P. 155 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 15401-15500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	abssuble0d 15401	Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (𝐵 − 𝐴))
Theorem	absdifltd 15402	The absolute value of a difference and 'less than' relation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) < 𝐶 ↔ ((𝐵 − 𝐶) < 𝐴 ∧ 𝐴 < (𝐵 + 𝐶))))
Theorem	absdifled 15403	The absolute value of a difference and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((abs‘(𝐴 − 𝐵)) ≤ 𝐶 ↔ ((𝐵 − 𝐶) ≤ 𝐴 ∧ 𝐴 ≤ (𝐵 + 𝐶))))
Theorem	icodiamlt 15404	Two elements in a half-open interval have separation strictly less than the difference between the endpoints. (Contributed by Stefan O'Rear, 12-Sep-2014.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵))) → (abs‘(𝐶 − 𝐷)) < (𝐵 − 𝐴))
Theorem	abscld 15405	Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ)
Theorem	sqrtcld 15406	Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (√‘𝐴) ∈ ℂ)
Theorem	sqrtrege0d 15407	The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → 0 ≤ (ℜ‘(√‘𝐴)))
Theorem	sqsqrtd 15408	Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((√‘𝐴)↑2) = 𝐴)
Theorem	msqsqrtd 15409	Square root theorem. Theorem I.35 of [Apostol] p. 29. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((√‘𝐴) · (√‘𝐴)) = 𝐴)
Theorem	sqr00d 15410	A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (√‘𝐴) = 0)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	absvalsqd 15411	Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴)))
Theorem	absvalsq2d 15412	Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴)↑2) = (((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2)))
Theorem	absge0d 15413	Absolute value is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → 0 ≤ (abs‘𝐴))
Theorem	absval2d 15414	Value of absolute value function. Definition 10.36 of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘𝐴) = (√‘(((ℜ‘𝐴)↑2) + ((ℑ‘𝐴)↑2))))
Theorem	abs00d 15415	The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (abs‘𝐴) = 0)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	absne0d 15416	The absolute value of a number is zero iff the number is zero. Proposition 10-3.7(c) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ≠ 0)
Theorem	absrpcld 15417	The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+)
Theorem	absnegd 15418	Absolute value of negative. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘-𝐴) = (abs‘𝐴))
Theorem	abscjd 15419	The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(∗‘𝐴)) = (abs‘𝐴))
Theorem	releabsd 15420	The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℜ‘𝐴) ≤ (abs‘𝐴))
Theorem	absexpd 15421	Absolute value of positive integer exponentiation. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (abs‘(𝐴↑𝑁)) = ((abs‘𝐴)↑𝑁))
Theorem	abssubd 15422	Swapping order of subtraction doesn't change the absolute value. Example of [Apostol] p. 363. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) = (abs‘(𝐵 − 𝐴)))
Theorem	absmuld 15423	Absolute value distributes over multiplication. Proposition 10-3.7(f) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (abs‘𝐵)))
Theorem	absdivd 15424	Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
Theorem	abstrid 15425	Triangle inequality for absolute value. Proposition 10-3.7(h) of [Gleason] p. 133. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 + 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
Theorem	abs2difd 15426	Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵)))
Theorem	abs2dif2d 15427	Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘𝐴) + (abs‘𝐵)))
Theorem	abs2difabsd 15428	Absolute value of difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵)))
Theorem	abs3difd 15429	Absolute value of differences around common element. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) ≤ ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐶 − 𝐵))))
Theorem	abs3lemd 15430	Lemma involving absolute value of differences. (Contributed by Mario Carneiro, 29-May-2016.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    &   ⊢ (𝜑 → (abs‘(𝐴 − 𝐶)) < (𝐷 / 2))    &   ⊢ (𝜑 → (abs‘(𝐶 − 𝐵)) < (𝐷 / 2))    ⇒   ⊢ (𝜑 → (abs‘(𝐴 − 𝐵)) < 𝐷)
Theorem	reusq0 15431*	A complex number is the square of exactly one complex number iff the given complex number is zero. (Contributed by AV, 21-Jun-2023.)
