P. 119 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 11801-11900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	minabs 11801	The minimum of two real numbers in terms of absolute value. (Contributed by Jim Kingdon, 15-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ, < ) = (((𝐴 + 𝐵) − (abs‘(𝐴 − 𝐵))) / 2))
Theorem	minclpr 11802	The minimum of two real numbers is one of those numbers if and only if dichotomy (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴) holds. For example, this can be combined with zletric 9523 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (inf({𝐴, 𝐵}, ℝ, < ) ∈ {𝐴, 𝐵} ↔ (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴)))
Theorem	rpmincl 11803	The minumum of two positive real numbers is a positive real number. (Contributed by Jim Kingdon, 25-Apr-2023.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ, < ) ∈ ℝ+)
Theorem	bdtrilem 11804	Lemma for bdtri 11805. (Contributed by Steven Nguyen and Jim Kingdon, 17-May-2023.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → ((abs‘(𝐴 − 𝐶)) + (abs‘(𝐵 − 𝐶))) ≤ (𝐶 + (abs‘((𝐴 + 𝐵) − 𝐶))))
Theorem	bdtri 11805	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵) ∧ 𝐶 ∈ ℝ+) → inf({(𝐴 + 𝐵), 𝐶}, ℝ, < ) ≤ (inf({𝐴, 𝐶}, ℝ, < ) + inf({𝐵, 𝐶}, ℝ, < )))
Theorem	mul0inf 11806	Equality of a product with zero. A bit of a curiosity, in the sense that theorems like abs00ap 11627 and mulap0bd 8837 may better express the ideas behind it. (Contributed by Jim Kingdon, 31-Jul-2023.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵) = 0 ↔ inf({(abs‘𝐴), (abs‘𝐵)}, ℝ, < ) = 0))
Theorem	mingeb 11807	Equivalence of ≤ and being equal to the minimum of two reals. (Contributed by Jim Kingdon, 14-Oct-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ inf({𝐴, 𝐵}, ℝ, < ) = 𝐴))
Theorem	2zinfmin 11808	Two ways to express the minimum of two integers. Because order of integers is decidable, we have more flexibility than for real numbers. (Contributed by Jim Kingdon, 14-Oct-2024.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → inf({𝐴, 𝐵}, ℝ, < ) = if(𝐴 ≤ 𝐵, 𝐴, 𝐵))
4.8.7  The maximum of two extended reals
Theorem	xrmaxleim 11809	Value of maximum when we know which extended real is larger. (Contributed by Jim Kingdon, 25-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 → sup({𝐴, 𝐵}, ℝ*, < ) = 𝐵))
Theorem	xrmaxiflemcl 11810	Lemma for xrmaxif 11816. Closure. (Contributed by Jim Kingdon, 29-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) ∈ ℝ*)
Theorem	xrmaxifle 11811	An upper bound for {𝐴, 𝐵} in the extended reals. (Contributed by Jim Kingdon, 26-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
Theorem	xrmaxiflemab 11812	Lemma for xrmaxif 11816. A variation of xrmaxleim 11809- that is, if we know which of two real numbers is larger, we know the maximum of the two. (Contributed by Jim Kingdon, 26-Apr-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = 𝐵)
Theorem	xrmaxiflemlub 11813	Lemma for xrmaxif 11816. A least upper bound for {𝐴, 𝐵}. (Contributed by Jim Kingdon, 28-Apr-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 < if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))    ⇒   ⊢ (𝜑 → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))
Theorem	xrmaxiflemcom 11814	Lemma for xrmaxif 11816. Commutativity of an expression which we will later show to be the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))) = if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, sup({𝐵, 𝐴}, ℝ, < ))))))
Theorem	xrmaxiflemval 11815*	Lemma for xrmaxif 11816. Value of the supremum. (Contributed by Jim Kingdon, 29-Apr-2023.)
