P. 461 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46001-46100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elioored 46001	A member of an open interval of reals is a real. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ (𝐵(,)𝐶))    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	ioogtlbd 46002	An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	ioofun 46003	(,) is a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ Fun (,)
Theorem	icomnfinre 46004	A left-closed, right-open, interval of extended reals, intersected with the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((-∞[,)𝐴) ∩ ℝ) = (-∞(,)𝐴))
Theorem	sqrlearg 46005	The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴↑2) ≤ 𝐴 ↔ 𝐴 ∈ (0[,]1)))
Theorem	ressiocsup 46006	If the supremum belongs to a set of reals, the set is a subset of the unbounded below, right-closed interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    &   ⊢ (𝜑 → 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (-∞(,]𝑆)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
Theorem	ressioosup 46007	If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = sup(𝐴, ℝ*, < )    &   ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (-∞(,)𝑆)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
Theorem	iooiinioc 46008*	A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))
Theorem	ressiooinf 46009	If the infimum does not belong to a set of reals, the set is a subset of the unbounded above, left-open interval, with lower bound equal to the infimum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝑆 = inf(𝐴, ℝ*, < )    &   ⊢ (𝜑 → ¬ 𝑆 ∈ 𝐴)    &   ⊢ 𝐼 = (𝑆(,)+∞)    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝐼)
Theorem	iocleubd 46010	An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))    ⇒   ⊢ (𝜑 → 𝐶 ≤ 𝐵)
Theorem	uzinico 46011	An upper interval of integers is the intersection of the integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞)))
Theorem	preimaiocmnf 46012*	Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (◡𝐹 “ (-∞(,]𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≤ 𝐵})
Theorem	uzinico2 46013	An upper interval of integers is the intersection of a larger upper interval of integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    ⇒   ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞)))
Theorem	uzinico3 46014	An upper interval of integers doesn't change when it's intersected with a left-closed, unbounded above interval, with the same lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞)))
Theorem	dmico 46015	The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ dom [,) = (ℝ* × ℝ*)
Theorem	ndmico 46016	The closed-below, open-above interval function's value is empty outside of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ∅)
Theorem	uzubioo 46017*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍)
Theorem	uzubico 46018*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘 ∈ 𝑍)
Theorem	uzubioo2 46019*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘 ∈ 𝑍)
Theorem	uzubico2 46020*	The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘 ∈ 𝑍)
Theorem	iocgtlbd 46021	An element of a left-open right-closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))    ⇒   ⊢ (𝜑 → 𝐴 < 𝐶)
Theorem	xrtgioo2 46022	The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ)
21.43.5  Finite sums
Theorem	fsummulc1f 46023*	Closure of a finite sum of complex numbers 𝐴(𝑘). A version of fsummulc1 15745 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (Σ𝑘 ∈ 𝐴 𝐵 · 𝐶) = Σ𝑘 ∈ 𝐴 (𝐵 · 𝐶))
Theorem	fsumnncl 46024*	Closure of a nonempty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℕ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℕ)
Theorem	fsumge0cl 46025*	The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞))    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞))
Theorem	fsumf1of 46026*	Re-index a finite sum using a bijection. Same as fsumf1o 15683, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑛𝜑    &   ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑛 ∈ 𝐶 𝐷)
Theorem	fsumiunss 46027*	Sum over a disjoint indexed union, intersected with a finite set 𝐷. Similar to fsumiun 15782, but here 𝐴 and 𝐵 need not be finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ Fin)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐷)𝐶 = Σ𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (𝐵 ∩ 𝐷) ≠ ∅}Σ𝑘 ∈ (𝐵 ∩ 𝐷)𝐶)
Theorem	fsumreclf 46028*	Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ)
