P. 157 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15601-15700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	climi 15601*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘(𝐵 − 𝐴)) < 𝐶))
Theorem	climi2 15602*	Convergence of a sequence of complex numbers. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘(𝐵 − 𝐴)) < 𝐶)
Theorem	climi0 15603*	Convergence of a sequence of complex numbers to zero. (Contributed by NM, 11-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ (𝜑 → 𝐹 ⇝ 0)    ⇒   ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝐶)
Theorem	rlimi 15604*	Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 28-Feb-2015.)
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))
Theorem	rlimi2 15605*	Convergence at infinity of a function on the reals. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∃𝑦 ∈ (𝐷[,)+∞)∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑅))
Theorem	ello1 15606*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚))
Theorem	ello12 15607*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)))
Theorem	ello12r 15608*	Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (𝐹‘𝑥) ≤ 𝑀)) → 𝐹 ∈ ≤𝑂(1))
Theorem	lo1f 15609	An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
Theorem	lo1dm 15610	An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
Theorem	lo1bdd 15611*	The defining property of an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:𝐴⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
Theorem	ello1mpt 15612*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
Theorem	ello1mpt2 15613*	Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
Theorem	ello1d 15614*	Sufficient condition for elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ≤ 𝑀)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))
Theorem	lo1bdd2 15615*	If an eventually bounded function is bounded on every interval 𝐴 ∩ (-∞, 𝑦) by a function 𝑀(𝑦), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    &   ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → 𝑀 ∈ ℝ)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝐵 ≤ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚)
Theorem	lo1bddrp 15616*	Refine o1bdd2 15632 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))    &   ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → 𝑀 ∈ ℝ)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝐵 ≤ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ ℝ+ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑚)
Theorem	elo1 15617*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))
Theorem	elo12 15618*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (abs‘(𝐹‘𝑦)) ≤ 𝑚)))
Theorem	elo12r 15619*	Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1))
Theorem	o1f 15620	An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)
Theorem	o1dm 15621	An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ)
Theorem	o1bdd 15622*	The defining property of an eventually bounded function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ ((𝐹 ∈ 𝑂(1) ∧ 𝐹:𝐴⟶ℂ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (abs‘(𝐹‘𝑦)) ≤ 𝑚))
Theorem	lo1o1 15623	A function is eventually bounded iff its absolute value is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ 𝑂(1) ↔ (abs ∘ 𝐹) ∈ ≤𝑂(1)))
Theorem	lo1o12 15624*	A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about ≤𝑂(1) to 𝑂(1).) (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1)))
Theorem	elo1mpt 15625*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑚)))
Theorem	elo1mpt2 15626*	Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 12-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ ∃𝑦 ∈ (𝐶[,)+∞)∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘𝐵) ≤ 𝑚)))
Theorem	elo1d 15627*	Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → (abs‘𝐵) ≤ 𝑀)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))
Theorem	o1lo1 15628*	A real function is eventually bounded iff it is eventually lower bounded and eventually upper bounded. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ∧ (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ ≤𝑂(1))))
Theorem	o1lo12 15629*	A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ≤ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1)))
Theorem	o1lo1d 15630*	A real eventually bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))
Theorem	icco1 15631*	Derive eventual boundedness from separate upper and lower eventual bounds. (Contributed by Mario Carneiro, 15-Apr-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ∈ (𝑀[,]𝑁))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))
Theorem	o1bdd2 15632*	If an eventually bounded function is bounded on every interval 𝐴 ∩ (-∞, 𝑦) by a function 𝑀(𝑦), then the function is bounded on the whole domain. (Contributed by Mario Carneiro, 9-Apr-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    &   ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → 𝑀 ∈ ℝ)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘𝐵) ≤ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (abs‘𝐵) ≤ 𝑚)
