P. 377 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37601-37700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opnmbllem0 37601*	Lemma for ismblfin 37606; could also be used to shorten proof of opnmbllem 25539. (Contributed by Brendan Leahy, 13-Jul-2018.)
⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴)
Theorem	mblfinlem1 37602*	Lemma for ismblfin 37606, ordering the sets of dyadic intervals that are antichains under subset and whose unions are contained entirely in 𝐴. (Contributed by Brendan Leahy, 13-Jul-2018.)
⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
Theorem	mblfinlem2 37603*	Lemma for ismblfin 37606, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different definition of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
Theorem	mblfinlem3 37604*	The difference between two sets measurable by the criterion in ismblfin 37606 is itself measurable by the same. Corollary 0.3 of [Viaclovsky7] p. 3. (Contributed by Brendan Leahy, 25-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))) → sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) = (vol*‘(𝐴 ∖ 𝐵)))
Theorem	mblfinlem4 37605*	Backward direction of ismblfin 37606. (Contributed by Brendan Leahy, 28-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))
Theorem	ismblfin 37606*	Measurability in terms of inner and outer measure. Proposition 7 of [Viaclovsky8] p. 3. (Contributed by Brendan Leahy, 4-Mar-2018.) (Revised by Brendan Leahy, 28-Mar-2018.)
⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ∈ dom vol ↔ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )))
Theorem	ovoliunnfl 37607*	ovoliun 25443 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)
⊢ ((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ))    ⇒   ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	ex-ovoliunnfl 37608*	Demonstration of ovoliunnfl 37607. (Contributed by Brendan Leahy, 21-Nov-2017.)
⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	voliunnfl 37609*	voliun 25492 is incompatible with the Feferman-Levy model; in that model, therefore, the Lebesgue measure as we've defined it isn't actually a measure. (Contributed by Brendan Leahy, 16-Dec-2017.)
⊢ 𝑆 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓‘𝑛)))    &   ⊢ ((∀𝑛 ∈ ℕ ((𝑓‘𝑛) ∈ dom vol ∧ (vol‘(𝑓‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)) → (vol‘∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) = sup(ran 𝑆, ℝ*, < ))    ⇒   ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	volsupnfl 37610*	volsup 25494 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 2-Jan-2018.)
⊢ ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))    ⇒   ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	mbfresfi 37611*	Measurability of a piecewise function across arbitrarily many subsets. (Contributed by Brendan Leahy, 31-Mar-2018.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝑆 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (𝐹 ↾ 𝑠) ∈ MblFn)    &   ⊢ (𝜑 → ∪ 𝑆 = 𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	mbfposadd 37612*	If the sum of two measurable functions is measurable, the sum of their nonnegative parts is measurable. (Contributed by Brendan Leahy, 2-Apr-2018.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn)
Theorem	cnambfre 37613	A real-valued, a.e. continuous function is measurable. (Contributed by Brendan Leahy, 4-Apr-2018.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → 𝐹 ∈ MblFn)
Theorem	dvtanlem 37614	Lemma for dvtan 37615- the domain of the tangent is open. (Contributed by Brendan Leahy, 8-Aug-2018.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ (◡cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
Theorem	dvtan 37615	Derivative of tangent. (Contributed by Brendan Leahy, 7-Aug-2018.)
⊢ (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
Theorem	itg2addnclem 37616*	An alternate expression for the ∫2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}    ⇒   ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))
Theorem	itg2addnclem2 37617*	Lemma for itg2addnc 37619. The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    ⇒   ⊢ (((𝜑 ∧ ℎ ∈ dom ∫1) ∧ 𝑣 ∈ ℝ+) → (𝑥 ∈ ℝ ↦ if(((((⌊‘((𝐹‘𝑥) / (𝑣 / 3))) − 1) · (𝑣 / 3)) ≤ (ℎ‘𝑥) ∧ (ℎ‘𝑥) ≠ 0), (((⌊‘((𝐹‘𝑥) / (𝑣 / 3))) − 1) · (𝑣 / 3)), (ℎ‘𝑥))) ∈ dom ∫1)
Theorem	itg2addnclem3 37618*	Lemma incomprehensible in isolation split off to shorten proof of itg2addnc 37619. (Contributed by Brendan Leahy, 11-Mar-2018.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) → ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
Theorem	itg2addnc 37619	Alternate proof of itg2add 25697 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 25646, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 10441, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))
Theorem	itg2gt0cn 37620*	itg2gt0 25698 holds on functions continuous on an open interval in the absence of ax-cc 10441. The fourth hypothesis is made unnecessary by the continuity hypothesis. (Contributed by Brendan Leahy, 16-Nov-2017.)