Theorem	fsumlessf 46029*	A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵)
Theorem	fsumsupp0 46030*	Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝐹 supp 0)(𝐹‘𝑘) = Σ𝑘 ∈ 𝐴 (𝐹‘𝑘))
Theorem	fsumsermpt 46031*	A finite sum expressed in terms of a partial sum of an infinite series. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ 𝐹 = (𝑛 ∈ 𝑍 ↦ Σ𝑘 ∈ (𝑀...𝑛)𝐴)    &   ⊢ 𝐺 = seq𝑀( + , (𝑘 ∈ 𝑍 ↦ 𝐴))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
21.43.6  Finite multiplication of numbers and finite multiplication of functions
Theorem	fmul01 46032*	Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ 𝐴 = seq𝐿( · , 𝐵)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿))    &   ⊢ (𝜑 → 𝐾 ∈ (𝐿...𝑀))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)    ⇒   ⊢ (𝜑 → (0 ≤ (𝐴‘𝐾) ∧ (𝐴‘𝐾) ≤ 1))
Theorem	fmulcl 46033*	If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))    &   ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑁)    &   ⊢ (𝜑 → 𝑁 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)    &   ⊢ (𝜑 → 𝑇 ∈ V)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝑌)
Theorem	fmuldfeqlem1 46034*	induction step for the proof of fmuldfeq 46035. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑓𝜑    &   ⊢ Ⅎ𝑔𝜑    &   ⊢ Ⅎ𝑡𝑌    &   ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))    &   ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))    &   ⊢ (𝜑 → 𝑇 ∈ V)    &   ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)    &   ⊢ (𝜑 → 𝑁 ∈ (1...𝑀))    &   ⊢ (𝜑 → (𝑁 + 1) ∈ (1...𝑀))    &   ⊢ (𝜑 → ((seq1(𝑃, 𝑈)‘𝑁)‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑁))    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)    ⇒   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((seq1(𝑃, 𝑈)‘(𝑁 + 1))‘𝑡) = (seq1( · , (𝐹‘𝑡))‘(𝑁 + 1)))
Theorem	fmuldfeq 46035*	X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑡𝑌    &   ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))    &   ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)    &   ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))    &   ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))    &   ⊢ (𝜑 → 𝑇 ∈ V)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
Theorem	fmul01lt1lem1 46036*	Given a finite multiplication of values between 0 and 1, a value larger than its first element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ 𝐴 = seq𝐿( · , 𝐵)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → (𝐵‘𝐿) < 𝐸)    ⇒   ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)
Theorem	fmul01lt1lem2 46037*	Given a finite multiplication of values between 0 and 1, a value 𝐸 larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ 𝐴 = seq𝐿( · , 𝐵)    &   ⊢ (𝜑 → 𝐿 ∈ ℤ)    &   ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝐿))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (𝐿...𝑀)) → (𝐵‘𝑖) ≤ 1)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑀))    &   ⊢ (𝜑 → (𝐵‘𝐽) < 𝐸)    ⇒   ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)
Theorem	fmul01lt1 46038*	Given a finite multiplication of values between 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑖𝐵    &   ⊢ Ⅎ𝑖𝜑    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ 𝐴 = seq1( · , 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝐵:(1...𝑀)⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ (𝐵‘𝑖))    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐵‘𝑖) ≤ 1)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → ∃𝑗 ∈ (1...𝑀)(𝐵‘𝑗) < 𝐸)    ⇒   ⊢ (𝜑 → (𝐴‘𝑀) < 𝐸)
Theorem	cncfmptss 46039*	A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵))
Theorem	rrpsscn 46040	The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ ℝ+ ⊆ ℂ
Theorem	mulc1cncfg 46041*	A version of mulc1cncf 24897 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 · (𝐹‘𝑥))) ∈ (𝐴–cn→ℂ))
Theorem	infrglb 46042*	The infimum of a nonempty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ∧ 𝐵 ∈ ℝ) → (inf(𝐴, ℝ, < ) < 𝐵 ↔ ∃𝑧 ∈ 𝐴 𝑧 < 𝐵))
Theorem	expcnfg 46043*	If 𝐹 is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 24918. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→ℂ))    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)↑𝑁)) ∈ (𝐴–cn→ℂ))
Theorem	prodeq2ad 46044*	Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)
Theorem	fprodsplit1 46045*	Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐶) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = (𝐷 · ∏𝑘 ∈ (𝐴 ∖ {𝐶})𝐵))
Theorem	fprodexp 46046*	Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁))
