum of dimensional volumes of n-dimensional half-open intervals is finite, then the sum stays finite if every half-open interval is intersected with a half-space. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑌 ∈ Fin)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑊 ∖ 𝑌))    &   ⊢ 𝑊 = (𝑌 ∪ {𝑍})    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))    &   ⊢ (𝜑 → 𝑆 ∈ ℝ)    ⇒   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑆)‘(𝐷‘𝑗))))) ∈ ℝ)
Theorem	hoidmvval0b 40804*	The dimensional volume of the (half-open interval) empty set. Definition 115A (c) of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐴) = 0)
Theorem	hoidmv1lelem1 40805*	The supremum of 𝑈 belongs to 𝑈. This is the last part of step (a) and the whole step (b) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)    &   ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)    &   ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}    &   ⊢ 𝑆 = sup(𝑈, ℝ, < )    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝑈 ∧ 𝐴 ∈ 𝑈 ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑈 𝑦 ≤ 𝑥))
Theorem	hoidmv1lelem2 40806*	This is the contradiction proven in step (c) in the proof of Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)    &   ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)    &   ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ≤ 𝑆)    &   ⊢ (𝜑 → 𝑆 < 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑆 ∈ ((𝐶‘𝐾)[,)(𝐷‘𝐾)))    &   ⊢ 𝑀 = if((𝐷‘𝐾) ≤ 𝐵, (𝐷‘𝐾), 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
Theorem	hoidmv1lelem3 40807*	The dimensional volume of a 1-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is the non-empty, finite generalized sum, sub case in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐶:ℕ⟶ℝ)    &   ⊢ (𝜑 → 𝐷:ℕ⟶ℝ)    &   ⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ∪ 𝑗 ∈ ℕ ((𝐶‘𝑗)[,)(𝐷‘𝑗)))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))) ∈ ℝ)    &   ⊢ 𝑈 = {𝑧 ∈ (𝐴[,]𝐵) ∣ (𝑧 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)if((𝐷‘𝑗) ≤ 𝑧, (𝐷‘𝑗), 𝑧)))))}    &   ⊢ 𝑆 = sup(𝑈, ℝ, < )    ⇒   ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (vol‘((𝐶‘𝑗)[,)(𝐷‘𝑗))))))
Theorem	hoidmv1le 40808*	The dimensional volume of a 1-dimensional half-open interval is less than or equal to the generalized sum of the dimensional volumes of countable half-open intervals that cover it. This is one of the two base cases of the induction of Lemma 115B of [Fremlin1] p. 29 (the other base case is the 0-dimensional case). This proof of the 1-dimensional case is given in Lemma 114B of [Fremlin1] p. 23. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ 𝑋 = {𝑍}    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
Theorem	hoidmvlelem1 40809*	The supremum of 𝑈 belongs to 𝑈. Step (c) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))    &   ⊢ 𝑊 = (𝑌 ∪ {𝑍})    &   ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))    &   ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}    &   ⊢ 𝑆 = sup(𝑈, ℝ, < )    &   ⊢ (𝜑 → (𝐴‘𝑍) < (𝐵‘𝑍))    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝑈)
Theorem	hoidmvlelem2 40810*	This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))    &   ⊢ 𝑊 = (𝑌 ∪ {𝑍})    &   ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)    &   ⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))    &   ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))    &   ⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑀)(𝑃‘𝑗)))    &   ⊢ 𝑂 = ran (𝑖 ∈ {𝑗 ∈ (1...𝑀) ∣ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))} ↦ ((𝐷‘𝑖)‘𝑍))    &   ⊢ 𝑉 = ({(𝐵‘𝑍)} ∪ 𝑂)    &   ⊢ 𝑄 = inf(𝑉, ℝ, < )    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
Theorem	hoidmvlelem3 40811*	This is the contradiction proven in step (d) in the proof of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))    &   ⊢ 𝑊 = (𝑌 ∪ {𝑍})    &   ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))    &   ⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))    &   ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}    &   ⊢ (𝜑 → 𝑆 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 < (𝐵‘𝑍))    &   ⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))    &   ⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))    &   ⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    &   ⊢ 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)))    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)
Theorem	hoidmvlelem4 40812*	The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29, case nonempty interval and dimension of the space greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))    &   ⊢ 𝑊 = (𝑌 ∪ {𝑍})    &   ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘))    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))))    &   ⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))}    &   ⊢ 𝑆 = sup(𝑈, ℝ, < )    &   ⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))    &   ⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝐸) · (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))