⊢ (𝜑 → 𝑋 < 𝑌)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 0 < (𝐹‘𝑥))    &   ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ))    ⇒   ⊢ (𝜑 → 0 < (∫2‘𝐹))
Theorem	ibladdnclem 37621*	Lemma for ibladdnc 37622; cf ibladdlem 25758, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 37619. (Contributed by Brendan Leahy, 31-Oct-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 = (𝐵 + 𝐶))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)    &   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)
Theorem	ibladdnc 37622*	Choice-free analogue of itgadd 25763. A measurability hypothesis is necessitated by the loss of mbfadd 25599; for large classes of functions, such as continuous functions, it should be relatively easy to show. (Contributed by Brendan Leahy, 1-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1)
Theorem	itgaddnclem1 37623*	Lemma for itgaddnc 37625; cf. itgaddlem1 25761. (Contributed by Brendan Leahy, 7-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Theorem	itgaddnclem2 37624*	Lemma for itgaddnc 37625; cf. itgaddlem2 25762. (Contributed by Brendan Leahy, 10-Nov-2017.) (Revised by Brendan Leahy, 3-Apr-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Theorem	itgaddnc 37625*	Choice-free analogue of itgadd 25763. (Contributed by Brendan Leahy, 11-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Theorem	iblsubnc 37626*	Choice-free analogue of iblsub 25760. (Contributed by Brendan Leahy, 11-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ 𝐿1)
Theorem	itgsubnc 37627*	Choice-free analogue of itgsub 25764. (Contributed by Brendan Leahy, 11-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 − 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 − ∫𝐴𝐶 d𝑥))
Theorem	iblabsnclem 37628*	Lemma for iblabsnc 37629; cf. iblabslem 25766. (Contributed by Brendan Leahy, 7-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐺 ∈ MblFn ∧ (∫2‘𝐺) ∈ ℝ))
Theorem	iblabsnc 37629*	Choice-free analogue of iblabs 25767. As with ibladdnc 37622, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1)
Theorem	iblmulc2nc 37630*	Choice-free analogue of iblmulc2 25769. (Contributed by Brendan Leahy, 17-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1)
Theorem	itgmulc2nclem1 37631*	Lemma for itgmulc2nc 37633; cf. itgmulc2lem1 25770. (Contributed by Brendan Leahy, 17-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Theorem	itgmulc2nclem2 37632*	Lemma for itgmulc2nc 37633; cf. itgmulc2lem2 25771. (Contributed by Brendan Leahy, 19-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Theorem	itgmulc2nc 37633*	Choice-free analogue of itgmulc2 25772. (Contributed by Brendan Leahy, 19-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Theorem	itgabsnc 37634*	Choice-free analogue of itgabs 25773. (Contributed by Brendan Leahy, 19-Nov-2017.) (Revised by Brendan Leahy, 19-Jun-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)
Theorem	itggt0cn 37635*	itggt0 25782 holds for continuous functions in the absence of ax-cc 10441. (Contributed by Brendan Leahy, 16-Nov-2017.)