Theorem	fprodabs2 46047*	The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))
Theorem	fprod0 46048*	A finite product with a zero term is zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐶    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 = 0)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0)
Theorem	mccllem 46049*	* Induction step for mccl 46050. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶))    &   ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m (𝐶 ∪ {𝐷})))    &   ⊢ (𝜑 → ∀𝑏 ∈ (ℕ0 ↑m 𝐶)((!‘Σ𝑘 ∈ 𝐶 (𝑏‘𝑘)) / ∏𝑘 ∈ 𝐶 (!‘(𝑏‘𝑘))) ∈ ℕ)    ⇒   ⊢ (𝜑 → ((!‘Σ𝑘 ∈ (𝐶 ∪ {𝐷})(𝐵‘𝑘)) / ∏𝑘 ∈ (𝐶 ∪ {𝐷})(!‘(𝐵‘𝑘))) ∈ ℕ)
Theorem	mccl 46050*	A multinomial coefficient, in its standard domain, is a positive integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ Ⅎ𝑘𝐵    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ (ℕ0 ↑m 𝐴))    ⇒   ⊢ (𝜑 → ((!‘Σ𝑘 ∈ 𝐴 (𝐵‘𝑘)) / ∏𝑘 ∈ 𝐴 (!‘(𝐵‘𝑘))) ∈ ℕ)
Theorem	fprodcnlem 46051*	A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.) Avoid ax-mulf 11116. (Revised by GG, 19-Apr-2025.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑍 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍))    &   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))
Theorem	fprodcn 46052*	A finite product of functions to complex numbers from a common topological space is continuous. The class expression for 𝐵 normally contains free variables 𝑘 and 𝑥 to index it. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
21.43.7  Limits
Theorem	clim1fr1 46053*	A class of sequences of fractions that converge to 1. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((𝐴 · 𝑛) + 𝐵) / (𝐴 · 𝑛)))    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 1)
Theorem	isumneg 46054*	Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 -𝐴 = -Σ𝑘 ∈ 𝑍 𝐴)
Theorem	climrec 46055*	Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0}))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘)))    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴))
Theorem	climmulf 46056*	A version of climmul 15593 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ Ⅎ𝑘𝐻    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵))
Theorem	climexp 46057*	The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐻    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘)↑𝑁))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐴↑𝑁))
Theorem	climinf 46058*	A bounded monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
Theorem	climsuselem1 46059*	The subsequence index 𝐼 has the expected properties: it belongs to the same upper integers as the original index, and it is always greater than or equal to the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ≥‘((𝐼‘𝑘) + 1)))    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐼‘𝐾) ∈ (ℤ≥‘𝐾))
Theorem	climsuse 46060*	A subsequence 𝐺 of a converging sequence 𝐹, converges to the same limit. 𝐼 is the strictly increasing and it is used to index the subsequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ Ⅎ𝑘𝐼    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → (𝐼‘𝑀) ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐼‘(𝑘 + 1)) ∈ (ℤ≥‘((𝐼‘𝑘) + 1)))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑌)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐹‘(𝐼‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
Theorem	climrecf 46061*	A version of climrec 46055 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ Ⅎ𝑘𝐻    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0}))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘)))    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴))
Theorem	climneg 46062*	Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -𝐴)
Theorem	climinff 46063*	A version of climinf 46058 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ, < ))
Theorem	climdivf 46064*	Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ Ⅎ𝑘𝐻    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0}))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) / (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐴 / 𝐵))
Theorem	climreeq 46065	If 𝐹 is a real function, then 𝐹 converges to 𝐴 with respect to the standard topology on the reals if and only if it converges to 𝐴 with respect to the standard topology on complex numbers. In the theorem, 𝑅 is defined to be convergence w.r.t. the standard topology on the reals and then 𝐹𝑅𝐴 represents the statement "𝐹 converges to 𝐴, with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that 𝐴 is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.)
