mmtheorems80 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 7901-8000 - Page 80 of 108

Type	Label	Description
Statement
Theorem	metcnpi2 7901	Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 7898.
⊢ X = dom dom C    &   ⊢ J = (Open ‘C)    &   ⊢ Y = dom dom D    &   ⊢ K = (Open ‘D)    ⇒   ⊢ (((C ∈ Met ⋀ D ∈ Met ⋀ P ∈ X) ⋀ (F ∈ ((J CnP K) ‘P) ⋀ A ∈ ℝ ⋀ 0 < A)) → ∃x ∈ ℝ (0 < x ⋀ ∀y ∈ X ((yCP) < x → ((F ‘y)D(F ‘P)) < A)))
Theorem	metcnpi3 7902	Epsilon-delta property of a metric space function continuous at P. A variation of metcnpi2 7901 with non-strict ordering.
⊢ X = dom dom C    &   ⊢ J = (Open ‘C)    &   ⊢ Y = dom dom D    &   ⊢ K = (Open ‘D)    ⇒   ⊢ (((C ∈ Met ⋀ D ∈ Met ⋀ P ∈ X) ⋀ (F ∈ ((J CnP K) ‘P) ⋀ A ∈ ℝ ⋀ 0 < A)) → ∃x ∈ ℝ (0 < x ⋀ ∀y ∈ X ((yCP) ≤ x → ((F ‘y)D(F ‘P)) ≤ A)))
Theorem	metcnpi4 7903	Epsilon-delta property of a metric space function continuous at P. A variation of metcnpi 7900 with non-strict ordering.
⊢ X = dom dom C    &   ⊢ J = (Open ‘C)    &   ⊢ Y = dom dom D    &   ⊢ K = (Open ‘D)    ⇒   ⊢ (((C ∈ Met ⋀ D ∈ Met ⋀ P ∈ X) ⋀ (F ∈ ((J CnP K) ‘P) ⋀ A ∈ ℝ ⋀ 0 < A)) → ∃x ∈ ℝ (0 < x ⋀ ∀y ∈ X ((PCy) ≤ x → ((F ‘P)D(F ‘y)) ≤ A)))
Theorem	metcni 7904	Epsilon-delta property of a continuous metric space function.
⊢ X = dom dom C    &   ⊢ J = (Open ‘C)    &   ⊢ Y = dom dom D    &   ⊢ K = (Open ‘D)    ⇒   ⊢ (((C ∈ Met ⋀ D ∈ Met ⋀ F ∈ (J Cn K)) ⋀ (P ∈ X ⋀ A ∈ ℝ ⋀ 0 < A)) → ∃x ∈ ℝ (0 < x ⋀ ∀y ∈ X ((PCy) < x → ((F ‘P)D(F ‘y)) < A)))
Theorem	metcni2 7905	Epsilon-delta property of a continuous metric space function.
⊢ X = dom dom C    &   ⊢ J = (Open ‘C)    &   ⊢ Y = dom dom D    &   ⊢ K = (Open ‘D)    ⇒   ⊢ (((C ∈ Met ⋀ D ∈ Met ⋀ F ∈ (J Cn K)) ⋀ (P ∈ X ⋀ A ∈ ℝ ⋀ 0 < A)) → ∃x ∈ ℝ (0 < x ⋀ ∀y ∈ X ((PCy) ≤ x → ((F ‘P)D(F ‘y)) ≤ A)))
Theorem	metcnp3 7906	Two ways to express that F is continuous at P for metric spaces. Proposition 14-4.2 of [Gleason] p. 240.
⊢ X = dom dom C    &   ⊢ J = (Open ‘C)    &   ⊢ Y = dom dom D    &   ⊢ K = (Open ‘D)    ⇒   ⊢ ((C ∈ Met ⋀ D ∈ Met ⋀ P ∈ X) → (F ∈ ((J CnP K) ‘P) ↔ (F:X–→Y ⋀ ∀y ∈ ℝ (0 < y → ∃z ∈ ℝ (0 < z ⋀ (F “ (P( ball ‘C)z)) ⊆ ((F ‘P)( ball ‘D)y))))))
Theorem	metcnco 7907	Composition of two continuous functions (metric space version of cnco 7778).
⊢ J = (Open ‘B)    &   ⊢ K = (Open ‘C)    &   ⊢ L = (Open ‘D)    ⇒   ⊢ (((B ∈ Met ⋀ C ∈ Met ⋀ D ∈ Met) ⋀ (F ∈ (J Cn K) ⋀ G ∈ (K Cn L))) → (G ∘ F) ∈ (J Cn L))
Theorem	metcnss 7908	Subset relationship for continuity of metric spaces.