Theorem	hoidmvlelem5 40813*	The dimensional volume of a multidimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Induction step of Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌))    &   ⊢ 𝑊 = (𝑌 ∪ {𝑍})    &   ⊢ (𝜑 → 𝐴:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑊⟶ℝ)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑊))    &   ⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))    &   ⊢ (𝜑 → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))
Theorem	hoidmvle 40814*	The dimensional volume of a n-dimensional half-open interval is less than or equal the generalized sum of the dimensional volumes of countable half-open intervals that cover it. Lemma 115B of [Fremlin1] p. 29. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
Theorem	ovnhoilem1 40815*	The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. First part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    &   ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}    &   ⊢ 𝐻 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
Theorem	ovnhoilem2 40816*	The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. Second part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}    &   ⊢ 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))))    &   ⊢ 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
Theorem	ovnhoi 40817*	The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	dmovn 40818	The domain of the Lebesgue outer measure is the power set of the n-dimensional Real numbers. Step (a)(i) of the proof of Proposition 115D (a) of [Fremlin1] p. 30 (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → dom (voln*‘𝑋) = 𝒫 (ℝ ↑𝑚 𝑋))
Theorem	hoicoto2 40819*	The half-open interval expressed using a composition of a function into (ℝ × ℝ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ))    &   ⊢ 𝐴 = (𝑘 ∈ 𝑋 ↦ (1st ‘(𝐼‘𝑘)))    &   ⊢ 𝐵 = (𝑘 ∈ 𝑋 ↦ (2nd ‘(𝐼‘𝑘)))    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
Theorem	dmvon 40820	Lebesgue measurable n-dimensional subsets of Reals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋)))
Theorem	hoi2toco 40821*	The half-open interval expressed using a composition of a function into (ℝ × ℝ) and using two distinct real-valued functions for the borders. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐼 = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
Theorem	hoidifhspval 40822*	𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐷‘𝑌) = (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑌), (𝑎‘𝑘)))))
Theorem	hspval 40823*	The value of the half-space of n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑖 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑖, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐼(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐼, (-∞(,)𝑌), ℝ))
Theorem	ovnlecvr2 40824*	Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))
Theorem	ovncvr2 40825*	𝐵 and 𝑇 are the left and right side of a cover of 𝐴. This cover is made of n-dimensional half open intervals, and approximates the n-dimensional Lebesgue outer volume of 𝐴. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})    &   ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))    &   ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))    &   ⊢ (𝜑 → 𝐼 ∈ ((𝐷‘𝐴)‘𝐸))    &   ⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))    &   ⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))    ⇒   ⊢ (𝜑 → (((𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋)) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
Theorem	dmovnsal 40826	The domain of the Lebesgue measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    ⇒   ⊢ (𝜑 → 𝑆 ∈ SAlg)
Theorem	unidmovn 40827	Base set of the n-dimensional Lebesgue outer measure (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = (ℝ ↑𝑚 𝑋))
Theorem	rrnmbl 40828	The set of n-dimensional Real numbers is Lebesgue measurable. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → (ℝ ↑𝑚 𝑋) ∈ dom (voln‘𝑋))
Theorem	hoidifhspval2 40829*	𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴) = (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑌 ≤ (𝐴‘𝑘), (𝐴‘𝑘), 𝑌), (𝐴‘𝑘))))
Theorem	hspdifhsp 40830*	A n-dimensional half-open interval is the intersection of the difference of half spaces. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) = ∩ 𝑖 ∈ 𝑋 ((𝑖(𝐻‘𝑋)(𝐵‘𝑖)) ∖ (𝑖(𝐻‘𝑋)(𝐴‘𝑖))))
Theorem	unidmvon 40831	Base set of the n-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    ⇒   ⊢ (𝜑 → ∪ 𝑆 = (ℝ ↑𝑚 𝑋))
Theorem	hoidifhspf 40832*	𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → ((𝐷‘𝑌)‘𝐴):𝑋⟶ℝ)