⊢ 𝑅 = (⇝𝑡‘(topGen‘ran (,)))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)    ⇒   ⊢ (𝜑 → (𝐹𝑅𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	ellimciota 46066*	An explicit value for the limit, when the limit exists at a limit point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐾)‘𝐴))    &   ⊢ (𝜑 → (𝐹 limℂ 𝐵) ≠ ∅)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    ⇒   ⊢ (𝜑 → (℩𝑥𝑥 ∈ (𝐹 limℂ 𝐵)) ∈ (𝐹 limℂ 𝐵))
Theorem	climaddf 46067*	A version of climadd 15592 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐹    &   ⊢ Ⅎ𝑘𝐺    &   ⊢ Ⅎ𝑘𝐻    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵))
Theorem	mullimc 46068*	Limit of the product of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 · 𝐶))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝐷))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐺 limℂ 𝐷))    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ (𝐻 limℂ 𝐷))
Theorem	ellimcabssub0 46069*	An equivalent condition for being a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶))    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐹 limℂ 𝐷) ↔ 0 ∈ (𝐺 limℂ 𝐷)))
Theorem	limcdm0 46070	If a function has empty domain, every complex number is a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:∅⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐹 limℂ 𝐵) = ℂ)
Theorem	islptre 46071*	An equivalence condition for a limit point w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐵 ∈ ((limPt‘𝐽)‘𝐴) ↔ ∀𝑎 ∈ ℝ* ∀𝑏 ∈ ℝ* (𝐵 ∈ (𝑎(,)𝑏) → ((𝑎(,)𝑏) ∩ (𝐴 ∖ {𝐵})) ≠ ∅)))
Theorem	limccog 46072	Limit of the composition of two functions. If the limit of 𝐹 at 𝐴 is 𝐵 and the limit of 𝐺 at 𝐵 is 𝐶, then the limit of 𝐺 ∘ 𝐹 at 𝐴 is 𝐶. With respect to limcco 25885 and limccnp 25883, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → ran 𝐹 ⊆ (dom 𝐺 ∖ {𝐵}))    &   ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐴))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐵))    ⇒   ⊢ (𝜑 → 𝐶 ∈ ((𝐺 ∘ 𝐹) limℂ 𝐴))
Theorem	limciccioolb 46073	The limit of a function at the lower bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴) = (𝐹 limℂ 𝐴))
Theorem	climf 46074*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. Similar to clim 15454, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑘𝐹    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))
Theorem	mullimcf 46075*	Limit of the multiplication of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝐴⟶ℂ)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥) · (𝐺‘𝑥)))    &   ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐷))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐺 limℂ 𝐷))    ⇒   ⊢ (𝜑 → (𝐵 · 𝐶) ∈ (𝐻 limℂ 𝐷))
Theorem	constlimc 46076*	Limit of constant function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → 𝐵 ∈ (𝐹 limℂ 𝐶))
Theorem	rexlim2d 46077*	Inference removing two restricted quantifiers. Same as rexlimdvv 3196, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))
Theorem	idlimc 46078*	Limit of the identity function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑥)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐹 limℂ 𝑋))
Theorem	divcnvg 46079*	The sequence of reciprocals of positive integers, multiplied by the factor 𝐴, converges to zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ) → (𝑛 ∈ (ℤ≥‘𝑀) ↦ (𝐴 / 𝑛)) ⇝ 0)
Theorem	limcperiod 46080*	If 𝐹 is a periodic function with period 𝑇, the limit doesn't change if we shift the limiting point by 𝑇. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:dom 𝐹⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)    &   ⊢ (𝜑 → 𝑇 ∈ ℂ)    &   ⊢ 𝐵 = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 + 𝑇)}    &   ⊢ (𝜑 → 𝐵 ⊆ dom 𝐹)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘(𝑦 + 𝑇)) = (𝐹‘𝑦))    &   ⊢ (𝜑 → 𝐶 ∈ ((𝐹 ↾ 𝐴) limℂ 𝐷))    ⇒   ⊢ (𝜑 → 𝐶 ∈ ((𝐹 ↾ 𝐵) limℂ (𝐷 + 𝑇)))
Theorem	limcrecl 46081	If 𝐹 is a real-valued function, 𝐵 is a limit point of its domain, and the limit of 𝐹 at 𝐵 exists, then this limit is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂfld))‘𝐴))    &   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))    ⇒   ⊢ (𝜑 → 𝐿 ∈ ℝ)