⊢ 𝑀 = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < )))))    ⇒   ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝑀 ∈ ℝ* ∧ ∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝑀 < 𝑥 ∧ ∀𝑥 ∈ ℝ* (𝑥 < 𝑀 → ∃𝑧 ∈ {𝐴, 𝐵}𝑥 < 𝑧)))
Theorem	xrmaxif 11816	Maximum of two extended reals in terms of if expressions. (Contributed by Jim Kingdon, 26-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → sup({𝐴, 𝐵}, ℝ*, < ) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, 𝐴, if(𝐴 = +∞, +∞, if(𝐴 = -∞, 𝐵, sup({𝐴, 𝐵}, ℝ, < ))))))
Theorem	xrmaxcl 11817	The maximum of two extended reals is an extended real. (Contributed by Jim Kingdon, 29-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → sup({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ*)
Theorem	xrmax1sup 11818	An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ sup({𝐴, 𝐵}, ℝ*, < ))
Theorem	xrmax2sup 11819	An extended real is less than or equal to the maximum of it and another. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 30-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐵 ≤ sup({𝐴, 𝐵}, ℝ*, < ))
Theorem	xrmaxrecl 11820	The maximum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 30-Apr-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → sup({𝐴, 𝐵}, ℝ*, < ) = sup({𝐴, 𝐵}, ℝ, < ))
Theorem	xrmaxleastlt 11821	The maximum as a least upper bound, in terms of less than. (Contributed by Jim Kingdon, 9-Feb-2022.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐶 < sup({𝐴, 𝐵}, ℝ*, < ))) → (𝐶 < 𝐴 ∨ 𝐶 < 𝐵))
Theorem	xrltmaxsup 11822	The maximum as a least upper bound. (Contributed by Jim Kingdon, 10-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 < sup({𝐴, 𝐵}, ℝ*, < ) ↔ (𝐶 < 𝐴 ∨ 𝐶 < 𝐵)))
Theorem	xrmaxltsup 11823	Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 30-Apr-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) < 𝐶 ↔ (𝐴 < 𝐶 ∧ 𝐵 < 𝐶)))
Theorem	xrmaxlesup 11824	Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 10-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (sup({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ 𝐵 ≤ 𝐶)))
Theorem	xrmaxaddlem 11825	Lemma for xrmaxadd 11826. The case where 𝐴 is real. (Contributed by Jim Kingdon, 11-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → sup({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒 sup({𝐵, 𝐶}, ℝ*, < )))
Theorem	xrmaxadd 11826	Distributing addition over maximum. (Contributed by Jim Kingdon, 11-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → sup({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒 sup({𝐵, 𝐶}, ℝ*, < )))
4.8.8  The minimum of two extended reals
Theorem	xrnegiso 11827	Negation is an order anti-isomorphism of the extended reals, which is its own inverse. (Contributed by Jim Kingdon, 2-May-2023.)
⊢ 𝐹 = (𝑥 ∈ ℝ* ↦ -𝑒𝑥)    ⇒   ⊢ (𝐹 Isom < , ◡ < (ℝ*, ℝ*) ∧ ◡𝐹 = 𝐹)
Theorem	infxrnegsupex 11828*	The infimum of a set of extended reals 𝐴 is the negative of the supremum of the negatives of its elements. (Contributed by Jim Kingdon, 2-May-2023.)
