P. 261 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mbfmulc2lem 26001	Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn)
Theorem	mbfmulc2re 26002	Multiplication by a constant preserves measurability. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 × {𝐵}) ∘f · 𝐹) ∈ MblFn)
Theorem	mbfmax 26003*	The maximum of two functions is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ if((𝐹‘𝑥) ≤ (𝐺‘𝑥), (𝐺‘𝑥), (𝐹‘𝑥)))    ⇒   ⊢ (𝜑 → 𝐻 ∈ MblFn)
Theorem	mbfneg 26004*	The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn)
Theorem	mbfpos 26005*	The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
Theorem	mbfposr 26006*	Converse to mbfpos 26005. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
Theorem	mbfposb 26007*	A function is measurable iff its positive and negative parts are measurable. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ MblFn)))
Theorem	ismbf3d 26008*	Simplified form of ismbfd 25993. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	mbfimaopnlem 26009*	Lemma for mbfimaopn 26010. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐺 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑥 + (i · 𝑦)))    &   ⊢ 𝐵 = ((,) “ (ℚ × ℚ))    &   ⊢ 𝐾 = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 × 𝑦))    ⇒   ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝐴) ∈ dom vol)
Theorem	mbfimaopn 26010	The preimage of any open set (in the complex topology) under a measurable function is measurable. (See also cncombf 26012, which explains why 𝐴 ∈ dom vol is too weak a condition for this theorem.) (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ ((𝐹 ∈ MblFn ∧ 𝐴 ∈ 𝐽) → (◡𝐹 “ 𝐴) ∈ dom vol)
Theorem	mbfimaopn2 26011	The preimage of any set open in the subspace topology of the range of the function is measurable. (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ 𝐾 = (𝐽 ↾t 𝐵)    ⇒   ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝐶 ∈ 𝐾) → (◡𝐹 “ 𝐶) ∈ dom vol)
Theorem	cncombf 26012	The composition of a continuous function with a measurable function is measurable. (More generally, 𝐺 can be a Borel-measurable function, but notably the condition that 𝐺 be only measurable is too weak, the usual counterexample taking 𝐺 to be the Cantor function and 𝐹 the indicator function of the 𝐺-image of a nonmeasurable set, which is a subset of the Cantor set and hence null and measurable.) (Contributed by Mario Carneiro, 25-Aug-2014.)
⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐺 ∈ (𝐵–cn→ℂ)) → (𝐺 ∘ 𝐹) ∈ MblFn)
Theorem	cnmbf 26013	A continuous function is measurable. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 26-Mar-2015.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐹 ∈ (𝐴–cn→ℂ)) → 𝐹 ∈ MblFn)
Theorem	mbfaddlem 26014	The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ MblFn)
Theorem	mbfadd 26015	The sum of two measurable functions is measurable. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ MblFn)
Theorem	mbfsub 26016	The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ MblFn)
Theorem	mbfmulc2 26017*	A complex constant times a measurable function is measurable. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)
Theorem	mbfsup 26018*	The supremum of a sequence of measurable, real-valued functions is measurable. Note that in this and related theorems, 𝐵(𝑛, 𝑥) is a function of both 𝑛 and 𝑥, since it is an 𝑛-indexed sequence of functions on 𝑥. (Contributed by Mario Carneiro, 14-Aug-2014.) (Revised by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦)    ⇒   ⊢ (𝜑 → 𝐺 ∈ MblFn)
Theorem	mbfinf 26019*	The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 13-Sep-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐺 ∈ MblFn)
Theorem	mbflimsup 26020*	The limit supremum of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)))    &   ⊢ 𝐻 = (𝑚 ∈ ℝ ↦ sup((((𝑛 ∈ 𝑍 ↦ 𝐵) “ (𝑚[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (lim sup‘(𝑛 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐺 ∈ MblFn)
Theorem	mbflimlem 26021*	The pointwise limit of a sequence of measurable real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
Theorem	mbflim 26022*	The pointwise limit of a sequence of measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
Syntax	c0p 26023	Extend class notation to include the zero polynomial.
class 0𝑝
Definition	df-0p 26024	Define the zero polynomial. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ 0𝑝 = (ℂ × {0})
Theorem	0pval 26025	The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.)
⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0)
Theorem	0plef 26026	Two ways to say that the function 𝐹 on the reals is nonnegative. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝐹:ℝ⟶(0[,)+∞) ↔ (𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹))
Theorem	0pledm 26027	Adjust the domain of the left argument to match the right, which works better in our theorems. (Contributed by Mario Carneiro, 28-Jul-2014.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    ⇒   ⊢ (𝜑 → (0𝑝 ∘r ≤ 𝐹 ↔ (𝐴 × {0}) ∘r ≤ 𝐹))
Theorem	isi1f 26028	The predicate "𝐹 is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom 𝐹 ∈ dom ∫1 to represent this concept because ∫1 is the first preparation function for our final definition ∫ (see df-itg 25977); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(◡𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
Theorem	i1fmbf 26029	Simple functions are measurable. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn)
Theorem	i1ff 26030	A simple function is a function on the reals. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ)
Theorem	i1frn 26031	A simple function has finite range. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
Theorem	i1fima 26032	Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol)
Theorem	i1fima2 26033	Any preimage of a simple function not containing zero has finite measure. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ ¬ 0 ∈ 𝐴) → (vol‘(◡𝐹 “ 𝐴)) ∈ ℝ)
Theorem	i1fima2sn 26034	Preimage of a singleton. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ (𝐵 ∖ {0})) → (vol‘(◡𝐹 “ {𝐴})) ∈ ℝ)
Theorem	i1fd 26035*	A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    &   ⊢ (𝜑 → ran 𝐹 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐹 ∈ dom ∫1)
Theorem	i1f0rn 26036	Any simple function takes the value zero on a set of unbounded measure, so in particular this set is not empty. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → 0 ∈ ran 𝐹)
Theorem	itg1val 26037*	The value of the integral on simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥}))))
Theorem	itg1val2 26038*	The value of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ (𝐴 ∈ Fin ∧ (ran 𝐹 ∖ {0}) ⊆ 𝐴 ∧ 𝐴 ⊆ (ℝ ∖ {0}))) → (∫1‘𝐹) = Σ𝑥 ∈ 𝐴 (𝑥 · (vol‘(◡𝐹 “ {𝑥}))))
Theorem	itg1cl 26039	Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝐹 ∈ dom ∫1 → (∫1‘𝐹) ∈ ℝ)
Theorem	itg1ge0 26040	Closure of the integral on positive simple functions. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝐹) → 0 ≤ (∫1‘𝐹))
Theorem	i1f0 26041	The zero function is simple. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (ℝ × {0}) ∈ dom ∫1
Theorem	itg10 26042	The zero function has zero integral. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (∫1‘(ℝ × {0})) = 0
Theorem	i1f1lem 26043*	Lemma for i1f1 26044 and itg11 26045. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0))    ⇒   ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴))
Theorem	i1f1 26044*	Base case simple functions are indicator functions of measurable sets. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0))    ⇒   ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐹 ∈ dom ∫1)
Theorem	itg11 26045*	The integral of an indicator function is the volume of the set. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0))    ⇒   ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (∫1‘𝐹) = (vol‘𝐴))
Theorem	itg1addlem1 26046*	Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
⊢ (𝜑 → 𝐹:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ (◡𝐹 “ {𝑘}))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ dom vol)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ)    ⇒   ⊢ (𝜑 → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵))
Theorem	i1faddlem 26047*	Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℂ) → (◡(𝐹 ∘f + 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ ran 𝐺((◡𝐹 “ {(𝐴 − 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
Theorem	i1fmullem 26048*	Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ (ℂ ∖ {0})) → (◡(𝐹 ∘f · 𝐺) “ {𝐴}) = ∪ 𝑦 ∈ (ran 𝐺 ∖ {0})((◡𝐹 “ {(𝐴 / 𝑦)}) ∩ (◡𝐺 “ {𝑦})))
Theorem	i1fadd 26049	The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    ⇒   ⊢ (𝜑 → (𝐹 ∘f + 𝐺) ∈ dom ∫1)
Theorem	i1fmul 26050	The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ dom ∫1)