Theorem	o1bddrp 15633*	Refine o1bdd2 15632 to give a strictly positive upper bound. (Contributed by Mario Carneiro, 25-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1))    &   ⊢ ((𝜑 ∧ (𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦)) → 𝑀 ∈ ℝ)    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ((𝑦 ∈ ℝ ∧ 𝐶 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘𝐵) ≤ 𝑀)    ⇒   ⊢ (𝜑 → ∃𝑚 ∈ ℝ+ ∀𝑥 ∈ 𝐴 (abs‘𝐵) ≤ 𝑚)
Theorem	climconst 15634*	An (eventually) constant sequence converges to its value. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
Theorem	rlimconst 15635*	A constant sequence converges to its value. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐵)
Theorem	rlimclim1 15636	Forward direction of rlimclim 15637. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐴)    &   ⊢ (𝜑 → 𝑍 ⊆ dom 𝐹)    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
Theorem	rlimclim 15637	A sequence on an upper integer set converges in the real sense iff it converges in the integer sense. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)    ⇒   ⊢ (𝜑 → (𝐹 ⇝𝑟 𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	climrlim2 15638*	Produce a real limit from an integer limit, where the real function is only dependent on the integer part of 𝑥. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝑛 = (⌊‘𝑥) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ≤ 𝑥)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
Theorem	climconst2 15639	A constant sequence converges to its value. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ (ℤ≥‘𝑀) ⊆ 𝑍    &   ⊢ 𝑍 ∈ V    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℤ) → (𝑍 × {𝐴}) ⇝ 𝐴)
Theorem	climz 15640	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ (ℤ × {0}) ⇝ 0
Theorem	rlimuni 15641	A real function whose domain is unbounded above converges to at most one limit. (Contributed by Mario Carneiro, 8-May-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐵)    &   ⊢ (𝜑 → 𝐹 ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	rlimdm 15642	Two ways to express that a function has a limit. (The expression ( ⇝𝑟 ‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 8-May-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (𝐹 ∈ dom ⇝𝑟 ↔ 𝐹 ⇝𝑟 ( ⇝𝑟 ‘𝐹)))
Theorem	climuni 15643	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 2-Oct-1999.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝐹 ⇝ 𝐴 ∧ 𝐹 ⇝ 𝐵) → 𝐴 = 𝐵)
Theorem	fclim 15644	The limit relation is function-like, and with codomain the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ⇝ :dom ⇝ ⟶ℂ
Theorem	climdm 15645	Two ways to express that a function has a limit. (The expression ( ⇝ ‘𝐹) is sometimes useful as a shorthand for "the unique limit of the function 𝐹"). (Contributed by Mario Carneiro, 18-Mar-2014.)
⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹))
Theorem	climeu 15646*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
⊢ (𝐹 ⇝ 𝐴 → ∃!𝑥 𝐹 ⇝ 𝑥)
Theorem	climreu 15647*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by NM, 25-Dec-2005.)
⊢ (𝐹 ⇝ 𝐴 → ∃!𝑥 ∈ ℂ 𝐹 ⇝ 𝑥)
Theorem	climmo 15648*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
⊢ ∃*𝑥 𝐹 ⇝ 𝑥
Theorem	rlimres 15649	The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝐹 ⇝𝑟 𝐴 → (𝐹 ↾ 𝐵) ⇝𝑟 𝐴)
Theorem	lo1res 15650	The restriction of an eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ 𝐴) ∈ ≤𝑂(1))
Theorem	o1res 15651	The restriction of an eventually bounded function is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝐹 ∈ 𝑂(1) → (𝐹 ↾ 𝐴) ∈ 𝑂(1))
Theorem	rlimres2 15652*	The restriction of a function converges if the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝𝑟 𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐷)
Theorem	lo1res2 15653*	The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ ≤𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1))
Theorem	o1res2 15654*	The restriction of a function is eventually bounded if the original is. (Contributed by Mario Carneiro, 21-May-2016.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝑂(1))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1))
Theorem	lo1resb 15655	The restriction of a function to an unbounded-above interval is eventually upper bounded iff the original is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶ℝ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ↾ (𝐵[,)+∞)) ∈ ≤𝑂(1)))
Theorem	rlimresb 15656	The restriction of a function to an unbounded-above interval converges iff the original converges. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 ↔ (𝐹 ↾ (𝐵[,)+∞)) ⇝𝑟 𝐶))
Theorem	o1resb 15657	The restriction of a function to an unbounded-above interval is eventually bounded iff the original is eventually bounded. (Contributed by Mario Carneiro, 9-Apr-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑂(1) ↔ (𝐹 ↾ (𝐵[,)+∞)) ∈ 𝑂(1)))
Theorem	climeq 15658*	Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 5-Nov-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))
Theorem	lo1eq 15659*	Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ≤ 𝑥)) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ ≤𝑂(1)))
