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Theorem List for Metamath Proof Explorer - 31201-31300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfsumcvg4 31201 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 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑆⟶ℂ)    &   (𝜑 → (𝐹 “ (ℂ ∖ {0})) ∈ Fin)       (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )

Theorempnfneige0 31202* A neighborhood of +∞ contains an unbounded interval based at a real number. See pnfnei 21804. (Contributed by Thierry Arnoux, 31-Jul-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))       ((𝐴𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)

Theoremlmxrge0 31203* 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*𝑠s (0[,]+∞)))    &   (𝜑𝐹:ℕ⟶(0[,]+∞))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) = 𝐴)       (𝜑 → (𝐹(⇝𝑡𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)𝑥 < 𝐴))

Theoremlmdvg 31204* If a monotonic sequence of real numbers diverges, it is unbounded. (Contributed by Thierry Arnoux, 4-Aug-2017.)
(𝜑𝐹:ℕ⟶(0[,)+∞))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   (𝜑 → ¬ 𝐹 ∈ dom ⇝ )       (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ𝑗)𝑥 < (𝐹𝑘))

Theoremlmdvglim 31205* 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*𝑠s (0[,]+∞)))    &   (𝜑𝐹:ℕ⟶(0[,)+∞))    &   ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ≤ (𝐹‘(𝑘 + 1)))    &   (𝜑 → ¬ 𝐹 ∈ dom ⇝ )       (𝜑𝐹(⇝𝑡𝐽)+∞)

20.3.12.15  Univariate polynomials

Theorempl1cn 31206 A univariate polynomial is continuous. (Contributed by Thierry Arnoux, 17-Sep-2018.)
𝑃 = (Poly1𝑅)    &   𝐸 = (eval1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐽 = (TopOpen‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑅 ∈ TopRing)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐸𝐹) ∈ (𝐽 Cn 𝐽))

20.3.13  Uniform Stuctures and Spaces

20.3.13.1  Hausdorff uniform completion

Syntaxchcmp 31207 Extend class notation with the Hausdorff uniform completion relation.
class HCmp

Definitiondf-hcmp 31208* 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

Theoremzringnm 31209 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)

Theoremzzsnm 31210 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

Theoremzlm0 31211 Zero of a -module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &    0 = (0g𝐺)        0 = (0g𝑊)

Theoremzlm1 31212 Unit of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &    1 = (1r𝐺)        1 = (1r𝑊)

Theoremzlmds 31213 Distance in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝐷 = (dist‘𝐺)       (𝐺𝑉𝐷 = (dist‘𝑊))

Theoremzlmtset 31214 Topology in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝐽 = (TopSet‘𝐺)       (𝐺𝑉𝐽 = (TopSet‘𝑊))

Theoremzlmnm 31215 Norm of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝑁 = (norm‘𝐺)       (𝐺𝑉𝑁 = (norm‘𝑊))

Theoremzhmnrg 31216 The -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)

Theoremnmmulg 31217 The norm of a group product, provided the -module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝐵 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &   𝑍 = (ℤMod‘𝑅)    &    · = (.g𝑅)       ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁𝑋)))

Theoremzrhnm 31218 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‘(𝐿𝑀)) = (abs‘𝑀))

Theoremcnzh 31219 The -module of is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
(ℤMod‘ℂfld) ∈ NrmMod

Theoremrezh 31220 The -module of 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

Syntaxcqqh 31221 Map the rationals into a field.
class ℚHom

Definitiondf-qqh 31222* 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)(/r𝑟)((ℤRHom‘𝑟)‘𝑦))⟩))

Theoremqqhval 31223* Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)
/ = (/r𝑅)    &    1 = (1r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))

Theoremzrhf1ker 31224 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 = (0g𝑅)       (𝑅 ∈ Ring → (𝐿:ℤ–1-1𝐵 ↔ (𝐿 “ { 0 }) = {0}))

Theoremzrhchr 31225 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 = (0g𝑅)       (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ 𝐿:ℤ–1-1𝐵))

Theoremzrhker 31226 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 = (0g𝑅)       (𝑅 ∈ Ring → ((chr‘𝑅) = 0 ↔ (𝐿 “ { 0 }) = {0}))

Theoremzrhunitpreima 31227 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 = (0g𝑅)       ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0}))

Theoremelzrhunit 31228 Condition for the image by ℤRHom to be a unit. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → (𝐿𝑀) ∈ (Unit‘𝑅))

Theoremelzdif0 31229 Lemma for qqhval2 31231. (Contributed by Thierry Arnoux, 29-Oct-2017.)
(𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))

Theoremqqhval2lem 31230 Lemma for qqhval2 31231. (Contributed by Thierry Arnoux, 29-Oct-2017.)
𝐵 = (Base‘𝑅)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0)) → ((𝐿‘(numer‘(𝑋 / 𝑌))) / (𝐿‘(denom‘(𝑋 / 𝑌)))) = ((𝐿𝑋) / (𝐿𝑌)))