Theorem	hoidifhspval3 40833*	𝐷 is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑋) ↦ (𝑘 ∈ 𝑋 ↦ if(𝑘 = 𝐾, if(𝑥 ≤ (𝑎‘𝑘), (𝑎‘𝑘), 𝑥), (𝑎‘𝑘)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)‘𝐽) = if(𝐽 = 𝐾, if(𝑌 ≤ (𝐴‘𝐽), (𝐴‘𝐽), 𝑌), (𝐴‘𝐽)))
Theorem	hoidifhspdmvle 40834*	The dimensional volume of the difference of a half-open interval and a half-space is less than or equal to the dimensional volume of the whole half-open interval. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ 𝐷 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (((𝐷‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵) ≤ (𝐴(𝐿‘𝑋)𝐵))
Theorem	voncmpl 40835	The Lebesgue measure is complete. See Definition 112Df of [Fremlin1] p. 19. This is an observation written after Definition 115E of [Fremlin1] p. 31 (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐸 ∈ dom (voln‘𝑋))    &   ⊢ (𝜑 → ((voln‘𝑋)‘𝐸) = 0)    &   ⊢ (𝜑 → 𝐹 ⊆ 𝐸)    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑆)
Theorem	hoiqssbllem1 40836*	The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑖𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐶:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(#‘𝑋)))))(,)(𝑌‘𝑖)))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(#‘𝑋)))))))    ⇒   ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))
Theorem	hoiqssbllem2 40837*	The center of the n-dimensional ball belongs to the half-open interval. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑖𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐶:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐷:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐶‘𝑖) ∈ (((𝑌‘𝑖) − (𝐸 / (2 · (√‘(#‘𝑋)))))(,)(𝑌‘𝑖)))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐷‘𝑖) ∈ ((𝑌‘𝑖)(,)((𝑌‘𝑖) + (𝐸 / (2 · (√‘(#‘𝑋)))))))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸))
Theorem	hoiqssbllem3 40838*	A n-dimensional ball contains a non-empty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑𝑚 𝑋)∃𝑑 ∈ (ℚ ↑𝑚 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))
Theorem	hoiqssbl 40839*	A n-dimensional ball contains a non-empty half-open interval with vertices with rational components. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ (ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ (ℚ ↑𝑚 𝑋)∃𝑑 ∈ (ℚ ↑𝑚 𝑋)(𝑌 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑌(ball‘(dist‘(ℝ^‘𝑋)))𝐸)))
Theorem	hspmbllem1 40840*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. This is Step (a) of Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))    &   ⊢ 𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))    ⇒   ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) = ((𝐴(𝐿‘𝑋)((𝑇‘𝑌)‘𝐵)) +𝑒 (((𝑆‘𝑌)‘𝐴)(𝐿‘𝑋)𝐵)))
Theorem	hspmbllem2 40841*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. This is Step (b) of Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))    &   ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸))    &   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)    &   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ)    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    &   ⊢ 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))    &   ⊢ 𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ)))))    ⇒   ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸))
Theorem	hspmbllem3 40842*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. This proof handles the non-trivial cases (nonzero dimension and finite outer measure) (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    &   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})    &   ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))    &   ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))    &   ⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑘))))    &   ⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑘))))    ⇒   ⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) +𝑒 ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ ((voln*‘𝑋)‘𝐴))
Theorem	hspmbl 40843*	Any half-space of the n-dimensional Real numbers is Lebesgue measurable. Lemma 115F of [Fremlin1] p. 31. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈ 𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ)))    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) ∈ dom (voln‘𝑋))
Theorem	hoimbllem 40844*	Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑖 ∈ 𝑥 if(𝑖 = 𝑙, (-∞(,)𝑦), ℝ)))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆)
Theorem	hoimbl 40845*	Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,)(𝐵‘𝑖)) ∈ 𝑆)