⊢ J = (Open ‘B)    &   ⊢ K = (Open ‘C)    &   ⊢ L = (Open ‘D)    ⇒   ⊢ (((B ∈ Met ⋀ C ∈ Met ⋀ D ∈ Met) ⋀ C ⊆ D) → (J Cn K) ⊆ (J Cn L))
Theorem	metcnss2 7909	Subset relationship for continuity of metric spaces.
⊢ X = dom dom B    &   ⊢ J = (Open ‘B)    &   ⊢ K = (Open ‘C)    &   ⊢ L = (Open ‘D)    ⇒   ⊢ (((B ∈ Met ⋀ C ∈ Met ⋀ D ∈ Met) ⋀ (B ⊆ C ⋀ F ∈ (K Cn L))) → (F ↾ X) ∈ (J Cn L))
Theorem	metidcn 7910	The identity function is continuous (metric space version of idcn 7776).
⊢ X = dom dom C    &   ⊢ J = (Open ‘C)    &   ⊢ K = (Open ‘D)    ⇒   ⊢ ((C ∈ Met ⋀ D ∈ Met ⋀ C ⊆ D) → (I ↾ X) ∈ (J Cn K))
Theorem	metdnsconst 7911	If a continuous mapping to a metric space is constant on a dense subset, it is constant on the entire space (metric space version of dnsconst 7798).
⊢ X = dom dom C    &   ⊢ Y = dom dom D    &   ⊢ J = (Open ‘C)    &   ⊢ K = (Open ‘D)    ⇒   ⊢ (((C ∈ Met ⋀ D ∈ Met ⋀ F ∈ (J Cn K)) ⋀ (P ∈ Y ⋀ A ⊆ (◡F “ {P}) ⋀ ((cls ‘J) ‘A) = X)) → F:X–→{P})
Examples of metric spaces
Theorem	cnmetdval 7912	Value of the distance function of the metric space of complex numbers.
⊢ D = (abs ∘ − )    ⇒   ⊢ ((A ∈ ℂ ⋀ B ∈ ℂ) → (ADB) = (abs ‘(A − B)))
Theorem	cnmetba 7913	The base set of the metric for complex numbers.
⊢ D = (abs ∘ − )    ⇒   ⊢ ℂ = dom dom D
Theorem	cnmet 7914	The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
⊢ D = (abs ∘ − )    ⇒   ⊢ D ∈ Met
Theorem	cncfmet 7915	Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.)
⊢ C = ((abs ∘ − ) ↾ (A × A))    &   ⊢ D = ((abs ∘ − ) ↾ (B × B))    &   ⊢ J = (Open ‘C)    &   ⊢ K = (Open ‘D)    ⇒   ⊢ ((A ⊆ ℂ ⋀ B ⊆ ℂ) → (A–cn→B) = (J Cn K))
Theorem	cncfmet1 7916	Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 28-Nov-2007.)
⊢ D = (abs ∘ − )    &   ⊢ J = (Open ‘D)    ⇒   ⊢ (ℂ–cn→ℂ) = (J Cn J)
Theorem	cn2met 7917	The standard metric space on ℂ × ℂ.
⊢ C = (abs ∘ − )    &   ⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℂ × ℂ) ⋀ y ∈ (ℂ × ℂ)) ⋀ z = sup({((1st ‘x)C(1st ‘y)), ((2nd ‘x)C(2nd ‘y))}, ℝ, < ))}    ⇒   ⊢ D ∈ Met
Theorem	remetdval 7918	Value of the distance function of the metric space of real numbers.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ) → (ADB) = (abs ‘(A − B)))
Theorem	remetba 7919	The base set for the metric for real numbers.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ℝ = dom dom D
Theorem	remet 7920	The absolute value metric determines a metric space on the reals.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ D ∈ Met
Theorem	bl2ioo 7921	A ball in terms of an open interval of reals.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ ⋀ 0 < B) → (A( ball ‘D)B) = ((A − B)(,)(A + B)))
Theorem	ioo2bl 7922	An open interval of reals in terms of a ball.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((A ∈ ℝ ⋀ B ∈ ℝ ⋀ A < B) → (A(,)B) = (((A + B) / 2)( ball ‘D)((B − A) / 2)))
Theorem	blssioo 7923	The balls of the standard real metric space are included in the open real intervals.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ran ( ball ‘D) ⊆ ran (,)
Theorem	tgioolem 7924	Lemma for tgioo 7925. An open interval includes a ball around any of its points. Warning: The HTML proof page is 0.6MB in size.