⊢ (𝜑 → 𝑋 < 𝑌)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ))    ⇒   ⊢ (𝜑 → 0 < ∫(𝑋(,)𝑌)𝐵 d𝑥)
Theorem	ftc1cnnclem 37636*	Lemma for ftc1cnnc 37637; cf. ftc1lem4 25983. The stronger assumptions of ftc1cn 25987 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝑐 ∈ (𝐴(,)𝐵))    &   ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝑐)) < 𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (abs‘(𝑌 − 𝑐)) < 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐))) < 𝐸)
Theorem	ftc1cnnc 37637*	Choice-free proof of ftc1cn 25987. (Contributed by Brendan Leahy, 20-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)
Theorem	ftc1anclem1 37638	Lemma for ftc1anc 37646- the absolute value of a real-valued measurable function is measurable. Would be trivial with cncombf 25596, but this proof avoids ax-cc 10441. (Contributed by Brendan Leahy, 18-Jun-2018.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐹 ∈ MblFn) → (abs ∘ 𝐹) ∈ MblFn)
Theorem	ftc1anclem2 37639*	Lemma for ftc1anc 37646- restriction of an integrable function to the absolute value of its real or imaginary part. (Contributed by Brendan Leahy, 19-Jun-2018.) (Revised by Brendan Leahy, 8-Aug-2018.)
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1 ∧ 𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝐴, (abs‘(𝐺‘(𝐹‘𝑡))), 0))) ∈ ℝ)
Theorem	ftc1anclem3 37640	Lemma for ftc1anc 37646- the absolute value of the sum of a simple function and i times another simple function is itself a simple function. (Contributed by Brendan Leahy, 27-May-2018.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (abs ∘ (𝐹 ∘f + ((ℝ × {i}) ∘f · 𝐺))) ∈ dom ∫1)
Theorem	ftc1anclem4 37641*	Lemma for ftc1anc 37646. (Contributed by Brendan Leahy, 17-Jun-2018.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ 𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ)
Theorem	ftc1anclem5 37642*	Lemma for ftc1anc 37646, the existence of a simple function the integral of whose pointwise difference from the function is less than a given positive real. (Contributed by Brendan Leahy, 17-Jun-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0)) − (𝑓‘𝑡))))) < 𝑌)
Theorem	ftc1anclem6 37643*	Lemma for ftc1anc 37646- construction of simple functions within an arbitrary absolute distance of the given function. Similar to Lemma 565Ib of [Fremlin5] p. 218, but without Fremlin's additional step of converting the simple function into a continuous one, which is unnecessary to this lemma's use; also, two simple functions are used to allow for complex-valued 𝐹. (Contributed by Brendan Leahy, 31-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < 𝑌)
Theorem	ftc1anclem7 37644*	Lemma for ftc1anc 37646. (Contributed by Brendan Leahy, 13-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)))) < ((𝑦 / 2) + (𝑦 / 2)))
Theorem	ftc1anclem8 37645*	Lemma for ftc1anc 37646. (Contributed by Brendan Leahy, 29-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) + (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < 𝑦)
Theorem	ftc1anc 37646*	ftc1a 25981 holds for functions that obey the triangle inequality in the absence of ax-cc 10441. Theorem 565Ma of [Fremlin5] p. 220. (Contributed by Brendan Leahy, 11-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    &   ⊢ (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠(𝐹‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘(𝐹‘𝑡)), 0))))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Theorem	ftc2nc 37647*	Choice-free proof of ftc2 25988. (Contributed by Brendan Leahy, 19-Jun-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
Theorem	asindmre 37648	Real part of domain of differentiability of arcsine. (Contributed by Brendan Leahy, 19-Dec-2018.)
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))    ⇒   ⊢ (𝐷 ∩ ℝ) = (-1(,)1)
Theorem	dvasin 37649*	Derivative of arcsine. (Contributed by Brendan Leahy, 18-Dec-2018.)
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))    ⇒   ⊢ (ℂ D (arcsin ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2)))))
Theorem	dvacos 37650*	Derivative of arccosine. (Contributed by Brendan Leahy, 18-Dec-2018.)
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))    ⇒   ⊢ (ℂ D (arccos ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (-1 / (√‘(1 − (𝑥↑2)))))
Theorem	dvreasin 37651	Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.) (Proof shortened by Brendan Leahy, 18-Dec-2018.)
⊢ (ℝ D (arcsin ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (1 / (√‘(1 − (𝑥↑2)))))
Theorem	dvreacos 37652	Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.) (Proof shortened by Brendan Leahy, 18-Dec-2018.)