Theorem	opnvonmbllem1 40846*	The half-open interval expressed using a composition of a function (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ Ⅎ𝑖𝜑    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶:𝑋⟶ℚ)    &   ⊢ (𝜑 → 𝐷:𝑋⟶ℚ)    &   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)) ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ X𝑖 ∈ 𝑋 ((𝐶‘𝑖)[,)(𝐷‘𝑖)))    &   ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑𝑚 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺}    &   ⊢ 𝐻 = (𝑖 ∈ 𝑋 ↦ ⟨(𝐶‘𝑖), (𝐷‘𝑖)⟩)    ⇒   ⊢ (𝜑 → ∃ℎ ∈ 𝐾 𝑌 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
Theorem	opnvonmbllem2 40847*	An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))    &   ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑𝑚 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺}    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝑆)
Theorem	opnvonmbl 40848	An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ 𝑆)
Theorem	opnssborel 40849	Open sets of a generalized real Euclidean space are Borel sets (notice that this theorem is even more general, because 𝑋 is not required to be a set). (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ 𝐴 = (TopOpen‘(ℝ^‘𝑋))    &   ⊢ 𝐵 = (SalGen‘𝐴)    ⇒   ⊢ 𝐴 ⊆ 𝐵
Theorem	borelmbl 40850	All Borel subsets of the n-dimensional Real numbers are Lebesgue measurable. This is Proposition 115G (b) of [Fremlin1] p. 32. See also Definition 111G (d) of [Fremlin1] p. 13. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ 𝐵 = (SalGen‘(TopOpen‘(ℝ^‘𝑋)))    ⇒   ⊢ (𝜑 → 𝐵 ⊆ 𝑆)
Theorem	volicorege0 40851	The Lebesgue measure of a left-closed right-open interval with real bounds, is a nonnegative real number. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) ∈ (0[,)+∞))
Theorem	isvonmbl 40852*	The predicate "𝐴 is measurable w.r.t. the n-dimensional Lebesgue measure". A set is measurable if it splits every other set 𝑥 in a "nice" way, that is, if the measure of the pieces 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴 sum up to the measure of 𝑥. Definition 114E of [Fremlin1] p. 25. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → (𝐸 ∈ dom (voln‘𝑋) ↔ (𝐸 ⊆ (ℝ ↑𝑚 𝑋) ∧ ∀𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎))))
Theorem	mblvon 40853	The n-dimensional Lebesgue measure of a measurable set is the same as its n-dimensional Lebesgue outer measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ dom (voln‘𝑋))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) = ((voln*‘𝑋)‘𝐴))
Theorem	vonmblss 40854	n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    ⇒   ⊢ (𝜑 → dom (voln‘𝑋) ⊆ 𝒫 (ℝ ↑𝑚 𝑋))
Theorem	volico2 40855	The measure of left closed, right open interval. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (vol‘(𝐴[,)𝐵)) = if(𝐴 ≤ 𝐵, (𝐵 − 𝐴), 0))
Theorem	vonmblss2 40856	n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑌 ∈ dom (voln‘𝑋))    ⇒   ⊢ (𝜑 → 𝑌 ⊆ (ℝ ↑𝑚 𝑋))
Theorem	ovolval2lem 40857*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))    ⇒   ⊢ (𝜑 → ran seq1( + , ((abs ∘ − ) ∘ 𝐹)) = ran (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ (1...𝑛)(vol‘(([,) ∘ 𝐹)‘𝑘))))
Theorem	ovolval2 40858*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^. See ovolval 23242 for an alternative expression. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((abs ∘ − ) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovnsubadd2lem 40859*	(voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑𝑚 𝑋))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ if(𝑛 = 1, 𝐴, if(𝑛 = 2, 𝐵, ∅)))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
Theorem	ovnsubadd2 40860	(voln*‘𝑋) is subadditive. Proposition 115D (a)(iv) of [Fremlin1] p. 31 . The special case of the union of 2 sets. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))    &   ⊢ (𝜑 → 𝐵 ⊆ (ℝ ↑𝑚 𝑋))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∪ 𝐵)) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 ((voln*‘𝑋)‘𝐵)))
Theorem	ovolval3 40861*	The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^ and vol ∘ (,). See ovolval 23242 and ovolval2 40858 for alternative expressions. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovnsplit 40862	The n-dimensional Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))    ⇒   ⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (((voln*‘𝑋)‘(𝐴 ∩ 𝐵)) +𝑒 ((voln*‘𝑋)‘(𝐴 ∖ 𝐵))))
Theorem	ovolval4lem1 40863*	|- ( ( ph /\ n e. A ) -> ( ( (,) o. G ) 𝑛) = (((,) ∘ 𝐹) n ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝐹‘𝑛)), if((1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛)), (2nd ‘(𝐹‘𝑛)), (1st ‘(𝐹‘𝑛)))⟩)    &   ⊢ 𝐴 = {𝑛 ∈ ℕ ∣ (1st ‘(𝐹‘𝑛)) ≤ (2nd ‘(𝐹‘𝑛))}    ⇒   ⊢ (𝜑 → (∪ ran ((,) ∘ 𝐹) = ∪ ran ((,) ∘ 𝐺) ∧ (vol ∘ ((,) ∘ 𝐹)) = (vol ∘ ((,) ∘ 𝐺))))
