P. 257 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25601-25700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	itg1addlem5 25601*	Lemma for itg1add 25602. (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 25602	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 25603	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 25604	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 25605	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 25606*	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 25607*	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 25608*	Deduction form of i1fposd 25608. (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 25609	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 25610	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 25611*	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 25612*	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 25613*	Approximate version of itg1le 25614. 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 25614	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 25615*	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 25616*	Lemma for mbfi1fseq 25622. (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 25617*	Lemma for mbfi1fseq 25622. (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 25618*	Lemma for mbfi1fseq 25622. (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 25619*	Lemma for mbfi1fseq 25622. 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 25620*	Lemma for mbfi1fseq 25622. 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 25621*	Lemma for mbfi1fseq 25622. 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 25622*	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 25623*	Lemma for mbfi1flim 25624. (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 25624*	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 25625*	Lemma for mbfmul 25627. (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 25626	Lemma for mbfmul 25627. (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 25627	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 25628*	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 25629*	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 25630*	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 25631*	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 25632	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 25633	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 25634	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 25635*	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 25636	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 25637	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 25638	The integral of the zero function. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (∫2‘(ℝ × {0})) = 0
Theorem	itg2lecl 25639	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 25640	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 25641*	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 25642*	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 25643*	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 25656, 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 25644*	Approximate version of itg2ub 25634. 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 25645*	Approximate version of itg2le 25640. 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 25646*	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 25647	Lemma for itg2mulc 25648. (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)))
Theorem	itg2mulc 25648	The integral of a nonnegative constant times a function is the constant times the integral of the original function. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → (∫2‘((ℝ × {𝐴}) ∘f · 𝐹)) = (𝐴 · (∫2‘𝐹)))
Theorem	itg2splitlem 25649*	Lemma for itg2split 25650. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ (𝜑 → 𝐴 ∈ dom vol)    &   ⊢ (𝜑 → 𝐵 ∈ dom vol)    &   ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (0[,]+∞))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐶, 0))    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, 𝐶, 0))    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, 𝐶, 0))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫2‘𝐻) ≤ ((∫2‘𝐹) + (∫2‘𝐺)))
Theorem	itg2split 25650*	The ∫2 integral splits under an almost disjoint union. The proof avoids the use of itg2add 25660, which requires countable choice. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ (𝜑 → 𝐴 ∈ dom vol)    &   ⊢ (𝜑 → 𝐵 ∈ dom vol)    &   ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0)    &   ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ (0[,]+∞))    &   ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐶, 0))    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐵, 𝐶, 0))    &   ⊢ 𝐻 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑈, 𝐶, 0))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫2‘𝐻) = ((∫2‘𝐹) + (∫2‘𝐺)))
Theorem	itg2monolem1 25651*	Lemma for itg2mono 25654. We show that for any constant 𝑡 less than one, 𝑡 · ∫1𝐻 is less than 𝑆, and so ∫1𝐻 ≤ 𝑆, which is one half of the equality in itg2mono 25654. Consider the sequence 𝐴(𝑛) = {𝑥 ∣ 𝑡 · 𝐻 ≤ 𝐹(𝑛)}. This is an increasing sequence of measurable sets whose union is ℝ, and so 𝐻 ↾ 𝐴(𝑛) has an integral which equals ∫1𝐻 in the limit, by itg1climres 25615. Then by taking the limit in (𝑡 · 𝐻) ↾ 𝐴(𝑛) ≤ 𝐹(𝑛), we get 𝑡 · ∫1𝐻 ≤ 𝑆 as desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)    &   ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )    &   ⊢ (𝜑 → 𝑇 ∈ (0(,)1))    &   ⊢ (𝜑 → 𝐻 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝐻 ∘r ≤ 𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ)    &   ⊢ 𝐴 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ ℝ ∣ (𝑇 · (𝐻‘𝑥)) ≤ ((𝐹‘𝑛)‘𝑥)})    ⇒   ⊢ (𝜑 → (𝑇 · (∫1‘𝐻)) ≤ 𝑆)
Theorem	itg2monolem2 25652*	Lemma for itg2mono 25654. (Contributed by Mario Carneiro, 16-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)    &   ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )    &   ⊢ (𝜑 → 𝑃 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝑃 ∘r ≤ 𝐺)    &   ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆)    ⇒   ⊢ (𝜑 → 𝑆 ∈ ℝ)
Theorem	itg2monolem3 25653*	Lemma for itg2mono 25654. (Contributed by Mario Carneiro, 16-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)    &   ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )    &   ⊢ (𝜑 → 𝑃 ∈ dom ∫1)    &   ⊢ (𝜑 → 𝑃 ∘r ≤ 𝐺)    &   ⊢ (𝜑 → ¬ (∫1‘𝑃) ≤ 𝑆)    ⇒   ⊢ (𝜑 → (∫1‘𝑃) ≤ 𝑆)
Theorem	itg2mono 25654*	The Monotone Convergence Theorem for nonnegative functions. If {(𝐹‘𝑛):𝑛 ∈ ℕ} is a monotone increasing sequence of positive, measurable, real-valued functions, and 𝐺 is the pointwise limit of the sequence, then (∫2‘𝐺) is the limit of the sequence {(∫2‘(𝐹‘𝑛)):𝑛 ∈ ℕ}. (Contributed by Mario Carneiro, 16-Aug-2014.)