⊢ (ℝ D (arccos ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (-1 / (√‘(1 − (𝑥↑2)))))
Theorem	areacirclem1 37653*	Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ (𝑅 ∈ ℝ+ → (ℝ D (𝑡 ∈ (-𝑅(,)𝑅) ↦ ((𝑅↑2) · ((arcsin‘(𝑡 / 𝑅)) + ((𝑡 / 𝑅) · (√‘(1 − ((𝑡 / 𝑅)↑2)))))))) = (𝑡 ∈ (-𝑅(,)𝑅) ↦ (2 · (√‘((𝑅↑2) − (𝑡↑2))))))
Theorem	areacirclem2 37654*	Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
Theorem	areacirclem3 37655*	Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (2 · (√‘((𝑅↑2) − (𝑡↑2))))) ∈ 𝐿1)
Theorem	areacirclem4 37656*	Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ (𝑅 ∈ ℝ+ → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) · ((arcsin‘(𝑡 / 𝑅)) + ((𝑡 / 𝑅) · (√‘(1 − ((𝑡 / 𝑅)↑2))))))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
Theorem	areacirclem5 37657*	Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}    ⇒   ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑆 “ {𝑡}) = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))
Theorem	areacirc 37658*	The area of a circle of radius 𝑅 is π · 𝑅↑2. This is Metamath 100 proof #9. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}    ⇒   ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (area‘𝑆) = (π · (𝑅↑2)))
21.24  Mathbox for Jeff Madsen
21.24.1  Logic and set theory
Theorem	unirep 37659*	Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ (𝑦 = 𝐷 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐷 → 𝐵 = 𝐶)    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → 𝐵 = 𝐹)    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝜑 ∧ 𝜒) → 𝐵 = 𝐹) ∧ (𝐷 ∈ 𝐴 ∧ 𝜓)) → (℩𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵)) = 𝐶)
Theorem	cover2 37660*	Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑". Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐴 = ∪ 𝐵    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑))
Theorem	cover2g 37661*	Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑". Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 21-Jun-2010.)
⊢ 𝐴 = ∪ 𝐵    ⇒   ⊢ (𝐵 ∈ 𝐶 → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∧ 𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵(∪ 𝑧 = 𝐴 ∧ ∀𝑦 ∈ 𝑧 𝜑)))
Theorem	brabg2 37662*	Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    &   ⊢ (𝜒 → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒))
Theorem	opelopab3 37663*	Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝐵 ∈ 𝐷 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
Theorem	cocanfo 37664	Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺 Fn 𝐵 ∧ 𝐻 Fn 𝐵) ∧ (𝐺 ∘ 𝐹) = (𝐻 ∘ 𝐹)) → 𝐺 = 𝐻)
Theorem	brresi2 37665	Restriction of a binary relation. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴(𝑅 ↾ 𝐶)𝐵 → 𝐴𝑅𝐵)
Theorem	fnopabeqd 37666*	Equality deduction for function abstractions. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)})
Theorem	fvopabf4g 37667*	Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
⊢ 𝐶 ∈ V    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑m 𝐷) ↦ 𝐵)    ⇒   ⊢ ((𝐷 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌 ∧ 𝐴:𝐷⟶𝑅) → (𝐹‘𝐴) = 𝐶)
Theorem	fnopabco 37668*	Composition of a function with a function abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}    &   ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐻‘𝐵))}    ⇒   ⊢ (𝐻 Fn 𝐶 → 𝐺 = (𝐻 ∘ 𝐹))
Theorem	opropabco 37669*	Composition of an operator with a function abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑅)    &   ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑆)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = ⟨𝐵, 𝐶⟩)}    &   ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐵𝑀𝐶))}    ⇒   ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))
Theorem	cocnv 37670	Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ 𝐺) ∘ ◡𝐺) = (𝐹 ↾ ran 𝐺))
Theorem	f1ocan1fv 37671	Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐹 ∘ 𝐺)‘(◡𝐺‘𝑋)) = (𝐹‘𝑋))
Theorem	f1ocan2fv 37672	Cancel a composition by the converse of a bijection by preapplying the bijection. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((Fun 𝐹 ∧ 𝐺:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ ◡𝐺)‘(𝐺‘𝑋)) = (𝐹‘𝑋))
Theorem	inixp 37673*	Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) = X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)
Theorem	upixp 37674*	Universal property of the indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
⊢ 𝑋 = X𝑏 ∈ 𝐴 (𝐶‘𝑏)    &   ⊢ 𝑃 = (𝑤 ∈ 𝐴 ↦ (𝑥 ∈ 𝑋 ↦ (𝑥‘𝑤)))    ⇒   ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎):𝐵⟶(𝐶‘𝑎)) → ∃!ℎ(ℎ:𝐵⟶𝑋 ∧ ∀𝑎 ∈ 𝐴 (𝐹‘𝑎) = ((𝑃‘𝑎) ∘ ℎ)))