⊢ (𝑋 ∈ ℂ → (∃!𝑥 ∈ ℂ (𝑥↑2) = 𝑋 ↔ 𝑋 = 0))
Theorem	bhmafibid1cn 15432	The Brahmagupta-Fibonacci identity for complex numbers. Express the product of two sums of two squares as a sum of two squares. First result. (Contributed by Thierry Arnoux, 1-Feb-2020.) Generalization for complex numbers proposed by GL. (Revised by AV, 8-Jun-2023.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) − (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) + (𝐵 · 𝐶))↑2)))
Theorem	bhmafibid2cn 15433	The Brahmagupta-Fibonacci identity for complex numbers. Express the product of two sums of two squares as a sum of two squares. Second result. (Contributed by Thierry Arnoux, 1-Feb-2020.) Generalization for complex numbers proposed by GL. (Revised by AV, 8-Jun-2023.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) + (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) − (𝐵 · 𝐶))↑2)))
Theorem	bhmafibid1 15434	The Brahmagupta-Fibonacci identity. Express the product of two sums of two squares as a sum of two squares. First result. Remark: The proof uses a different approach than the proof of bhmafibid1cn 15432, and is a little bit shorter. (Contributed by Thierry Arnoux, 1-Feb-2020.) (Proof modification is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) − (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) + (𝐵 · 𝐶))↑2)))
Theorem	bhmafibid2 15435	The Brahmagupta-Fibonacci identity. Express the product of two sums of two squares as a sum of two squares. Second result. (Contributed by Thierry Arnoux, 1-Feb-2020.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → (((𝐴↑2) + (𝐵↑2)) · ((𝐶↑2) + (𝐷↑2))) = ((((𝐴 · 𝐶) + (𝐵 · 𝐷))↑2) + (((𝐴 · 𝐷) − (𝐵 · 𝐶))↑2)))
5.10  Elementary limits and convergence
5.10.1  Superior limit (lim sup)
Syntax	clsp 15436	Extend class notation to include the limsup function.
class lim sup
Definition	df-limsup 15437*	Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 15440 for its value. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 11-Sep-2020.)
⊢ lim sup = (𝑥 ∈ V ↦ inf(ran (𝑘 ∈ ℝ ↦ sup(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ))
Theorem	limsupgord 15438	Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → sup(((𝐹 “ (𝐵[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ sup(((𝐹 “ (𝐴[,)+∞)) ∩ ℝ*), ℝ*, < ))
Theorem	limsupcl 15439	Closure of the superior limit. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2020.)
⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) ∈ ℝ*)
Theorem	limsupval 15440*	The superior limit of an infinite sequence 𝐹 of extended real numbers, which is the infimum of the set of suprema of all upper infinite subsequences of 𝐹. Definition 12-4.1 of [Gleason] p. 175. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 12-Sep-2014.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    ⇒   ⊢ (𝐹 ∈ 𝑉 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
Theorem	limsupgf 15441*	Closure of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    ⇒   ⊢ 𝐺:ℝ⟶ℝ*
Theorem	limsupgval 15442*	Value of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    ⇒   ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = sup(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < ))
Theorem	limsupgle 15443*	The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    ⇒   ⊢ (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ*) → ((𝐺‘𝐶) ≤ 𝐴 ↔ ∀𝑗 ∈ 𝐵 (𝐶 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)))
Theorem	limsuple 15444*	The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    ⇒   ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑗 ∈ ℝ 𝐴 ≤ (𝐺‘𝑗)))
Theorem	limsuplt 15445*	The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    ⇒   ⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*) → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑗 ∈ ℝ (𝐺‘𝑗) < 𝐴))
Theorem	limsupval2 15446*	The superior limit, relativized to an unbounded set. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞)    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) = inf((𝐺 “ 𝐴), ℝ*, < ))
Theorem	limsupgre 15447*	If a sequence of real numbers has upper bounded limit supremum, then all the partial suprema are real. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))    &   ⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℝ ∧ (lim sup‘𝐹) < +∞) → 𝐺:ℝ⟶ℝ)
Theorem	limsupbnd1 15448*	If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence 1 / 𝑛 which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴))    ⇒   ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴)
Theorem	limsupbnd2 15449*	If a sequence is eventually greater than 𝐴, then the limsup is also greater than 𝐴. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
⊢ (𝜑 → 𝐵 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → sup(𝐵, ℝ*, < ) = +∞)    &   ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)))    ⇒   ⊢ (𝜑 → 𝐴 ≤ (lim sup‘𝐹))
5.10.2  Limits
Syntax	cli 15450	Extend class notation with convergence relation for limits.
class ⇝
Syntax	crli 15451	Extend class notation with real convergence relation for limits.
class ⇝𝑟
Syntax	co1 15452	Extend class notation with the set of all eventually bounded functions.
class 𝑂(1)
Syntax	clo1 15453	Extend class notation with the set of all eventually upper bounded functions.
class ≤𝑂(1)
Definition	df-clim 15454*	Define the limit relation for complex number sequences. See clim 15460 for its relational expression. (Contributed by NM, 28-Aug-2005.)