Theorem	itg1addlem2 26051*	Lemma for itg1add 26055. The function 𝐼 represents the pieces into which we will break up the domain of the sum. Since it is infinite only when both 𝑖 and 𝑗 are zero, we arbitrarily define it to be zero there to simplify the sums that are manipulated in itg1addlem4 26053 and itg1addlem5 26054. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    &   ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))    ⇒   ⊢ (𝜑 → 𝐼:(ℝ × ℝ)⟶ℝ)
Theorem	itg1addlem3 26052*	Lemma for itg1add 26055. (Contributed by Mario Carneiro, 26-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    &   ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))    ⇒   ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) → (𝐴𝐼𝐵) = (vol‘((◡𝐹 “ {𝐴}) ∩ (◡𝐺 “ {𝐵}))))
Theorem	itg1addlem4 26053*	Lemma for itg1add 26055. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof shortened by SN, 3-Oct-2024.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    &   ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))    &   ⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))    ⇒   ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
Theorem	itg1addlem5 26054*	Lemma for itg1add 26055. (Contributed by Mario Carneiro, 27-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    &   ⊢ 𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))    &   ⊢ 𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))    ⇒   ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = ((∫1‘𝐹) + (∫1‘𝐺)))
Theorem	itg1add 26055	The integral of a sum of simple functions is the sum of the integrals. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    ⇒   ⊢ (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = ((∫1‘𝐹) + (∫1‘𝐺)))
Theorem	i1fmulclem 26056	Decompose the preimage of a constant times a function. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℝ) → (◡((ℝ × {𝐴}) ∘f · 𝐹) “ {𝐵}) = (◡𝐹 “ {(𝐵 / 𝐴)}))
Theorem	i1fmulc 26057	A nonnegative constant times a simple function gives another simple function. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((ℝ × {𝐴}) ∘f · 𝐹) ∈ dom ∫1)
Theorem	itg1mulc 26058	The integral of a constant times a simple function is the constant times the original integral. (Contributed by Mario Carneiro, 25-Jun-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫1‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫1‘𝐹)))
Theorem	i1fres 26059*	The "restriction" of a simple function to a measurable subset is simple. (It's not actually a restriction because it is zero instead of undefined outside 𝐴.) (Contributed by Mario Carneiro, 29-Jun-2014.)
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (𝐹‘𝑥), 0))    ⇒   ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐴 ∈ dom vol) → 𝐺 ∈ dom ∫1)
Theorem	i1fpos 26060*	The positive part of a simple function is simple. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(0 ≤ (𝐹‘𝑥), (𝐹‘𝑥), 0))    ⇒   ⊢ (𝐹 ∈ dom ∫1 → 𝐺 ∈ dom ∫1)
Theorem	i1fposd 26061*	Deduction form of i1fposd 26061. (Contributed by Mario Carneiro, 6-Aug-2014.)
⊢ (𝜑 → (𝑥 ∈ ℝ ↦ 𝐴) ∈ dom ∫1)    ⇒   ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(0 ≤ 𝐴, 𝐴, 0)) ∈ dom ∫1)
Theorem	i1fsub 26062	The difference of two simple functions is a simple function. (Contributed by Mario Carneiro, 6-Aug-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (𝐹 ∘f − 𝐺) ∈ dom ∫1)
Theorem	itg1sub 26063	The integral of a difference of two simple functions. (Contributed by Mario Carneiro, 6-Aug-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (∫1‘(𝐹 ∘f − 𝐺)) = ((∫1‘𝐹) − (∫1‘𝐺)))
Theorem	itg10a 26064*	The integral of a simple function supported on a nullset is zero. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = 0)    ⇒   ⊢ (𝜑 → (∫1‘𝐹) = 0)
Theorem	itg1ge0a 26065*	The integral of an almost positive simple function is positive. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → 0 ≤ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → 0 ≤ (∫1‘𝐹))
Theorem	itg1lea 26066*	Approximate version of itg1le 26067. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫1𝐹 ≤ ∫1𝐺. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (∫1‘𝐹) ≤ (∫1‘𝐺))
Theorem	itg1le 26067	If one simple function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 6-Aug-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1 ∧ 𝐹 ∘r ≤ 𝐺) → (∫1‘𝐹) ≤ (∫1‘𝐺))
Theorem	itg1climres 26068*	Restricting the simple function 𝐹 to the increasing sequence 𝐴(𝑛) of measurable sets whose union is ℝ yields a sequence of simple functions whose integrals approach the integral of 𝐹. (Contributed by Mario Carneiro, 15-Aug-2014.)