Theorem	abrexdom 37675*	An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)
Theorem	abrexdom2 37676*	An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = 𝐵} ≼ 𝐴)
Theorem	ac6gf 37677*	Axiom of Choice. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ Ⅎ𝑦𝜓    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	indexa 37678*	If for every element of an indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a subset of 𝐵 consisting only of those elements which are indexed by 𝐴. Used to avoid the Axiom of Choice in situations where only the range of the choice function is needed. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐵 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐(𝑐 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑))
Theorem	indexdom 37679*	If for every element of an indexing set 𝐴 there exists a corresponding element of another set 𝐵, then there exists a subset of 𝐵 consisting only of those elements which are indexed by 𝐴, and which is dominated by the set 𝐴. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ 𝑀 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑐((𝑐 ≼ 𝐴 ∧ 𝑐 ⊆ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑐 𝜑 ∧ ∀𝑦 ∈ 𝑐 ∃𝑥 ∈ 𝐴 𝜑)))
Theorem	frinfm 37680*	A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 Fr 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
Theorem	welb 37681*	A nonempty subset of a well-ordered set has a lower bound. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 We 𝐴 ∧ (𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → (◡𝑅 Or 𝐵 ∧ ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
Theorem	supex2g 37682	Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)
Theorem	supclt 37683*	Closure of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
Theorem	supubt 37684*	Upper bound property of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 Or 𝐴 ∧ ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
21.24.2  Real and complex numbers; integers
Theorem	filbcmb 37685*	Combine a finite set of lower bounds. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐵 ⊆ ℝ) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 ≤ 𝑧 → 𝜑) → ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 ≤ 𝑧 → ∀𝑥 ∈ 𝐴 𝜑)))
Theorem	fzmul 37686	Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℕ) → (𝐽 ∈ (𝑀...𝑁) → (𝐾 · 𝐽) ∈ ((𝐾 · 𝑀)...(𝐾 · 𝑁))))
21.24.3  Sequences and sums
Theorem	sdclem2 37687*	Lemma for sdc 37689. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏))    &   ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))    &   ⊢ 𝐽 = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)}    &   ⊢ 𝐹 = (𝑤 ∈ 𝑍, 𝑥 ∈ 𝐽 ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝐽)    &   ⊢ (𝜑 → (𝐺‘𝑀):(𝑀...𝑀)⟶𝐴)    &   ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → (𝐺‘(𝑤 + 1)) ∈ (𝑤𝐹(𝐺‘𝑤)))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))
Theorem	sdclem1 37688*	Lemma for sdc 37689. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏))    &   ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))    &   ⊢ 𝐽 = {𝑔 ∣ ∃𝑛 ∈ 𝑍 (𝑔:(𝑀...𝑛)⟶𝐴 ∧ 𝜓)}    &   ⊢ 𝐹 = (𝑤 ∈ 𝑍, 𝑥 ∈ 𝐽 ↦ {ℎ ∣ ∃𝑘 ∈ 𝑍 (ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑥 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)})    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))
Theorem	sdc 37689*	Strong dependent choice. Suppose we may choose an element of 𝐴 such that property 𝜓 holds, and suppose that if we have already chosen the first 𝑘 elements (represented here by a function from 1...𝑘 to 𝐴), we may choose another element so that all 𝑘 + 1 elements taken together have property 𝜓. Then there exists an infinite sequence of elements of 𝐴 such that the first 𝑛 terms of this sequence satisfy 𝜓 for all 𝑛. This theorem allows to construct infinite sequences where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜏))    &   ⊢ (𝑛 = 𝑘 → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑔 = ℎ ∧ 𝑛 = (𝑘 + 1)) → (𝜓 ↔ 𝜎))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴 ∧ 𝜏))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴 ∧ 𝜃) → ∃ℎ(ℎ:(𝑀...(𝑘 + 1))⟶𝐴 ∧ 𝑔 = (ℎ ↾ (𝑀...𝑘)) ∧ 𝜎)))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑍⟶𝐴 ∧ ∀𝑛 ∈ 𝑍 𝜒))
Theorem	fdc 37690*	Finite version of dependent choice. Construct a function whose value depends on the previous function value, except at a final point at which no new value can be chosen. The final hypothesis ensures that the process will terminate. The proof does not use the Axiom of Choice. (Contributed by Jeff Madsen, 18-Jun-2010.)