Theorem	sumnnodd 46082*	A series indexed by ℕ with only odd terms. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:ℕ⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ (𝑘 / 2) ∈ ℕ) → (𝐹‘𝑘) = 0)    &   ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝐵)    ⇒   ⊢ (𝜑 → (seq1( + , (𝑘 ∈ ℕ ↦ (𝐹‘((2 · 𝑘) − 1)))) ⇝ 𝐵 ∧ Σ𝑘 ∈ ℕ (𝐹‘𝑘) = Σ𝑘 ∈ ℕ (𝐹‘((2 · 𝑘) − 1))))
Theorem	lptioo2 46083	The upper bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))
Theorem	lptioo1 46084	The lower bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))
Theorem	limcmptdm 46085*	The domain of a maps-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ ℂ)
Theorem	clim2f 46086*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 15454. Similar to clim2 15464, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝑥))))
Theorem	limcicciooub 46087	The limit of a function at the upper bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵) = (𝐹 limℂ 𝐵))
Theorem	ltmod 46088	A sufficient condition for a "less than" relationship for the mod operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ((𝐴 − (𝐴 mod 𝐵))[,)𝐴))    ⇒   ⊢ (𝜑 → (𝐶 mod 𝐵) < (𝐴 mod 𝐵))
Theorem	islpcn 46089*	A characterization for a limit point for the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝑃 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂfld))‘𝑆) ↔ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒))
Theorem	lptre2pt 46090*	If a set in the real line has a limit point than it contains two distinct points that are closer than a given distance. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → ((limPt‘𝐽)‘𝐴) ≠ ∅)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (abs‘(𝑥 − 𝑦)) < 𝐸))
Theorem	limsupre 46091*	If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → sup(𝐵, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → 𝐹:𝐵⟶ℝ)    &   ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
Theorem	limcresiooub 46092	The left limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    &   ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐷 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) limℂ 𝐶) = ((𝐹 ↾ (𝐷(,)𝐶)) limℂ 𝐶))
Theorem	limcresioolb 46093	The right limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    &   ⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ≤ 𝐷)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) limℂ 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) limℂ 𝐵))
Theorem	limcleqr 46094	If the left and the right limits are equal, the limit of the function exits and the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐿 ∈ ((𝐹 ↾ (-∞(,)𝐵)) limℂ 𝐵))    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐹 ↾ (𝐵(,)+∞)) limℂ 𝐵))    &   ⊢ (𝜑 → 𝐿 = 𝑅)    ⇒   ⊢ (𝜑 → 𝐿 ∈ (𝐹 limℂ 𝐵))
Theorem	lptioo2cn 46095	The upper bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))
Theorem	lptioo1cn 46096	The lower bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐽 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ((limPt‘𝐽)‘(𝐴(,)𝐵)))
Theorem	neglimc 46097*	Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ (𝐹 limℂ 𝐷))    ⇒   ⊢ (𝜑 → -𝐶 ∈ (𝐺 limℂ 𝐷))
Theorem	addlimc 46098*	Sum of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐸 ∈ (𝐹 limℂ 𝐷))    &   ⊢ (𝜑 → 𝐼 ∈ (𝐺 limℂ 𝐷))    ⇒   ⊢ (𝜑 → (𝐸 + 𝐼) ∈ (𝐻 limℂ 𝐷))
Theorem	0ellimcdiv 46099*	If the numerator converges to 0 and the denominator converges to a nonzero number, then the fraction converges to 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ 𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐵 / 𝐶))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ (ℂ ∖ {0}))    &   ⊢ (𝜑 → 0 ∈ (𝐹 limℂ 𝐸))    &   ⊢ (𝜑 → 𝐷 ∈ (𝐺 limℂ 𝐸))    &   ⊢ (𝜑 → 𝐷 ≠ 0)    ⇒   ⊢ (𝜑 → 0 ∈ (𝐻 limℂ 𝐸))
Theorem	clim2cf 46100*	Express the predicate 𝐹 converges to 𝐴. Similar to clim2 15464, but without the disjoint var constraint 𝐹𝑘. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ Ⅎ𝑘𝐹    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝑥))