P. 340 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33901-34000   *Has distinct variable group(s)

Type	Label	Description
Statement
21.3.13.16  Univariate polynomials
Theorem	pl1cn 33901	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))
21.3.14  Uniform Stuctures and Spaces
21.3.14.1  Hausdorff uniform completion
Syntax	chcmp 33902	Extend class notation with the Hausdorff uniform completion relation.
class HCmp
Definition	df-hcmp 33903*	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))}
21.3.15  Topology and algebraic structures
21.3.15.1  The norm on the ring of the integer numbers
Theorem	zringnm 33904	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 33905	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))
21.3.15.2  Topological ` ZZ ` -modules
Theorem	zlm0 33906	Zero of a ℤ-module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ 0 = (0g‘𝑊)
Theorem	zlm1 33907	Unity element of a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 1 = (1r‘𝐺)    ⇒   ⊢ 1 = (1r‘𝑊)
Theorem	zlmds 33908	Distance in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 11-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
Theorem	zlmdsOLD 33909	Obsolete proof of zlmds 33908 as of 11-Nov-2024. Distance in a ℤ -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊))
Theorem	zlmtset 33910	Topology in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof shortened by AV, 12-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐽 = (TopSet‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))
Theorem	zlmtsetOLD 33911	Obsolete proof of zlmtset 33910 as of 11-Nov-2024. Topology in a ℤ -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐽 = (TopSet‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝐽 = (TopSet‘𝑊))
Theorem	zlmnm 33912	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 33913	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 33914	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 33915	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 33916	The ℤ-module of ℂ is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
⊢ (ℤMod‘ℂfld) ∈ NrmMod
Theorem	rezh 33917	The ℤ-module of ℝ is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)
⊢ (ℤMod‘ℝfld) ∈ NrmMod
21.3.15.3  Canonical embedding of the field of the rational numbers into a division ring
Syntax	cqqh 33918	Map the rationals into a field.
class ℚHom
Definition	df-qqh 33919*	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 33920*	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 33921	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 33922	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 33923	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 33924	The preimage by ℤRHom of the units 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 33925	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 33926	Lemma for qqhval2 33928. (Contributed by Thierry Arnoux, 29-Oct-2017.)
⊢ (𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))
Theorem	qqhval2lem 33927	Lemma for qqhval2 33928. (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 33928*	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 33929	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 33930	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 33931	The image of 1 by the ℚHom homomorphism is the ring unity. (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 33932	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 33933	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 33934	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 33935	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 33936	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 33937	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 33938	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))
21.3.15.4  Canonical embedding of the real numbers into a complete ordered field
Syntax	crrh 33939	Map the real numbers into a complete field.
class ℝHom
Syntax	crrext 33940	Extend class notation with the class of extension fields of ℝ.
class ℝExt
Definition	df-rrh 33941	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 33942	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 33943	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 33944	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 33945	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 33946 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 33946	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 33947	An extension of ℝ is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
Theorem	rrextdrg 33948	An extension of ℝ is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)
Theorem	rrextnlm 33949	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 33950	The ring characteristic of an extension of ℝ is zero. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → (chr‘𝑅) = 0)
Theorem	rrextcusp 33951	An extension of ℝ is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)
Theorem	rrexttps 33952	An extension of ℝ is a topological space. (Contributed by Thierry Arnoux, 7-Sep-2018.)
⊢ (𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)
Theorem	rrexthaus 33953	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 33954	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 33955	The field of the real numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ ℝfld ∈ ℝExt
Theorem	cnrrext 33956	The field of the complex numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
⊢ ℂfld ∈ ℝExt
Theorem	qqtopn 33957	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 33958	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 33959	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 33960	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 33961	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))
21.3.15.5  Embedding from the extended real numbers into a complete lattice
Syntax	cxrh 33962	Map the extended real numbers into a complete lattice.
class ℝ*Hom
Definition	df-xrh 33963*	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 33964*	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)))))
21.3.15.6  Canonical embeddings into the ordered field of the real numbers
Theorem	zrhre 33965	The ℤRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
⊢ (ℤRHom‘ℝfld) = ( I ↾ ℤ)
Theorem	qqhre 33966	The ℚHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
⊢ (ℚHom‘ℝfld) = ( I ↾ ℚ)
Theorem	rrhre 33967	The ℝHom homomorphism for the real numbers structure is the identity. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ (ℝHom‘ℝfld) = ( I ↾ ℝ)
21.3.15.7  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 33968	The class of n-manifold topologies.
class ManTop
Definition	df-mntop 33969*	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 33970	Manifold is a relation. (Contributed by Thierry Arnoux, 28-Dec-2019.)
⊢ Rel ManTop
Theorem	ismntoplly 33971	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 33972*	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))))))
21.3.16  Real and complex functions
21.3.16.1  Integer powers - misc. additions
Theorem	nexple 33973	A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵↑𝐴))
21.3.16.2  Indicator Functions
Syntax	cind 33974	Extend class notation with the indicator function generator.
class 𝟭
Definition	df-ind 33975*	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 33976*	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 33977*	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 33978	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 33979	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 33980	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 33981	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 33982	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 33983	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 33984	Preimage of the singleton {1} by the indicator function. See i1f1lem 25743. (Contributed by Thierry Arnoux, 21-Aug-2017.)
⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)
Theorem	indsum 33985*	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 33986*	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 33987*	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 33988	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 33989	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 33990*	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})
21.3.16.3  Extended sum
Syntax	cesum 33991	Extend class notation to include infinite summations.
class Σ*𝑘 ∈ 𝐴𝐵
Definition	df-esum 33992	Define a short-hand for the possibly infinite sum over the extended nonnegative reals. Σ* is relying on the properties of the tsums, developed by Mario Carneiro. (Contributed by Thierry Arnoux, 21-Sep-2016.)
⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))
Theorem	esumex 33993	An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
⊢ Σ*𝑘 ∈ 𝐴𝐵 ∈ V
Theorem	esumcl 33994*	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 33995	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 33996*	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)
Theorem	esumeq12d 33997*	Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐷)
Theorem	esumeq1 33998*	Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
⊢ (𝐴 = 𝐵 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)
Theorem	esumeq1d 33999	Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
⊢ Ⅎ𝑘𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐶 = Σ*𝑘 ∈ 𝐵𝐶)
Theorem	esumeq2 34000*	Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑘 ∈ 𝐴𝐶)