Theorem	tgioo 7925	The topology generated by open intervals of reals is the same as the open sets of the standard metric space on the reals.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ J = (Open ‘D)    ⇒   ⊢ (topGen ‘ran (,)) = J
Theorem	qdensere2 7926	ℚ is dense in ℝ.
⊢ D = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   ⊢ J = (Open ‘D)    ⇒   ⊢ ((cls ‘J) ‘ℚ) = ℝ
Theorem	rehaus 7927	The standard topology on the reals is Hausdorff.
⊢ (topGen ‘ran (,)) ∈ Haus
Theorem	dscmet 7928	The discrete metric on any set X. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
⊢ X ∈ V    &   ⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ X ⋀ y ∈ X) ⋀ z = if(x = y, 0, 1))}    ⇒   ⊢ D ∈ Met
Convergence and completeness
Syntax	clm 7929	Extend class notation with a function on metric spaces whose value is the convergence relation for limit sequences in the space.
class ⇝m
Syntax	cca 7930	Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
class Cau
Syntax	cms 7931	Extend class notation with class of complete metric spaces.
class CMet
Definition	df-lm 7932	Define a function on metric spaces whose value is the convergence relation for the space. Although f is typically a function from upper integers to the metric space, it doesn't have to be. Unfortunately, the expression after "w =" must exist to use fvopab4 3789, and we use the otherwise unnecessary conjunct f ⊆ (ℂ × dom dom z) to ensure that. This could be changed to the more liberal (but more complex) f ⊆ (ℂ × (dom dom z ∪ {∅})) if we want to allow for functions with undefined values.
⊢ ⇝m = {⟨z, w⟩∣(z ∈ Met ⋀ w = {⟨f, y⟩∣(f ⊆ (ℂ × dom dom z) ⋀ y ∈ dom dom z ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ ℤ ∀k ∈ ℤ (j ≤ k → ((f ‘k) ∈ dom dom z ⋀ ((f ‘k)Dy) < x))))})}
Definition	df-cau 7933	Define a function on metric spaces whose value is the set of Cauchy sequences of the space.
⊢ Cau = {⟨z, w⟩∣(z ∈ Met ⋀ w = {f∣(f ⊆ (ℂ × dom dom z) ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ ℤ ∀k ∈ ℤ ∀m ∈ ℤ ((j ≤ k ⋀ j ≤ m) → ((f ‘k) ∈ dom dom z ⋀ (f ‘m) ∈ dom dom z ⋀ ((f ‘k)D(f ‘m)) < x))))})}
Definition	df-cmet 7934	Define the class of complete metrics.
⊢ CMet = {x ∈ Met∣∀f ∈ (Cau ‘x)∃y ∈ dom dom x f(⇝m ‘x)y}
Theorem	lmfval 7935	The relation "sequence f converges to point y" in a metric space.
⊢ X = dom dom D    ⇒   ⊢ (D ∈ Met → (⇝m ‘D) = {⟨f, y⟩∣(f ⊆ (ℂ × X) ⋀ y ∈ X ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ ℤ ∀k ∈ ℤ (j ≤ k → ((f ‘k) ∈ X ⋀ ((f ‘k)Dy) < x))))})
Theorem	caufval 7936	The set of Cauchy sequences on a metric space.
⊢ X = dom dom D    ⇒   ⊢ (D ∈ Met → (Cau ‘D) = {f∣(f ⊆ (ℂ × X) ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ ℤ ∀k ∈ ℤ ∀m ∈ ℤ ((j ≤ k ⋀ j ≤ m) → ((f ‘k) ∈ X ⋀ (f ‘m) ∈ X ⋀ ((f ‘k)D(f ‘m)) < x))))})
Theorem	lmrel 7937	The metric space convergence relation is a relation.
⊢ (D ∈ Met → Rel (⇝m ‘D))
Theorem	lmbr 7938	Express the binary relation "sequence F converges to point P " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition F ⊆ (ℂ × X) allows us to use objects more general than sequences when convenient; see the comment in df-lm 7932.
