P. 319 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rge0scvg 31801	Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 16550. (Contributed by Thierry Arnoux, 28-Jul-2017.)
⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → sup(ran seq1( + , 𝐹), ℝ, < ) ∈ ℝ)
Theorem	fsumcvg4 31802	A serie with finite support is a finite sum, and therefore converges. (Contributed by Thierry Arnoux, 6-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝑆 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑆⟶ℂ)    &   ⊢ (𝜑 → (◡𝐹 “ (ℂ ∖ {0})) ∈ Fin)    ⇒   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
Theorem	pnfneige0 31803*	A neighborhood of +∞ contains an unbounded interval based at a real number. See pnfnei 22279. (Contributed by Thierry Arnoux, 31-Jul-2017.)
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞)))    ⇒   ⊢ ((𝐴 ∈ 𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
Theorem	lmxrge0 31804*	Express "sequence 𝐹 converges to plus infinity" (i.e. diverges), for a sequence of nonnegative extended real numbers. (Contributed by Thierry Arnoux, 2-Aug-2017.)
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞)))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = 𝐴)    ⇒   ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < 𝐴))
Theorem	lmdvg 31805*	If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.)
⊢ (𝜑 → 𝐹:ℕ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   ⊢ (𝜑 → ¬ 𝐹 ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘))
Theorem	lmdvglim 31806*	If a monotonic real number sequence 𝐹 diverges, it converges in the extended real numbers and its limit is plus infinity. (Contributed by Thierry Arnoux, 3-Aug-2017.)
⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞)))    &   ⊢ (𝜑 → 𝐹:ℕ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   ⊢ (𝜑 → ¬ 𝐹 ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)+∞)
20.3.12.16  Univariate polynomials
Theorem	pl1cn 31807	A univariate polynomial is continuous. (Contributed by Thierry Arnoux, 17-Sep-2018.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐸 = (eval1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ TopRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐸‘𝐹) ∈ (𝐽 Cn 𝐽))
20.3.13  Uniform Stuctures and Spaces
20.3.13.1  Hausdorff uniform completion
Syntax	chcmp 31808	Extend class notation with the Hausdorff uniform completion relation.
class HCmp
Definition	df-hcmp 31809*	Definition of the Hausdorff completion. In this definition, a structure 𝑤 is a Hausdorff completion of a uniform structure 𝑢 if 𝑤 is a complete uniform space, in which 𝑢 is dense, and which admits the same uniform structure. Theorem 3 of [BourbakiTop1] p. II.21. states the existence and uniqueness of such a completion. (Contributed by Thierry Arnoux, 5-Mar-2018.)
⊢ HCmp = {⟨𝑢, 𝑤⟩ ∣ ((𝑢 ∈ ∪ ran UnifOn ∧ 𝑤 ∈ CUnifSp) ∧ ((UnifSt‘𝑤) ↾t dom ∪ 𝑢) = 𝑢 ∧ ((cls‘(TopOpen‘𝑤))‘dom ∪ 𝑢) = (Base‘𝑤))}
20.3.14  Topology and algebraic structures
20.3.14.1  The norm on the ring of the integer numbers
Theorem	zringnm 31810	The norm (function) for a ring of integers is the absolute value function (restricted to the integers). (Contributed by AV, 13-Jun-2019.)
⊢ (norm‘ℤring) = (abs ↾ ℤ)
Theorem	zzsnm 31811	The norm of the ring of the integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 13-Jun-2019.)
⊢ (𝑀 ∈ ℤ → (abs‘𝑀) = ((norm‘ℤring)‘𝑀))
20.3.14.2  Topological ` ZZ ` -modules
Theorem	zlm0 31812	Zero of a ℤ-module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ 0 = (0g‘𝑊)
Theorem	zlm1 31813	Unit of a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 1 = (1r‘𝐺)    ⇒   ⊢ 1 = (1r‘𝑊)
Theorem	zlmds 31814	Distance in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
Theorem	zlmtset 31815	Topology in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐽 = (TopSet‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))
Theorem	zlmnm 31816	Norm of a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝑁 = (norm‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝑁 = (norm‘𝑊))
Theorem	zhmnrg 31817	The ℤ-module built from a normed ring is also a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ NrmRing → 𝑊 ∈ NrmRing)
Theorem	nmmulg 31818	The norm of a group product, provided the ℤ-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑁 = (norm‘𝑅)    &   ⊢ 𝑍 = (ℤMod‘𝑅)    &   ⊢ · = (.g‘𝑅)    ⇒   ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁‘𝑋)))
Theorem	zrhnm 31819	The norm of the image by ℤRHom of an integer in a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑁 = (norm‘𝑅)    &   ⊢ 𝑍 = (ℤMod‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (abs‘𝑀))
Theorem	cnzh 31820	The ℤ-module of ℂ is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
⊢ (ℤMod‘ℂfld) ∈ NrmMod
Theorem	rezh 31821	The ℤ-module of ℝ is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)
⊢ (ℤMod‘ℝfld) ∈ NrmMod
20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring
Syntax	cqqh 31822	Map the rationals into a field.
class ℚHom
Definition	df-qqh 31823*	Define the canonical homomorphism from the rationals into any field. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
⊢ ℚHom = (𝑟 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡(ℤRHom‘𝑟) “ (Unit‘𝑟)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑟)‘𝑥)(/r‘𝑟)((ℤRHom‘𝑟)‘𝑦))⟩))
Theorem	qqhval 31824*	Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ / = (/r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
Theorem	zrhf1ker 31825	The kernel of the homomorphism from the integers to a ring, if it is injective. (Contributed by Thierry Arnoux, 26-Oct-2017.) (Revised by Thierry Arnoux, 23-May-2023.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐿:ℤ–1-1→𝐵 ↔ (◡𝐿 “ { 0 }) = {0}))
Theorem	zrhchr 31826	The kernel of the homomorphism from the integers to a ring is injective if and only if the ring has characteristic 0 . (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ 𝐿:ℤ–1-1→𝐵))
Theorem	zrhker 31827	The kernel of the homomorphism from the integers to a ring with characteristic 0. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ (◡𝐿 “ { 0 }) = {0}))
Theorem	zrhunitpreima 31828	The preimage by ℤRHom of the unit of a division ring is (ℤ ∖ {0}). (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0}))
Theorem	elzrhunit 31829	Condition for the image by ℤRHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝐿‘𝑀) ∈ (Unit‘𝑅))
Theorem	elzdif0 31830	Lemma for qqhval2 31832. (Contributed by Thierry Arnoux, 29-Oct-2017.)
⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))
Theorem	qqhval2lem 31831	Lemma for qqhval2 31832. (Contributed by Thierry Arnoux, 29-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0)) → ((𝐿‘(numer‘(𝑋 / 𝑌))) / (𝐿‘(denom‘(𝑋 / 𝑌)))) = ((𝐿‘𝑋) / (𝐿‘𝑌)))
Theorem	qqhval2 31832*	Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))
Theorem	qqhvval 31833	Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 𝑄 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑄) = ((𝐿‘(numer‘𝑄)) / (𝐿‘(denom‘𝑄))))
Theorem	qqh0 31834	The image of 0 by the ℚHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g‘𝑅))
Theorem	qqh1 31835	The image of 1 by the ℚHom homomorphism is the ring's unit. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1r‘𝑅))
Theorem	qqhf 31836	ℚHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)
Theorem	qqhvq 31837	The image of a quotient by the ℚHom homomorphism. (Contributed by Thierry Arnoux, 28-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0)) → ((ℚHom‘𝑅)‘(𝑋 / 𝑌)) = ((𝐿‘𝑋) / (𝐿‘𝑌)))
Theorem	qqhghm 31838	The ℚHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))
Theorem	qqhrhm 31839	The ℚHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝐿 = (ℤRHom‘𝑅)    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    ⇒   ⊢ ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅))
Theorem	qqhnm 31840	The norm of the image by ℚHom of a rational number in a topological division ring. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑁 = (norm‘𝑅)    &   ⊢ 𝑍 = (ℤMod‘𝑅)    ⇒   ⊢ (((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ 𝑄 ∈ ℚ) → (𝑁‘((ℚHom‘𝑅)‘𝑄)) = (abs‘𝑄))
Theorem	qqhcn 31841	The ℚHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ 𝐽 = (TopOpen‘𝑄)    &   ⊢ 𝑍 = (ℤMod‘𝑅)    &   ⊢ 𝐾 = (TopOpen‘𝑅)    ⇒   ⊢ ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾))
Theorem	qqhucn 31842	The ℚHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ 𝑈 = (UnifSt‘𝑄)    &   ⊢ 𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))    &   ⊢ 𝑍 = (ℤMod‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NrmRing)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑍 ∈ NrmMod)    &   ⊢ (𝜑 → (chr‘𝑅) = 0)    ⇒   ⊢ (𝜑 → (ℚHom‘𝑅) ∈ (𝑈 Cnu𝑉))
20.3.14.4  Canonical embedding of the real numbers into a complete ordered field
Syntax	crrh 31843	Map the real numbers into a complete field.
class ℝHom
Syntax	crrext 31844	Extend class notation with the class of extension fields of ℝ.
class ℝExt
Definition	df-rrh 31845	Define the canonical homomorphism from the real numbers to any complete field, as the extension by continuity of the canonical homomorphism from the rational numbers. (Contributed by Mario Carneiro, 22-Oct-2017.) (Revised by Thierry Arnoux, 23-Oct-2017.)
⊢ ℝHom = (𝑟 ∈ V ↦ (((topGen‘ran (,))CnExt(TopOpen‘𝑟))‘(ℚHom‘𝑟)))
Theorem	rrhval 31846	Value of the canonical homormorphism from the real numbers to a complete space. (Contributed by Thierry Arnoux, 2-Nov-2017.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐾 = (TopOpen‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))
Theorem	rrhcn 31847	If the topology of 𝑅 is Hausdorff, and 𝑅 is a complete uniform space, then the canonical homomorphism from the real numbers to 𝑅 is continuous. (Contributed by Thierry Arnoux, 17-Jan-2018.)
⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (TopOpen‘𝑅)    &   ⊢ 𝑍 = (ℤMod‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑅 ∈ NrmRing)    &   ⊢ (𝜑 → 𝑍 ∈ NrmMod)    &   ⊢ (𝜑 → (chr‘𝑅) = 0)    &   ⊢ (𝜑 → 𝑅 ∈ CUnifSp)    &   ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷))    ⇒   ⊢ (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))
Theorem	rrhf 31848	If the topology of 𝑅 is Hausdorff, Cauchy sequences have at most one limit, i.e. the canonical homomorphism of ℝ into 𝑅 is a function. (Contributed by Thierry Arnoux, 2-Nov-2017.)
⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))    &   ⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (TopOpen‘𝑅)    &   ⊢ 𝑍 = (ℤMod‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑅 ∈ NrmRing)    &   ⊢ (𝜑 → 𝑍 ∈ NrmMod)    &   ⊢ (𝜑 → (chr‘𝑅) = 0)    &   ⊢ (𝜑 → 𝑅 ∈ CUnifSp)    &   ⊢ (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷))    ⇒   ⊢ (𝜑 → (ℝHom‘𝑅):ℝ⟶𝐵)
Definition	df-rrext 31849	Define the class of extensions of ℝ. This is a shorthand for listing the necessary conditions for a structure to admit a canonical embedding of ℝ into it. Interestingly, this is not coming from a mathematical reference, but was from the necessary conditions to build the embedding at each step (ℤ, ℚ and ℝ). It would be interesting see if this is formally treated in the literature. See isrrext 31850 for a better readable version. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ ℝExt = {𝑟 ∈ (NrmRing ∩ DivRing) ∣ (((ℤMod‘𝑟) ∈ NrmMod ∧ (chr‘𝑟) = 0) ∧ (𝑟 ∈ CUnifSp ∧ (UnifSt‘𝑟) = (metUnif‘((dist‘𝑟) ↾ ((Base‘𝑟) × (Base‘𝑟))))))}
Theorem	isrrext 31850	Express the property "𝑅 is an extension of ℝ". (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))    &   ⊢ 𝑍 = (ℤMod‘𝑅)    ⇒   ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘𝐷))))
Theorem	rrextnrg 31851	An extension of ℝ is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
Theorem	rrextdrg 31852	An extension of ℝ is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)
Theorem	rrextnlm 31853	The norm of an extension of ℝ is absolutely homogeneous. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ 𝑍 = (ℤMod‘𝑅)    ⇒   ⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)
Theorem	rrextchr 31854	The ring characteristic of an extension of ℝ is zero. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → (chr‘𝑅) = 0)
Theorem	rrextcusp 31855	An extension of ℝ is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)
Theorem	rrexttps 31856	An extension of ℝ is a topological space. (Contributed by Thierry Arnoux, 7-Sep-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)
Theorem	rrexthaus 31857	The topology of an extension of ℝ is Hausdorff. (Contributed by Thierry Arnoux, 7-Sep-2018.)
⊢ 𝐾 = (TopOpen‘𝑅)    ⇒   ⊢ (𝑅 ∈ ℝExt → 𝐾 ∈ Haus)
Theorem	rrextust 31858	The uniformity of an extension of ℝ is the uniformity generated by its distance. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐷 = ((dist‘𝑅) ↾ (𝐵 × 𝐵))    ⇒   ⊢ (𝑅 ∈ ℝExt → (UnifSt‘𝑅) = (metUnif‘𝐷))
Theorem	rerrext 31859	The field of the real numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ ℝfld ∈ ℝExt
Theorem	cnrrext 31860	The field of the complex numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ ℂfld ∈ ℝExt
Theorem	qqtopn 31861	The topology of the field of the rational numbers. (Contributed by Thierry Arnoux, 29-Aug-2020.)
⊢ ((TopOpen‘ℝfld) ↾t ℚ) = (TopOpen‘(ℂfld ↾s ℚ))
Theorem	rrhfe 31862	If 𝑅 is an extension of ℝ, then the canonical homomorphism of ℝ into 𝑅 is a function. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ ℝExt → (ℝHom‘𝑅):ℝ⟶𝐵)
Theorem	rrhcne 31863	If 𝑅 is an extension of ℝ, then the canonical homomorphism of ℝ into 𝑅 is continuous. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ 𝐽 = (topGen‘ran (,))    &   ⊢ 𝐾 = (TopOpen‘𝑅)    ⇒   ⊢ (𝑅 ∈ ℝExt → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))
Theorem	rrhqima 31864	The ℝHom homomorphism leaves rational numbers unchanged. (Contributed by Thierry Arnoux, 27-Mar-2018.)
⊢ ((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((ℝHom‘𝑅)‘𝑄) = ((ℚHom‘𝑅)‘𝑄))
Theorem	rrh0 31865	The image of 0 by the ℝHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ (𝑅 ∈ ℝExt → ((ℝHom‘𝑅)‘0) = (0g‘𝑅))
20.3.14.5  Embedding from the extended real numbers into a complete lattice
Syntax	cxrh 31866	Map the extended real numbers into a complete lattice.
class ℝ*Hom
Definition	df-xrh 31867*	Define an embedding from the extended real number into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ ℝ*Hom = (𝑟 ∈ V ↦ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))))))
Theorem	xrhval 31868*	The value of the embedding from the extended real numbers into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
⊢ 𝐵 = ((ℝHom‘𝑅) “ ℝ)    &   ⊢ 𝐿 = (glb‘𝑅)    &   ⊢ 𝑈 = (lub‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈‘𝐵), (𝐿‘𝐵)))))
20.3.14.6  Canonical embeddings into the ordered field of the real numbers
Theorem	zrhre 31869	The ℤRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
⊢ (ℤRHom‘ℝfld) = ( I ↾ ℤ)
Theorem	qqhre 31870	The ℚHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
⊢ (ℚHom‘ℝfld) = ( I ↾ ℚ)
Theorem	rrhre 31871	The ℝHom homomorphism for the real numbers structure is the identity. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ (ℝHom‘ℝfld) = ( I ↾ ℝ)
20.3.14.7  Topological Manifolds Found this and was curious about how manifolds would be expressed in set.mm: https://mathoverflow.net/questions/336367/real-manifolds-in-a-theorem-prover This chapter proposes to define first manifold topologies, which characterize topological manifolds, and then to extend the structure with presentations, i.e., equivalence classes of atlases for a given topological space. We suggest to use the extensible structures to define the "topological space" aspect of topological manifolds, and then extend it with charts/presentations.
Syntax	cmntop 31872	The class of n-manifold topologies.
class ManTop
Definition	df-mntop 31873*	Define the class of 𝑁-manifold topologies, as second countable Hausdorff topologies locally homeomorphic to a ball of the Euclidean space of dimension 𝑁. (Contributed by Thierry Arnoux, 22-Dec-2019.)
⊢ ManTop = {⟨𝑛, 𝑗⟩ ∣ (𝑛 ∈ ℕ0 ∧ (𝑗 ∈ 2ndω ∧ 𝑗 ∈ Haus ∧ 𝑗 ∈ Locally [(TopOpen‘(𝔼hil‘𝑛))] ≃ ))}
Theorem	relmntop 31874	Manifold is a relation. (Contributed by Thierry Arnoux, 28-Dec-2019.)
⊢ Rel ManTop
Theorem	ismntoplly 31875	Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil‘𝑁))] ≃ )))
Theorem	ismntop 31876*	Property of being a manifold. (Contributed by Thierry Arnoux, 5-Jan-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐽 ∈ 𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ≃ (TopOpen‘(𝔼hil‘𝑁))))))
20.3.15  Real and complex functions
20.3.15.1  Integer powers - misc. additions
Theorem	nexple 31877	A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))
20.3.15.2  Indicator Functions
Syntax	cind 31878	Extend class notation with the indicator function generator.
class 𝟭
Definition	df-ind 31879*	Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
⊢ 𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥 ∈ 𝑜 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indv 31880*	Value of the indicator function generator with domain 𝑂. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
Theorem	indval 31881*	Value of the indicator function generator for a set 𝐴 and a domain 𝑂. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝐴, 1, 0)))
Theorem	indval2 31882	Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂 ∖ 𝐴) × {0})))
Theorem	indf 31883	An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1})
Theorem	indfval 31884	Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋 ∈ 𝐴, 1, 0))
Theorem	ind1 31885	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 1)
Theorem	ind0 31886	Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ (𝑂 ∖ 𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0)
Theorem	ind1a 31887	Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑋 ∈ 𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑋) = 1 ↔ 𝑋 ∈ 𝐴))
Theorem	indpi1 31888	Preimage of the singleton {1} by the indicator function. See i1f1lem 24758. (Contributed by Thierry Arnoux, 21-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)
Theorem	indsum 31889*	Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ (𝜑 → 𝑂 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑥 ∈ 𝑂 ((((𝟭‘𝑂)‘𝐴)‘𝑥) · 𝐵) = Σ𝑥 ∈ 𝐴 𝐵)
Theorem	indsumin 31890*	Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑂)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶)
Theorem	prodindf 31891*	The product of indicators is one if and only if all values are in the set. (Contributed by Thierry Arnoux, 11-Dec-2021.)
⊢ (𝜑 → 𝑂 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑂)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑂)    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (((𝟭‘𝑂)‘𝐵)‘(𝐹‘𝑘)) = if(ran 𝐹 ⊆ 𝐵, 1, 0))
Theorem	indf1o 31892	The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑m 𝑂))
Theorem	indpreima 31893	A function with range {0, 1} as an indicator of the preimage of {1}. (Contributed by Thierry Arnoux, 23-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐹:𝑂⟶{0, 1}) → 𝐹 = ((𝟭‘𝑂)‘(◡𝐹 “ {1})))
Theorem	indf1ofs 31894*	The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin})
20.3.15.3  Extended sum
Syntax	cesum 31895	Extend class notation to include infinite summations.
class Σ*𝑘 ∈ 𝐴𝐵
Definition	df-esum 31896	Define a short-hand for the possibly infinite sum over the extended nonnegative reals. Σ* is relying on the properties of the tsums, developped by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)
⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
Theorem	esumex 31897	An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V
Theorem	esumcl 31898*	Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.)
⊢ Ⅎ𝑘𝐴    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞))
Theorem	esumeq12dvaf 31899	Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)
Theorem	esumeq12dva 31900*	Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)