⊢ (𝜑 → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ* (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))    &   ⊢ (𝜑 → 𝐴 ⊆ ℝ*)    ⇒   ⊢ (𝜑 → inf(𝐴, ℝ*, < ) = -𝑒sup({𝑧 ∈ ℝ* ∣ -𝑒𝑧 ∈ 𝐴}, ℝ*, < ))
Theorem	xrnegcon1d 11829	Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    ⇒   ⊢ (𝜑 → (-𝑒𝐴 = 𝐵 ↔ -𝑒𝐵 = 𝐴))
Theorem	xrminmax 11830	Minimum expressed in terms of maximum. (Contributed by Jim Kingdon, 2-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) = -𝑒sup({-𝑒𝐴, -𝑒𝐵}, ℝ*, < ))
Theorem	xrmincl 11831	The minumum of two extended reals is an extended real. (Contributed by Jim Kingdon, 3-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ*)
Theorem	xrmin1inf 11832	The minimum of two extended reals is less than or equal to the first. (Contributed by Jim Kingdon, 3-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐴)
Theorem	xrmin2inf 11833	The minimum of two extended reals is less than or equal to the second. (Contributed by Jim Kingdon, 3-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → inf({𝐴, 𝐵}, ℝ*, < ) ≤ 𝐵)
Theorem	xrmineqinf 11834	The minimum of two extended reals is equal to the second if the first is bigger. (Contributed by Mario Carneiro, 25-Mar-2015.) (Revised by Jim Kingdon, 3-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐵 ≤ 𝐴) → inf({𝐴, 𝐵}, ℝ*, < ) = 𝐵)
Theorem	xrltmininf 11835	Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 < inf({𝐵, 𝐶}, ℝ*, < ) ↔ (𝐴 < 𝐵 ∧ 𝐴 < 𝐶)))
Theorem	xrlemininf 11836	Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by Mario Carneiro, 18-Jun-2014.) (Revised by Jim Kingdon, 4-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ≤ inf({𝐵, 𝐶}, ℝ*, < ) ↔ (𝐴 ≤ 𝐵 ∧ 𝐴 ≤ 𝐶)))
Theorem	xrminltinf 11837	Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (inf({𝐵, 𝐶}, ℝ*, < ) < 𝐴 ↔ (𝐵 < 𝐴 ∨ 𝐶 < 𝐴)))
Theorem	xrminrecl 11838	The minimum of two real numbers is the same when taken as extended reals or as reals. (Contributed by Jim Kingdon, 18-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → inf({𝐴, 𝐵}, ℝ*, < ) = inf({𝐴, 𝐵}, ℝ, < ))
Theorem	xrminrpcl 11839	The minimum of two positive reals is a positive real. (Contributed by Jim Kingdon, 4-May-2023.)
⊢ ((𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → inf({𝐴, 𝐵}, ℝ*, < ) ∈ ℝ+)
Theorem	xrminadd 11840	Distributing addition over minimum. (Contributed by Jim Kingdon, 10-May-2023.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → inf({(𝐴 +𝑒 𝐵), (𝐴 +𝑒 𝐶)}, ℝ*, < ) = (𝐴 +𝑒 inf({𝐵, 𝐶}, ℝ*, < )))
Theorem	xrbdtri 11841	Triangle inequality for bounded values. (Contributed by Jim Kingdon, 15-May-2023.)
⊢ (((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵) ∧ (𝐶 ∈ ℝ* ∧ 0 < 𝐶)) → inf({(𝐴 +𝑒 𝐵), 𝐶}, ℝ*, < ) ≤ (inf({𝐴, 𝐶}, ℝ*, < ) +𝑒 inf({𝐵, 𝐶}, ℝ*, < )))
Theorem	iooinsup 11842	Intersection of two open intervals of extended reals. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 22-May-2023.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴(,)𝐵) ∩ (𝐶(,)𝐷)) = (sup({𝐴, 𝐶}, ℝ*, < )(,)inf({𝐵, 𝐷}, ℝ*, < )))
4.9  Elementary limits and convergence
4.9.1  Limits
Syntax	cli 11843	Extend class notation with convergence relation for limits.
class ⇝
Definition	df-clim 11844*	Define the limit relation for complex number sequences. See clim 11846 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))}
Theorem	climrel 11845	The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ Rel ⇝
Theorem	clim 11846*	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	climcl 11847	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	clim2 11848*	Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴, with more general quantifier restrictions than clim 11846. (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 11849*	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 11850*	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 11851*	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	climi 11852*	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 11853*	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 11854*	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	climconst 11855*	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	climconst2 11856	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 11857	The zero sequence converges to zero. (Contributed by NM, 2-Oct-1999.) (Revised by Mario Carneiro, 31-Jan-2014.)
⊢ (ℤ × {0}) ⇝ 0
Theorem	climuni 11858	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 11859	The limit relation is function-like, and with codomian the complex numbers. (Contributed by Mario Carneiro, 31-Jan-2014.)
⊢ ⇝ :dom ⇝ ⟶ℂ
Theorem	climdm 11860	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 11861*	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 11862*	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 11863*	An infinite sequence of complex numbers converges to at most one limit. (Contributed by Mario Carneiro, 13-Jul-2013.)
⊢ ∃*𝑥 𝐹 ⇝ 𝑥
Theorem	climeq 11864*	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	climmpt 11865*	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 11866*	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	climshftlemg 11867	A shifted function converges if the original function converges. (Contributed by Mario Carneiro, 5-Nov-2013.)
⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ 𝑉) → (𝐹 ⇝ 𝐴 → (𝐹 shift 𝑀) ⇝ 𝐴))
Theorem	climres 11868	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 11869	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 11870	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	climshft2 11871*	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	climabs0 11872*	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	climcn1 11873*	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 11874*	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 11875*	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 and df-cncf are not yet available to us. See addcncntop 15292 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 11876*	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 11877*	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	reccn2ap 11878*	The reciprocal function is continuous. The class 𝑇 is just for convenience in writing the proof and typically would be passed in as an instance of eqid 2231. (Contributed by Mario Carneiro, 9-Feb-2014.) Using apart, infimum of pair. (Revised by Jim Kingdon, 26-May-2023.)
⊢ 𝑇 = (inf({1, ((abs‘𝐴) · 𝐵)}, ℝ, < ) · ((abs‘𝐴) / 2))    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0 ∧ 𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ {𝑤 ∈ ℂ ∣ 𝑤 # 0} ((abs‘(𝑧 − 𝐴)) < 𝑦 → (abs‘((1 / 𝑧) − (1 / 𝐴))) < 𝐵))
Theorem	cn1lem 11879*	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 11880*	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 11881*	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 11882*	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 11883*	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 11884*	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 11885*	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 11886*	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 11887*	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 11888*	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	climrecl 11889*	The limit of a convergent real sequence is real. Corollary 12-2.5 of [Gleason] p. 172. (Contributed by NM, 10-Sep-2005.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℝ)
Theorem	climge0 11890*	A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.)
⊢ 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	climadd 11891*	Limit of the sum of two converging sequences. Proposition 12-2.1(a) of [Gleason] p. 168. (Contributed by NM, 24-Sep-2005.) (Proof shortened by Mario Carneiro, 31-Jan-2014.)
⊢ 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	climmul 11892*	Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
⊢ 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	climsub 11893*	Limit of the difference of two converging sequences. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by NM, 4-Aug-2007.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
⊢ 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	climaddc1 11894*	Limit of a constant 𝐶 added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) + 𝐶))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐴 + 𝐶))
Theorem	climaddc2 11895*	Limit of a constant 𝐶 added to each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 + (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐶 + 𝐴))
Theorem	climmulc2 11896*	Limit of a sequence multiplied by a constant 𝐶. Corollary 12-2.2 of [Gleason] p. 171. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 · (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐶 · 𝐴))
Theorem	climsubc1 11897*	Limit of a constant 𝐶 subtracted from each term of a sequence. (Contributed by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = ((𝐹‘𝑘) − 𝐶))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐴 − 𝐶))
Theorem	climsubc2 11898*	Limit of a constant 𝐶 minus each term of a sequence. (Contributed by NM, 24-Sep-2005.) (Revised by Mario Carneiro, 9-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐶 − (𝐹‘𝑘)))    ⇒   ⊢ (𝜑 → 𝐺 ⇝ (𝐶 − 𝐴))
Theorem	climle 11899*	Comparison of the limits of two sequences. (Contributed by Paul Chapman, 10-Sep-2007.) (Revised by Mario Carneiro, 1-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ⇝ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	climsqz 11900*	Convergence of a sequence sandwiched between another converging sequence and its limit. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ (𝐺‘𝑘))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ≤ 𝐴)    ⇒   ⊢ (𝜑 → 𝐺 ⇝ 𝐴)