⊢ X = dom dom D    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ A) → (F(⇝m ‘D)P ↔ (F ⊆ (ℂ × X) ⋀ P ∈ X ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ ℤ ∀k ∈ ℤ (j ≤ k → ((F ‘k) ∈ X ⋀ ((F ‘k)DP) < x))))))
Theorem	lmbr2 7939	Express the binary relation "sequence F converges to point P " in a metric space using an abitrary set of upper integers.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ A) → (F(⇝m ‘D)P ↔ (F ⊆ (ℂ × X) ⋀ P ∈ X ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ Z ∀k ∈ Z (j ≤ k → ((F ‘k) ∈ X ⋀ ((F ‘k)DP) < x))))))
Theorem	lmbrf 7940	Express the binary relation "sequence F converges to point P " in a metric space using an abitrary set of upper integers. This version of lmbr2 7939 presupposes that F is a function.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ A ⋀ F:Z–→X) → (F(⇝m ‘D)P ↔ (P ∈ X ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ Z ∀k ∈ Z (j ≤ k → ((F ‘k)DP) < x)))))
Theorem	lmbrf2 7941	Express the binary relation "sequence F converges to point P " in a metric space using an abitrary set of upper integers. This version of lmbrf 7940 presupposes that the convergence point is in the metric space.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ X ⋀ F:Z–→X) → (F(⇝m ‘D)P ↔ ∀x ∈ ℝ (0 < x → ∃j ∈ Z ∀k ∈ Z (j ≤ k → ((F ‘k)DP) < x))))
Theorem	lmcvg 7942	Convergence property of a converging sequence.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ (((D ∈ Met ⋀ P ∈ A ⋀ F(⇝m ‘D)P) ⋀ (R ∈ ℝ ⋀ 0 < R)) → ∃j ∈ Z ∀k ∈ Z (j ≤ k → ((F ‘k) ∈ X ⋀ ((F ‘k)DP) < R)))
Theorem	lmcvg2 7943	Convergence property of a converging sequence.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ (((D ∈ Met ⋀ P ∈ A ⋀ F(⇝m ‘D)P) ⋀ (R ∈ ℝ ⋀ 0 < R)) → ∃j ∈ Z ∀k ∈ Z (j ≤ k → ((F ‘k)DP) < R))
Theorem	lmconst 7944	A constant sequence converges to its value.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ X) → (Z × {P})(⇝m ‘D)P)
Theorem	lmnn 7945	A condition that implies convergence.
⊢ X = dom dom D    ⇒   ⊢ (((D ∈ Met ⋀ P ∈ X) ⋀ (F:ℕ–→X ⋀ ∀k ∈ ℕ ((F ‘k)DP) < (1 / k))) → F(⇝m ‘D)P)
Theorem	iscau 7946	Express the property "F is a Cauchy sequence of metric D." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition F ⊆ (ℂ × X) allows us to use objects more general than sequences when convenient; see the comment in df-lm 7932.
⊢ X = dom dom D    ⇒   ⊢ (D ∈ Met → (F ∈ (Cau ‘D) ↔ (F ⊆ (ℂ × X) ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ ℤ ∀k ∈ ℤ ∀m ∈ ℤ ((j ≤ k ⋀ j ≤ m) → ((F ‘k) ∈ X ⋀ (F ‘m) ∈ X ⋀ ((F ‘k)D(F ‘m)) < x))))))
Theorem	iscau2 7947	Express the property "F is a Cauchy sequence of metric D," using an abitrary set of upper integers.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ (D ∈ Met → (F ∈ (Cau ‘D) ↔ (F ⊆ (ℂ × X) ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ Z ∀k ∈ Z ∀m ∈ Z ((j ≤ k ⋀ j ≤ m) → ((F ‘k) ∈ X ⋀ (F ‘m) ∈ X ⋀ ((F ‘k)D(F ‘m)) < x))))))
Theorem	iscau3 7948	Express the property "F is a Cauchy sequence of metric D " with one less quantifier.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ (D ∈ Met → (F ∈ (Cau ‘D) ↔ (F ⊆ (ℂ × X) ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ Z ∀k ∈ Z (j ≤ k → ((F ‘j) ∈ X ⋀ (F ‘k) ∈ X ⋀ ((F ‘j)D(F ‘k)) < x))))))
Theorem	iscauf 7949	Express the property "F is a Cauchy sequence of metric D " presupposing F is a function.
⊢ X = dom dom D    &   ⊢ N ∈ ℤ    &   ⊢ Z = (ℤ≥ ‘N)    ⇒   ⊢ ((D ∈ Met ⋀ F:Z–→X) → (F ∈ (Cau ‘D) ↔ ∀x ∈ ℝ (0 < x → ∃j ∈ Z ∀k ∈ Z (j ≤ k → ((F ‘j)D(F ‘k)) < x))))
Theorem	iscau4 7950	Express the property "F is a Cauchy sequence of metric D."