⊢ ⇝ = {⟨𝑓, 𝑦⟩ ∣ (𝑦 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)((𝑓‘𝑘) ∈ ℂ ∧ (abs‘((𝑓‘𝑘) − 𝑦)) < 𝑥))}
Definition	df-rlim 15455*	Define the limit relation for partial functions on the reals. See rlim 15461 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ ⇝𝑟 = {⟨𝑓, 𝑥⟩ ∣ ((𝑓 ∈ (ℂ ↑pm ℝ) ∧ 𝑥 ∈ ℂ) ∧ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ ∀𝑤 ∈ dom 𝑓(𝑧 ≤ 𝑤 → (abs‘((𝑓‘𝑤) − 𝑥)) < 𝑦))}
Definition	df-o1 15456*	Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of 𝑂(1) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
Definition	df-lo1 15457*	Define the set of eventually upper bounded real functions. This fills a gap in 𝑂(1) coverage, to express statements like 𝑓(𝑥) ≤ 𝑔(𝑥) + 𝑂(𝑥) via (𝑥 ∈ ℝ+ ↦ (𝑓(𝑥) − 𝑔(𝑥)) / 𝑥) ∈ ≤𝑂(1). (Contributed by Mario Carneiro, 25-May-2016.)
⊢ ≤𝑂(1) = {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ 𝑚}
Theorem	climrel 15458	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ Rel ⇝
Theorem	rlimrel 15459	The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)
⊢ Rel ⇝𝑟
Theorem	clim 15460*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑗 such that the absolute difference of any later complex number in the sequence and the limit is less than 𝑥. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (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	rlim 15461*	Express the predicate: The limit of complex number function 𝐹 is 𝐶, or 𝐹 converges to 𝐶, in the real sense. This means that for any real 𝑥, no matter how small, there always exists a number 𝑦 such that the absolute difference of any number in the function beyond 𝑦 and the limit is less than 𝑥. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ⇝𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥))))
Theorem	rlim2 15462*	Rewrite rlim 15461 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
Theorem	rlim2lt 15463*	Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.)
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 < 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
Theorem	rlim3 15464*	Restrict the range of the domain bound to reals greater than some 𝐷 ∈ ℝ. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ (𝐷[,)+∞)∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
Theorem	climcl 15465	Closure of the limit of a sequence of complex numbers. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
Theorem	rlimpm 15466	Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ))
Theorem	rlimf 15467	Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹:dom 𝐹⟶ℂ)
Theorem	rlimss 15468	Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
Theorem	rlimcl 15469	Closure of the limit of a sequence of complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝐹 ⇝𝑟 𝐴 → 𝐴 ∈ ℂ)
Theorem	clim2 15470*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 15460. (Contributed by NM, 6-Jan-2007.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 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	clim2c 15471*	Express the predicate 𝐹 converges to 𝐴. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 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))
Theorem	clim0 15472*	Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵 ∈ ℂ ∧ (abs‘𝐵) < 𝑥)))
Theorem	clim0c 15473*	Express the predicate 𝐹 converges to 0. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘𝐵) < 𝑥))
Theorem	rlim0 15474*	Express the predicate 𝐵(𝑧) converges to 0. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
Theorem	rlim0lt 15475*	Use strictly less-than in place of less equal in the real limit predicate. (Contributed by Mario Carneiro, 18-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ)    ⇒   ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 < 𝑧 → (abs‘𝐵) < 𝑥)))
Theorem	climi 15476*	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 15477*	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 15478*	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 15479*	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 15480*	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 15481*	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 15482*	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 15483*	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 15484	An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
Theorem	lo1dm 15485	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 15486*	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 15487*	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 15488*	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 15489*	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 15490*	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 15491*	Refine o1bdd2 15507 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 15492*	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 15493*	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 15494*	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 15495	An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.)
⊢ (𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ)
Theorem	o1dm 15496	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 15497*	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 15498	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 15499*	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 15500*	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)))