⊢ 𝐺 = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛):ℝ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∘r ≤ (𝐹‘(𝑛 + 1)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℕ ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)    &   ⊢ 𝑆 = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝐹‘𝑛))), ℝ*, < )    ⇒   ⊢ (𝜑 → (∫2‘𝐺) = 𝑆)
Theorem	itg2i1fseqle 25655*	Subject to the conditions coming from mbfi1fseq 25622, the sequence of simple functions are all less than the target function 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) ∘r ≤ 𝐹)
Theorem	itg2i1fseq 25656*	Subject to the conditions coming from mbfi1fseq 25622, the integral of the sequence of simple functions converges to the integral of the target function. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚)))    ⇒   ⊢ (𝜑 → (∫2‘𝐹) = sup(ran 𝑆, ℝ*, < ))
Theorem	itg2i1fseq2 25657*	In an extension to the results of itg2i1fseq 25656, if there is an upper bound on the integrals of the simple functions approaching 𝐹, then ∫2𝐹 is real and the standard limit relation applies. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚)))    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∫1‘(𝑃‘𝑘)) ≤ 𝑀)    ⇒   ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹))
Theorem	itg2i1fseq3 25658*	Special case of itg2i1fseq2 25657: if the integral of 𝐹 is a real number, then the standard limit relation holds on the integrals of simple functions approaching 𝐹. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))    &   ⊢ 𝑆 = (𝑚 ∈ ℕ ↦ (∫1‘(𝑃‘𝑚)))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝑆 ⇝ (∫2‘𝐹))
Theorem	itg2addlem 25659*	Lemma for itg2add 25660. (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    &   ⊢ (𝜑 → 𝑃:ℕ⟶dom ∫1)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑃‘𝑛) ∧ (𝑃‘𝑛) ∘r ≤ (𝑃‘(𝑛 + 1))))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑃‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))    &   ⊢ (𝜑 → 𝑄:ℕ⟶dom ∫1)    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ (0𝑝 ∘r ≤ (𝑄‘𝑛) ∧ (𝑄‘𝑛) ∘r ≤ (𝑄‘(𝑛 + 1))))    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑄‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))
Theorem	itg2add 25660	The ∫2 integral is linear. (Measurability is an essential component of this theorem; otherwise consider the characteristic function of a nonmeasurable set and its complement.) (Contributed by Mario Carneiro, 17-Aug-2014.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐺 ∈ MblFn)    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))
Theorem	itg2gt0 25661*	If the function 𝐹 is strictly positive on a set of positive measure, then the integral of the function is positive. (Contributed by Mario Carneiro, 30-Aug-2014.)
⊢ (𝜑 → 𝐴 ∈ dom vol)    &   ⊢ (𝜑 → 0 < (vol‘𝐴))    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 < (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → 0 < (∫2‘𝐹))
Theorem	itg2cnlem1 25662*	Lemma for itgcn 25746. (Contributed by Mario Carneiro, 30-Aug-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    ⇒   ⊢ (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑛, (𝐹‘𝑥), 0)))), ℝ*, < ) = (∫2‘𝐹))
Theorem	itg2cnlem2 25663*	Lemma for itgcn 25746. (Contributed by Mario Carneiro, 31-Aug-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → ¬ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹‘𝑥) ≤ 𝑀, (𝐹‘𝑥), 0))) ≤ ((∫2‘𝐹) − (𝐶 / 2)))    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑢, (𝐹‘𝑥), 0))) < 𝐶))
Theorem	itg2cn 25664*	A sort of absolute continuity of the Lebesgue integral (this is the core of ftc1a 25944 which is about actual absolute continuity). (Contributed by Mario Carneiro, 1-Sep-2014.)
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑢 ∈ dom vol((vol‘𝑢) < 𝑑 → (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝑢, (𝐹‘𝑥), 0))) < 𝐶))
Theorem	ibllem 25665	Conditioned equality theorem for the if statement. (Contributed by Mario Carneiro, 31-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
Theorem	isibl 25666*	The predicate "𝐹 is integrable". The "integrable" predicate corresponds roughly to the range of validity of ∫𝐴𝐵 d𝑥, which is to say that the expression ∫𝐴𝐵 d𝑥 doesn't make sense unless (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))    &   ⊢ (𝜑 → dom 𝐹 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝐿1 ↔ (𝐹 ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))
Theorem	isibl2 25667*	The predicate "𝐹 is integrable" when 𝐹 is a mapping operation. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝑘))))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ ∀𝑘 ∈ (0...3)(∫2‘𝐺) ∈ ℝ)))
Theorem	iblmbf 25668	An integrable function is measurable. (Contributed by Mario Carneiro, 7-Jul-2014.)