⊢ X = dom dom D    ⇒   ⊢ (D ∈ Met → (F ∈ (Cau ‘D) ↔ (F ⊆ (ℂ × X) ⋀ ∀x ∈ ℝ (0 < x → ∃j ∈ ℤ ∀k ∈ ℤ (j ≤ k → ((F ‘j) ∈ X ⋀ (F ‘k) ∈ X ⋀ ((F ‘j)D(F ‘k)) < x))))))
Theorem	iscau5 7951	Express the property "F is a Cauchy sequence of metric D."
⊢ X = dom dom D    ⇒   ⊢ ((D ∈ Met ⋀ F:ℕ–→X) → (F ∈ (Cau ‘D) ↔ ∀x ∈ ℝ+ ∃j ∈ ℕ ∀k ∈ ℕ (j ≤ k → ((F ‘j)D(F ‘k)) < x)))
Theorem	lmbrnns 7952	Express the binary relation "sequence F converges to point P " in a metric space."
⊢ X = dom dom D    &   ⊢ (k ∈ ℕ → A = (F ‘k))    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ X ⋀ F:ℕ–→X) → (F(⇝m ‘D)P ↔ ∀x ∈ ℝ+ ∃j ∈ ℕ ∀k ∈ ℕ (j ≤ k → (ADP) < x)))
Theorem	lmcvgnns 7953	Convergence property of a converging sequence.
⊢ X = dom dom D    &   ⊢ (k ∈ ℕ → A = (F ‘k))    ⇒   ⊢ (((D ∈ Met ⋀ P ∈ B) ⋀ (F(⇝m ‘D)P ⋀ R ∈ ℝ+)) → ∃j ∈ ℕ ∀k ∈ ℕ (j ≤ k → (ADP) < R))
Theorem	iscaunns 7954	Express the property "F is a Cauchy sequence of metric D."
⊢ X = dom dom D    &   ⊢ (k ∈ ℕ → A = (F ‘k))    ⇒   ⊢ ((D ∈ Met ⋀ F:ℕ–→X) → (F ∈ (Cau ‘D) ↔ ∀x ∈ ℝ+ ∃j ∈ ℕ ∀k ∈ ℕ (j ≤ k → ([j / k]ADA) < x)))
Theorem	caun0 7955	A metric with a Cauchy sequence cannot be empty.
⊢ X = dom dom D    ⇒   ⊢ ((D ∈ Met ⋀ F ∈ (Cau ‘D)) → X ≠ ∅)
Theorem	iscms 7956	The property "D is a complete metric," meaning all Cauchy sequences converge to a point in the space. Part of Definition 1.4-3 of [Kreyszig] p. 28.
⊢ X = dom dom D    ⇒   ⊢ (D ∈ CMet ↔ (D ∈ Met ⋀ ∀f ∈ (Cau ‘D)∃x ∈ X f(⇝m ‘D)x))
Theorem	cmscvg 7957	The convergence of a Cauchy sequence in a complete metric space.
⊢ X = dom dom D    ⇒   ⊢ ((D ∈ CMet ⋀ F ∈ (Cau ‘D)) → ∃x ∈ X F(⇝m ‘D)x)
Theorem	lmfss 7958	Inclusion of a function having a limit (used to ensure the limit relation is a set, under our definition).
⊢ X = dom dom D    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ A ⋀ F(⇝m ‘D)P) → F ⊆ (ℂ × X))
Theorem	lmcl 7959	Closure of a limit.
⊢ X = dom dom D    ⇒   ⊢ ((D ∈ Met ⋀ P ∈ A ⋀ F(⇝m ‘D)P) → P ∈ X)
Theorem	caufss 7960	Inclusion of a Cauchy sequence, under our definition.
⊢ X = dom dom D    ⇒   ⊢ ((D ∈ Met ⋀ F ∈ (Cau ‘D)) → F ⊆ (ℂ × X))
Theorem	lmuni 7961	A sequence converges to at most one limit. Part of Lemma 1.4-2(a) of [Kreyszig] p. 26.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ((D ∈ Met ⋀ F(⇝m ‘D)A ⋀ F(⇝m ‘D)B) → A = B)
Theorem	lmsslem 7962	Lemma for lmss 7963 and causs 7965.
Theorem	lmss 7963	Limit on a metric subspace.
⊢ ((D ∈ Met ⋀ P ∈ Y