Theorem	ovolval4lem2 40864*	The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 40861, but here 𝑓 is may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨(1st ‘(𝑓‘𝑛)), if((1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛)), (1st ‘(𝑓‘𝑛)))⟩)    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovolval4 40865*	The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 40861, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovolval5lem1 40866*	|- ( ph -> ( sum^ (𝑛 ∈ ℕ ↦ (vol ( ( A - ( W / ( 2 ^ n ) ) ) (,) B ) ) ) ) <_ ( ( sum^ (𝑛 ∈ ℕ ↦ (vol ( A [,) B ) ) ) ) +e W ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑊 ∈ ℝ+)    &   ⊢ 𝐶 = {𝑛 ∈ ℕ ∣ 𝐴 < 𝐵}    ⇒   ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘((𝐴 − (𝑊 / (2↑𝑛)))(,)𝐵)))) ≤ ((Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐴[,)𝐵)))) +𝑒 𝑊))
Theorem	ovolval5lem2 40867*	|- ( ( ph /\ n e. NN ) -> <. ( ( 1st (𝐹 n ) ) - ( W / ( 2 ^ n ) ) ) , ( 2nd (𝐹 n ) ) >. e. ( RR X. RR ) ) (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    &   ⊢ (𝜑 → 𝑌 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))    &   ⊢ 𝑍 = (Σ^‘((vol ∘ (,)) ∘ 𝐺))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))    &   ⊢ (𝜑 → 𝐴 ⊆ ∪ ran ([,) ∘ 𝐹))    &   ⊢ (𝜑 → 𝑊 ∈ ℝ+)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − (𝑊 / (2↑𝑛))), (2nd ‘(𝐹‘𝑛))⟩)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑄 𝑧 ≤ (𝑌 +𝑒 𝑊))
Theorem	ovolval5lem3 40868*	The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}    &   ⊢ 𝑄 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))}    ⇒   ⊢ inf(𝑄, ℝ*, < ) = inf(𝑀, ℝ*, < )
Theorem	ovolval5 40869*	The value of the Lebesgue outer measure for subsets of the reals, using covers of left-closed right-open intervals are used, instead of open intervals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < ))
Theorem	ovnovollem1 40870*	if 𝐹 is a cover of 𝐵 in ℝ, then 𝐼 is the corresponding cover in the space of 1-dimensional reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ ((ℝ × ℝ) ↑𝑚 ℕ))    &   ⊢ 𝐼 = (𝑗 ∈ ℕ ↦ {⟨𝐴, (𝐹‘𝑗)⟩})    &   ⊢ (𝜑 → 𝐵 ⊆ ∪ ran ([,) ∘ 𝐹))    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝐹)))    ⇒   ⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵 ↑𝑚 {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
Theorem	ovnovollem2 40871*	if 𝐼 is a cover of (𝐵 ↑𝑚 {𝐴}) in ℝ^1, then 𝐹 is the corresponding cover in the reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ))    &   ⊢ (𝜑 → (𝐵 ↑𝑚 {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝐼‘𝑗))‘𝑘))    &   ⊢ (𝜑 → 𝑍 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))    &   ⊢ 𝐹 = (𝑗 ∈ ℕ ↦ ((𝐼‘𝑗)‘𝐴))    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑍 = (Σ^‘((vol ∘ [,)) ∘ 𝑓))))
Theorem	ovnovollem3 40872*	The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑𝑚 {𝐴}) ↑𝑚 ℕ)((𝐵 ↑𝑚 {𝐴}) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝐴} (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ {𝐴} (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}    &   ⊢ 𝑁 = {𝑧 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐵 ⊆ ∪ ran ([,) ∘ 𝑓) ∧ 𝑧 = (Σ^‘((vol ∘ [,)) ∘ 𝑓)))}    ⇒   ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑𝑚 {𝐴})) = (vol*‘𝐵))
Theorem	ovnovol 40873	The 1-dimensional Lebesgue outer measure agrees with the Lebesgue outer measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝐵 ↑𝑚 {𝐴})) = (vol*‘𝐵))
Theorem	vonvolmbllem 40874*	If a subset 𝐵 of real numbers is Lebesgue measurable, then its corresponding 1-dimensional set is measurable w.r.t. the n-dimensional Lebesgue measure, (with 𝑛 equal to 1). (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))))    &   ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑𝑚 {𝐴}))    &   ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓    ⇒   ⊢ (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑𝑚 {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑𝑚 {𝐴})))) = ((voln*‘{𝐴})‘𝑋))
Theorem	vonvolmbl 40875	A subset of Real numbers is Lebesgue measurable if and only if its corresponding 1-dimensional set is measurable w.r.t. the 1-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ((𝐵 ↑𝑚 {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))
Theorem	vonvol 40876	The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ dom vol)    ⇒   ⊢ (𝜑 → ((voln‘{𝐴})‘(𝐵 ↑𝑚 {𝐴})) = (vol‘𝐵))
Theorem	vonvolmbl2 40877*	A subset 𝑋 of the space of 1-dimensional Real numbers is Lebesgue measurable if and only if its projection 𝑌 on the Real numbers is Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑓𝑌    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑𝑚 {𝐴}))    &   ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓    ⇒   ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol))
Theorem	vonvol2 40878*	The 1-dimensional Lebesgue measure agrees with the Lebesgue measure on subsets of Real numbers. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ Ⅎ𝑓𝑌    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ dom (voln‘{𝐴}))    &   ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓    ⇒   ⊢ (𝜑 → ((voln‘{𝐴})‘𝑋) = (vol‘𝑌))
Theorem	hoimbl2 40879*	Any n-dimensional half-open interval is Lebesgue measurable. This is a substep of Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → X𝑘 ∈ 𝑋 (𝐴[,)𝐵) ∈ 𝑆)