⊢ (𝐹 ∈ 𝐿1 → 𝐹 ∈ MblFn)
Theorem	iblitg 25669*	If a function is integrable, then the ∫2 integrals of the function's decompositions all exist. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (𝜑 → 𝐺 = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 = (ℜ‘(𝐵 / (i↑𝐾))))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ ℤ) → (∫2‘𝐺) ∈ ℝ)
Theorem	dfitg 25670*	Evaluate the class substitution in df-itg 25524. (Contributed by Mario Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝑇 = (ℜ‘(𝐵 / (i↑𝑘)))    ⇒   ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑇), 𝑇, 0))))
Theorem	itgex 25671	An integral is a set. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ ∫𝐴𝐵 d𝑥 ∈ V
Theorem	itgeq1f 25672	Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) Avoid axioms. (Revised by GG, 1-Sep-2025.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Theorem	itgeq1fOLD 25673	Obsolete version of itgeq1f 25672 as of 1-Sep-2025. (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Theorem	itgeq1 25674*	Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
Theorem	nfitg1 25675	Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ Ⅎ𝑥∫𝐴𝐵 d𝑥
Theorem	nfitg 25676*	Bound-variable hypothesis builder for an integral: if 𝑦 is (effectively) not free in 𝐴 and 𝐵, it is not free in ∫𝐴𝐵 d𝑥. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    ⇒   ⊢ Ⅎ𝑦∫𝐴𝐵 d𝑥
Theorem	cbvitg 25677*	Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦
Theorem	cbvitgv 25678*	Change bound variable in an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ⊢ ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦
Theorem	itgeq2 25679	Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
Theorem	itgresr 25680	The domain of an integral only matters in its intersection with ℝ. (Contributed by Mario Carneiro, 29-Jun-2014.)
⊢ ∫𝐴𝐵 d𝑥 = ∫(𝐴 ∩ ℝ)𝐵 d𝑥
Theorem	itg0 25681	The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014.)
⊢ ∫∅𝐴 d𝑥 = 0
Theorem	itgz 25682	The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ∫𝐴0 d𝑥 = 0
Theorem	itgeq2dv 25683*	Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
Theorem	itgmpt 25684*	Change bound variable in an integral. (Contributed by Mario Carneiro, 29-Jun-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) d𝑦)
Theorem	itgcl 25685*	The integral of an integrable function is a complex number. This is Metamath 100 proof #86. (Contributed by Mario Carneiro, 29-Jun-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℂ)
Theorem	itgvallem 25686*	Substitution lemma. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ (i↑𝐾) = 𝑇    ⇒   ⊢ (𝑘 = 𝐾 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / 𝑇))), (ℜ‘(𝐵 / 𝑇)), 0))))
Theorem	itgvallem3 25687*	Lemma for itgposval 25697 and itgreval 25698. (Contributed by Mario Carneiro, 7-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 0)    ⇒   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) = 0)
Theorem	ibl0 25688	The zero function is integrable on any measurable set. (Unlike iblconst 25719, this does not require 𝐴 to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.)
⊢ (𝐴 ∈ dom vol → (𝐴 × {0}) ∈ 𝐿1)
Theorem	iblcnlem1 25689*	Lemma for iblcnlem 25690. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))    &   ⊢ 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))    &   ⊢ 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))    &   ⊢ 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
Theorem	iblcnlem 25690*	Expand out the universal quantifier in isibl2 25667. (Contributed by Mario Carneiro, 6-Aug-2014.)
⊢ 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))    &   ⊢ 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))    &   ⊢ 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))    &   ⊢ 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
Theorem	itgcnlem 25691*	Expand out the sum in dfitg 25670. (Contributed by Mario Carneiro, 1-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))    &   ⊢ 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))    &   ⊢ 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))    &   ⊢ 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((𝑅 − 𝑆) + (i · (𝑇 − 𝑈))))
Theorem	iblrelem 25692*	Integrability of a real function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) ∈ ℝ)))
Theorem	iblposlem 25693*	Lemma for iblpos 25694. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0))) = 0)
Theorem	iblpos 25694*	Integrability of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))) ∈ ℝ)))
Theorem	iblre 25695*	Integrability of a real function. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ if(0 ≤ -𝐵, -𝐵, 0)) ∈ 𝐿1)))
Theorem	itgrevallem1 25696*	Lemma for itgposval 25697 and itgreval 25698. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) − (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -𝐵), -𝐵, 0)))))
Theorem	itgposval 25697*	The integral of a nonnegative function. (Contributed by Mario Carneiro, 31-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫2‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 𝐵, 0))))
Theorem	itgreval 25698*	Decompose the integral of a real function into positive and negative parts. (Contributed by Mario Carneiro, 31-Jul-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = (∫𝐴if(0 ≤ 𝐵, 𝐵, 0) d𝑥 − ∫𝐴if(0 ≤ -𝐵, -𝐵, 0) d𝑥))
Theorem	itgrecl 25699*	Real closure of an integral. (Contributed by Mario Carneiro, 11-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    ⇒   ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 ∈ ℝ)
Theorem	iblcn 25700*	Integrability of a complex function. (Contributed by Mario Carneiro, 6-Aug-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1 ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ 𝐿1 ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ 𝐿1)))