Theorem	rlimeq 15660*	Two functions that are eventually equal to one another have the same limit. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ≤ 𝑥)) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐸 ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝐸))
Theorem	o1eq 15661*	Two functions that are eventually equal to one another are eventually bounded if one of them is. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐷 ≤ 𝑥)) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)))
Theorem	climmpt 15662*	Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))
Theorem	2clim 15663*	If two sequences converge to each other, they converge to the same limit. (Contributed by NM, 24-Dec-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)    &   ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − (𝐺‘𝑘))) < 𝑥)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    ⇒   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)
Theorem	climmpt2 15664*	Relate an integer limit on a not-quite-function to a real limit. (Contributed by Mario Carneiro, 17-Sep-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝑛 ∈ 𝑍 ↦ (𝐹‘𝑛)) ⇝𝑟 𝐴))
Theorem	climshftlem 15665	A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝑀 ∈ ℤ → (𝐹 ⇝ 𝐴 → (𝐹 shift 𝑀) ⇝ 𝐴))
Theorem	climres 15666	A function restricted to upper integers converges iff the original function converges. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → ((𝐹 ↾ (ℤ≥‘𝑀)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	climshft 15667	A shifted function converges iff the original function converges. (Contributed by NM, 16-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → ((𝐹 shift 𝑀) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴))
Theorem	serclim0 15668	The zero series converges to zero. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0)
Theorem	rlimcld2 15669*	If 𝐷 is a closed set in the topology of the complex numbers (stated here in basic form), and all the elements of the sequence lie in 𝐷, then the limit of the sequence also lies in 𝐷. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    &   ⊢ (𝜑 → 𝐷 ⊆ ℂ)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ (ℂ ∖ 𝐷)) → 𝑅 ∈ ℝ+)    &   ⊢ (((𝜑 ∧ 𝑦 ∈ (ℂ ∖ 𝐷)) ∧ 𝑧 ∈ 𝐷) → 𝑅 ≤ (abs‘(𝑧 − 𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	rlimrege0 15670*	The limit of a sequence of complex numbers with nonnegative real part has nonnegative real part. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (ℜ‘𝐵))    ⇒   ⊢ (𝜑 → 0 ≤ (ℜ‘𝐶))
Theorem	rlimrecl 15671*	The limit of a real sequence is real. (Contributed by Mario Carneiro, 9-May-2016.)
⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ℝ)
Theorem	rlimge0 15672*	The limit of a sequence of nonnegative reals is nonnegative. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → 0 ≤ 𝐶)
Theorem	climshft2 15673*	A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐾 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 𝐾)) = (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴))
Theorem	climrecl 15674*	The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	climge0 15675*	A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘))    ⇒   ⊢ (𝜑 → 0 ≤ 𝐴)
Theorem	climabs0 15676*	Convergence to zero of the absolute value is equivalent to convergence to zero. (Contributed by NM, 8-Jul-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ 𝐺 ⇝ 0))
Theorem	o1co 15677*	Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑂(1))    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ (𝐺‘𝑦)))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ 𝑂(1))
Theorem	o1compt 15678*	Sufficient condition for transforming the index set of an eventually bounded function. (Contributed by Mario Carneiro, 12-May-2016.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑂(1))    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑚 ≤ 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘ (𝑦 ∈ 𝐵 ↦ 𝐶)) ∈ 𝑂(1))
Theorem	rlimcn1 15679*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 17-Sep-2014.)
⊢ (𝜑 → 𝐺:𝐴⟶𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐺 ⇝𝑟 𝐶)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) ⇝𝑟 (𝐹‘𝐶))
Theorem	rlimcn1b 15680*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    &   ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝑋 ((abs‘(𝑧 − 𝐶)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐶))) < 𝑥))    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (𝐹‘𝐵)) ⇝𝑟 (𝐹‘𝐶))
Theorem	rlimcn3 15681*	Image of a limit under a continuous map, two-arg version. Originally a subproof of rlimcn2 15682. (Contributed by SN, 27-Sep-2024.)
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑌)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐵𝐹𝐶) ∈ ℂ)    &   ⊢ (𝜑 → (𝑅𝐹𝑆) ∈ ℂ)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝑅)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝑆)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((abs‘(𝑢 − 𝑅)) < 𝑟 ∧ (abs‘(𝑣 − 𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))    ⇒   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))
Theorem	rlimcn2 15682*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.)