⊢ (𝜑 → 𝐴:ℕ⟶dom vol)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ (𝐴‘(𝑛 + 1)))    &   ⊢ (𝜑 → ∪ ran 𝐴 = ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ dom ∫1)    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (𝐴‘𝑛), (𝐹‘𝑥), 0))    ⇒   ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ (∫1‘𝐺)) ⇝ (∫1‘𝐹))
Theorem	mbfi1fseqlem1 26069*	Lemma for mbfi1fseq 26075. (Contributed by Mario Carneiro, 16-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))    ⇒   ⊢ (𝜑 → 𝐽:(ℕ × ℝ)⟶(0[,)+∞))
Theorem	mbfi1fseqlem2 26070*	Lemma for mbfi1fseq 26075. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ ( -𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))    ⇒   ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ ( -𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
Theorem	mbfi1fseqlem3 26071*	Lemma for mbfi1fseq 26075. (Contributed by Mario Carneiro, 16-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ ( -𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐺‘𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
Theorem	mbfi1fseqlem4 26072*	Lemma for mbfi1fseq 26075. This lemma is not as interesting as it is long - it is simply checking that 𝐺 is in fact a sequence of simple functions, by verifying that its range is in (0...𝑛2↑𝑛) / (2↑𝑛) (which is to say, the numbers from 0 to 𝑛 in increments of 1 / (2↑𝑛)), and also that the preimage of each point 𝑘 is measurable, because it is equal to ( -𝑛[,]𝑛) ∩ (◡𝐹 “ (𝑘[,)𝑘 + 1 / (2↑𝑛))) for 𝑘 < 𝑛 and ( -𝑛[,]𝑛) ∩ (◡𝐹 “ (𝑘[,)+∞)) for 𝑘 = 𝑛. (Contributed by Mario Carneiro, 16-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ ( -𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))    ⇒   ⊢ (𝜑 → 𝐺:ℕ⟶dom ∫1)
Theorem	mbfi1fseqlem5 26073*	Lemma for mbfi1fseq 26075. Verify that 𝐺 describes an increasing sequence of positive functions. (Contributed by Mario Carneiro, 16-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ ( -𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (0𝑝 ∘r ≤ (𝐺‘𝐴) ∧ (𝐺‘𝐴) ∘r ≤ (𝐺‘(𝐴 + 1))))
Theorem	mbfi1fseqlem6 26074*	Lemma for mbfi1fseq 26075. Verify that 𝐺 converges pointwise to 𝐹, and wrap up the existential quantifier. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚)))    &   ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ ( -𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
Theorem	mbfi1fseq 26075*	A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function 𝐺 and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
Theorem	mbfi1flimlem 26076*	Lemma for mbfi1flim 26077. (Contributed by Mario Carneiro, 5-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶ℝ)    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
Theorem	mbfi1flim 26077*	Any real measurable function has a sequence of simple functions that converges to it. (Contributed by Mario Carneiro, 5-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑥 ∈ 𝐴 (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)))
Theorem	mbfmullem2 26078*	Lemma for mbfmul 26080. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))    &   ⊢ (𝜑 → 𝑄:ℕ⟶dom ∫1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ ℕ ↦ ((𝑄‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn)
Theorem	mbfmullem 26079	Lemma for mbfmul 26080. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐺:𝐴⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn)
Theorem	mbfmul 26080	The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝐹 ∘f · 𝐺) ∈ MblFn)
Theorem	itg2lcl 26081*	The set of lower sums is a set of extended reals. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}    ⇒   ⊢ 𝐿 ⊆ ℝ*
Theorem	itg2val 26082*	Value of the integral on nonnegative real functions. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}    ⇒   ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))
Theorem	itg2l 26083*	Elementhood in the set 𝐿 of lower sums of the integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}    ⇒   ⊢ (𝐴 ∈ 𝐿 ↔ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝐴 = (∫1‘𝑔)))
Theorem	itg2lr 26084*	Sufficient condition for elementhood in the set 𝐿. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}    ⇒   ⊢ ((𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ∈ 𝐿)
Theorem	xrge0f 26085	A real function is a nonnegative extended real function if all its values are greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 28-Jul-2014.)
⊢ ((𝐹:ℝ⟶ℝ ∧ 0𝑝 ∘r ≤ 𝐹) → 𝐹:ℝ⟶(0[,]+∞))
Theorem	itg2cl 26086	The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) ∈ ℝ*)
Theorem	itg2ub 26087	The integral of a nonnegative real function 𝐹 is an upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐺 ∈ dom ∫1 ∧ 𝐺 ∘r ≤ 𝐹) → (∫1‘𝐺) ≤ (∫2‘𝐹))
Theorem	itg2leub 26088*	Any upper bound on the integrals of all simple functions 𝐺 dominated by 𝐹 is greater than (∫2‘𝐹), the least upper bound. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ*) → ((∫2‘𝐹) ≤ 𝐴 ↔ ∀𝑔 ∈ dom ∫1(𝑔 ∘r ≤ 𝐹 → (∫1‘𝑔) ≤ 𝐴)))
Theorem	itg2ge0 26089	The integral of a nonnegative real function is greater than or equal to zero. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝐹:ℝ⟶(0[,]+∞) → 0 ≤ (∫2‘𝐹))
Theorem	itg2itg1 26090	The integral of a nonnegative simple function using ∫2 is the same as its value under ∫1. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 0𝑝 ∘r ≤ 𝐹) → (∫2‘𝐹) = (∫1‘𝐹))
Theorem	itg20 26091	The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (∫2‘(ℝ × {0})) = 0
Theorem	itg2lecl 26092	If an ∫2 integral is bounded above, then it is real. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐴 ∈ ℝ ∧ (∫2‘𝐹) ≤ 𝐴) → (∫2‘𝐹) ∈ ℝ)
Theorem	itg2le 26093	If one function dominates another, then the integral of the larger is also larger. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ((𝐹:ℝ⟶(0[,]+∞) ∧ 𝐺:ℝ⟶(0[,]+∞) ∧ 𝐹 ∘r ≤ 𝐺) → (∫2‘𝐹) ≤ (∫2‘𝐺))
Theorem	itg2const 26094*	Integral of a constant function. (Contributed by Mario Carneiro, 12-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ (0[,)+∞)) → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) = (𝐵 · (vol‘𝐴)))
Theorem	itg2const2 26095*	When the base set of a constant function has infinite volume, the integral is also infinite and vice-versa. (Contributed by Mario Carneiro, 30-Aug-2014.)
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ ℝ+) → ((vol‘𝐴) ∈ ℝ ↔ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ))
Theorem	itg2seq 26096*	Definitional property of the ∫2 integral: for any function 𝐹 there is a countable sequence 𝑔 of simple functions less than 𝐹 whose integrals converge to the integral of 𝐹. (This theorem is for the most part unnecessary in lieu of itg2i1fseq 26109, but unlike that theorem this one doesn't require 𝐹 to be measurable.) (Contributed by Mario Carneiro, 14-Aug-2014.)
⊢ (𝐹:ℝ⟶(0[,]+∞) → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧ ∀𝑛 ∈ ℕ (𝑔‘𝑛) ∘r ≤ 𝐹 ∧ (∫2‘𝐹) = sup(ran (𝑛 ∈ ℕ ↦ (∫1‘(𝑔‘𝑛))), ℝ*, < )))
Theorem	itg2uba 26097*	Approximate version of itg2ub 26087. If 𝐹 approximately dominates 𝐺, then ∫1𝐺 ≤ ∫2𝐹. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → (∫1‘𝐺) ≤ (∫2‘𝐹))
Theorem	itg2lea 26098*	Approximate version of itg2le 26093. If 𝐹 ≤ 𝐺 for almost all 𝑥, then ∫2𝐹 ≤ ∫2𝐺. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (∫2‘𝐹) ≤ (∫2‘𝐺))
Theorem	itg2eqa 26099*	Approximate equality of integrals. If 𝐹 = 𝐺 for almost all 𝑥, then ∫2𝐹 = ∫2𝐺. (Contributed by Mario Carneiro, 12-Aug-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → (vol*‘𝐴) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → (𝐹‘𝑥) = (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (∫2‘𝐹) = (∫2‘𝐺))
Theorem	itg2mulclem 26100	Lemma for itg2mulc 26101. (Contributed by Mario Carneiro, 8-Jul-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ+)    ⇒   ⊢ (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) ≤ (𝐴 · (∫2‘𝐹)))