Theorem	voncl 40880	The Lebesgue measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) ∈ (0[,]+∞))
Theorem	vonhoi 40881*	The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). A direct consequence of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	vonxrcl 40882	The Lebesgue measure of a set is an extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐴) ∈ ℝ*)
Theorem	ioosshoi 40883	A n-dimensional open interval is a subset of the half-open interval with the same bounds. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ X𝑘 ∈ 𝑋 (𝐴[,)𝐵)
Theorem	vonn0hoi 40884*	The Lebesgue outer measure of a multidimensional half-open interval when the dimension of the space is nonzero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
Theorem	von0val 40885	The Lebesgue measure (for the zero dimensional space of reals) of every measurable set is zero. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ dom (voln‘∅))    ⇒   ⊢ (𝜑 → ((voln‘∅)‘𝐴) = 0)
Theorem	vonhoire 40886*	The Lebesgue measure of a n-dimensional half-open interval is a real number. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (𝐴[,)𝐵)) ∈ ℝ)
Theorem	iinhoiicclem 40887*	A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))))    ⇒   ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
Theorem	iinhoiicc 40888*	A n-dimensional closed interval expressed as the indexed intersection of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (𝐴[,)(𝐵 + (1 / 𝑛))) = X𝑘 ∈ 𝑋 (𝐴[,]𝐵))
Theorem	iunhoiioolem 40889*	A n-dimensional open interval expressed as the indexed union of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))    &   ⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )    ⇒   ⊢ (𝜑 → 𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
Theorem	iunhoiioo 40890*	A n-dimensional open interval expressed as the indexed union of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) = X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
Theorem	ioovonmbl 40891*	Any n-dimensional open interval is Lebesgue measurable. This is the first statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ*)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ*)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∈ 𝑆)
Theorem	iccvonmbllem 40892*	Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐴‘𝑖) − (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑖 ∈ 𝑋 ↦ ((𝐵‘𝑖) + (1 / 𝑛))))    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆)
Theorem	iccvonmbl 40893*	Any n-dimensional closed interval is Lebesgue measurable. This is the second statement in Proposition 115G (c) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ 𝑆 = dom (voln‘𝑋)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    ⇒   ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐴‘𝑖)[,](𝐵‘𝑖)) ∈ 𝑆)
Theorem	vonioolem1 40894*	The sequence of the measures of the half-open intervals converges to the measure of their union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))    &   ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))    &   ⊢ 𝐸 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )    &   ⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1)    &   ⊢ 𝑍 = (ℤ≥‘𝑁)    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonioolem2 40895*	The n-dimensional Lebesgue measure of open intervals. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonioo 40896*	The n-dimensional Lebesgue measure of an open interval. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	vonicclem1 40897*	The sequence of the measures of the half-open intervals converges to the measure of their intersection. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))    &   ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))    ⇒   ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonicclem2 40898*	The n-dimensional Lebesgue measure of closed intervals. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐵‘𝑘))    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))    &   ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))    &   ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
Theorem	vonicc 40899*	The n-dimensional Lebesgue measure of a closed interval. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)    &   ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)    &   ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))    &   ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))    ⇒   ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
Theorem	snvonmbl 40900	A n-dimensional singleton is Lebesgue measurable. This is the first statement in Proposition 115G (e) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝑋 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ (ℝ ↑𝑚 𝑋))    ⇒   ⊢ (𝜑 → {𝐴} ∈ dom (voln‘𝑋))