⊢ 𝐴 ∈ V    &   ⊢ 𝑀 ∈ ℤ    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑁 = (𝑀 + 1)    &   ⊢ (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = (𝑓‘𝑘) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑎 = (𝑓‘𝑛) → (𝜃 ↔ 𝜏))    &   ⊢ (𝜂 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜂 → 𝑅 Fr 𝐴)    &   ⊢ ((𝜂 ∧ 𝑎 ∈ 𝐴) → (𝜃 ∨ ∃𝑏 ∈ 𝐴 𝜑))    &   ⊢ (((𝜂 ∧ 𝜑) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏𝑅𝑎)    ⇒   ⊢ (𝜂 → ∃𝑛 ∈ 𝑍 ∃𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ ((𝑓‘𝑀) = 𝐶 ∧ 𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒))
Theorem	fdc1 37691*	Variant of fdc 37690 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.)
⊢ 𝐴 ∈ V    &   ⊢ 𝑀 ∈ ℤ    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝑁 = (𝑀 + 1)    &   ⊢ (𝑎 = (𝑓‘𝑀) → (𝜁 ↔ 𝜎))    &   ⊢ (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = (𝑓‘𝑘) → (𝜓 ↔ 𝜒))    &   ⊢ (𝑎 = (𝑓‘𝑛) → (𝜃 ↔ 𝜏))    &   ⊢ (𝜂 → ∃𝑎 ∈ 𝐴 𝜁)    &   ⊢ (𝜂 → 𝑅 Fr 𝐴)    &   ⊢ ((𝜂 ∧ 𝑎 ∈ 𝐴) → (𝜃 ∨ ∃𝑏 ∈ 𝐴 𝜑))    &   ⊢ (((𝜂 ∧ 𝜑) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏𝑅𝑎)    ⇒   ⊢ (𝜂 → ∃𝑛 ∈ 𝑍 ∃𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ (𝜎 ∧ 𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒))
Theorem	seqpo 37692*	Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) → (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛)))
Theorem	incsequz 37693*	An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (ℤ≥‘𝐴))
Theorem	incsequz2 37694*	An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ≥‘𝐴))
Theorem	nnubfi 37695*	A bounded above set of positive integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
⊢ ((𝐴 ⊆ ℕ ∧ 𝐵 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ 𝑥 < 𝐵} ∈ Fin)
Theorem	nninfnub 37696*	An infinite set of positive integers is unbounded above. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
⊢ ((𝐴 ⊆ ℕ ∧ ¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑥} ≠ ∅)
21.24.4  Topology
Theorem	subspopn 37697	An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
⊢ (((𝐽 ∈ Top ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (𝐽 ↾t 𝐴))
Theorem	neificl 37698	Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)
⊢ (((𝐽 ∈ Top ∧ 𝑁 ⊆ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)) → ∩ 𝑁 ∈ ((nei‘𝐽)‘𝑆))
Theorem	lpss2 37699	Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((limPt‘𝐽)‘𝐵) ⊆ ((limPt‘𝐽)‘𝐴))
21.24.5  Metric spaces
Theorem	metf1o 37700*	Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ 𝑁 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)𝑀(𝐹‘𝑦)))    ⇒   ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁 ∈ (Met‘𝑌))