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑌)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝑅)    &   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝑆)    &   ⊢ (𝜑 → 𝐹:(𝑋 × 𝑌)⟶ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((abs‘(𝑢 − 𝑅)) < 𝑟 ∧ (abs‘(𝑣 − 𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))    ⇒   ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))
Theorem	climcn1 15683*	Image of a limit under a continuous map. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐹‘𝑧) ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ 𝐵 ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) < 𝑥))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐻 ⇝ (𝐹‘𝐴))
Theorem	climcn2 15684*	Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐶 ∧ 𝑣 ∈ 𝐷)) → (𝑢𝐹𝑣) ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐻 ⇝ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ 𝐶 ∀𝑣 ∈ 𝐷 (((abs‘(𝑢 − 𝐴)) < 𝑦 ∧ (abs‘(𝑣 − 𝐵)) < 𝑧) → (abs‘((𝑢𝐹𝑣) − (𝐴𝐹𝐵))) < 𝑥))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾‘𝑘) = ((𝐺‘𝑘)𝐹(𝐻‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐾 ⇝ (𝐴𝐹𝐵))
Theorem	addcn2 15685*	Complex number addition is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (We write out the definition directly because df-cn 23458 and df-cncf 25112 are not yet available to us. See addcn 25098 for the abbreviated version.) (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 + 𝑣) − (𝐵 + 𝐶))) < 𝐴))
Theorem	subcn2 15686*	Complex number subtraction is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 − 𝑣) − (𝐵 − 𝐶))) < 𝐴))
Theorem	mulcn2 15687*	Complex number multiplication is a continuous function. Part of Proposition 14-4.16 of [Gleason] p. 243. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ∃𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑢 ∈ ℂ ∀𝑣 ∈ ℂ (((abs‘(𝑢 − 𝐵)) < 𝑦 ∧ (abs‘(𝑣 − 𝐶)) < 𝑧) → (abs‘((𝑢 · 𝑣) − (𝐵 · 𝐶))) < 𝐴))
Theorem	reccn2 15688*	The reciprocal function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.) (Revised by Mario Carneiro, 22-Sep-2014.)
⊢ 𝑇 = (if(1 ≤ ((abs‘𝐴) · 𝐵), 1, ((abs‘𝐴) · 𝐵)) · ((abs‘𝐴) / 2))    ⇒   ⊢ ((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ (ℂ ∖ {0})((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
Theorem	cn1lem 15689*	A sufficient condition for a function to be continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ 𝐹:ℂ⟶ℂ    &   ⊢ ((𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) ≤ (abs‘(𝑧 − 𝐴)))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐹‘𝑧) − (𝐹‘𝐴))) < 𝑥))
Theorem	abscn2 15690*	The absolute value function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((abs‘𝑧) − (abs‘𝐴))) < 𝑥))
Theorem	cjcn2 15691*	The complex conjugate function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((∗‘𝑧) − (∗‘𝐴))) < 𝑥))
Theorem	recn2 15692*	The real part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((ℜ‘𝑧) − (ℜ‘𝐴))) < 𝑥))
Theorem	imcn2 15693*	The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((ℑ‘𝑧) − (ℑ‘𝐴))) < 𝑥))
Theorem	climcn1lem 15694*	The limit of a continuous function, theorem form. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ 𝐻:ℂ⟶ℂ    &   ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ ℂ ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((𝐻‘𝑧) − (𝐻‘𝐴))) < 𝑥))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐻‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐻‘𝐴))
Theorem	climabs 15695*	Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (abs‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (abs‘𝐴))
Theorem	climcj 15696*	Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (∗‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (∗‘𝐴))
Theorem	climre 15697*	Limit of the real part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℜ‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (ℜ‘𝐴))
Theorem	climim 15698*	Limit of the imaginary part of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by NM, 7-Jun-2006.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (ℑ‘(𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (ℑ‘𝐴))
Theorem	rlimmptrcl 15699*	Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
Theorem	rlimabs 15700*	Limit of the absolute value of a sequence. Proposition 12-2.4(c) of [Gleason] p. 172. (Contributed by Mario Carneiro, 10-May-2016.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ (abs‘𝐵)) ⇝𝑟 (abs‘𝐶))