Theoremqqhval2 31231* 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)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))

Theoremqqhvval 31232 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)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ 𝑄 ∈ ℚ) → ((ℚHom‘𝑅)‘𝑄) = ((𝐿‘(numer‘𝑄)) / (𝐿‘(denom‘𝑄))))

Theoremqqh0 31233 The image of 0 by the ℚHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝐵 = (Base‘𝑅)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘0) = (0g𝑅))

Theoremqqh1 31234 The image of 1 by the ℚHom homomorphism is the ring's unit. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝐵 = (Base‘𝑅)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ((ℚHom‘𝑅)‘1) = (1r𝑅))

Theoremqqhf 31235 ℚHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.)
𝐵 = (Base‘𝑅)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅):ℚ⟶𝐵)

Theoremqqhvq 31236 The image of a quotient by the ℚHom homomorphism. (Contributed by Thierry Arnoux, 28-Oct-2017.)
𝐵 = (Base‘𝑅)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)       (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ ∧ 𝑌 ≠ 0)) → ((ℚHom‘𝑅)‘(𝑋 / 𝑌)) = ((𝐿𝑋) / (𝐿𝑌)))

Theoremqqhghm 31237 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)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &   𝑄 = (ℂflds ℚ)       ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 GrpHom 𝑅))

Theoremqqhrhm 31238 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)    &    / = (/r𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &   𝑄 = (ℂflds ℚ)       ((𝑅 ∈ Field ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝑄 RingHom 𝑅))

Theoremqqhnm 31239 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))

Theoremqqhcn 31240 The ℚHom homomorphism is a continuous function. (Contributed by Thierry Arnoux, 9-Nov-2017.)
𝑄 = (ℂflds ℚ)    &   𝐽 = (TopOpen‘𝑄)    &   𝑍 = (ℤMod‘𝑅)    &   𝐾 = (TopOpen‘𝑅)       ((𝑅 ∈ (NrmRing ∩ DivRing) ∧ 𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Theoremqqhucn 31241 The ℚHom homomorphism is uniformly continuous. (Contributed by Thierry Arnoux, 28-Jan-2018.)
𝐵 = (Base‘𝑅)    &   𝑄 = (ℂflds ℚ)    &   𝑈 = (UnifSt‘𝑄)    &   𝑉 = (metUnif‘((dist‘𝑅) ↾ (𝐵 × 𝐵)))    &   𝑍 = (ℤMod‘𝑅)    &   (𝜑𝑅 ∈ NrmRing)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑍 ∈ 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

Syntaxcrrh 31242 Map the real numbers into a complete field.
class ℝHom

Syntaxcrrext 31243 Extend class notation with the class of extension fields of .
class ℝExt

Definitiondf-rrh 31244 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)))

Theoremrrhval 31245 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)       (𝑅𝑉 → (ℝHom‘𝑅) = ((𝐽CnExt𝐾)‘(ℚHom‘𝑅)))

Theoremrrhcn 31246 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)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑅 ∈ NrmRing)    &   (𝜑𝑍 ∈ NrmMod)    &   (𝜑 → (chr‘𝑅) = 0)    &   (𝜑𝑅 ∈ CUnifSp)    &   (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷))       (𝜑 → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Theoremrrhf 31247 If the topology of 𝑅 is Hausdorff, Cauchy sequences have at most one limit, i.e. the canonical homomorphism of 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)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑅 ∈ NrmRing)    &   (𝜑𝑍 ∈ NrmMod)    &   (𝜑 → (chr‘𝑅) = 0)    &   (𝜑𝑅 ∈ CUnifSp)    &   (𝜑 → (UnifSt‘𝑅) = (metUnif‘𝐷))       (𝜑 → (ℝHom‘𝑅):ℝ⟶𝐵)

Definitiondf-rrext 31248 Define the class of extensions of . This is a shorthand for listing the necessary conditions for a structure to admit a canonical embedding of into it. Interestingly, this is not coming from a mathematical reference, but was from the necessary conditions to build the embedding at each step (, and ). It would be interesting see if this is formally treated in the literature. See isrrext 31249 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))))))}

Theoremisrrext 31249 Express the property "𝑅 is an extension of ". (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))))

Theoremrrextnrg 31250 An extension of is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)

Theoremrrextdrg 31251 An extension of is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)

Theoremrrextnlm 31252 The norm of an extension of is absolutely homogeneous. (Contributed by Thierry Arnoux, 2-May-2018.)
𝑍 = (ℤMod‘𝑅)       (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)

Theoremrrextchr 31253 The ring characteristic of an extension of is zero. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → (chr‘𝑅) = 0)

Theoremrrextcusp 31254 An extension of is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)

Theoremrrexttps 31255 An extension of is a topological space. (Contributed by Thierry Arnoux, 7-Sep-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Theoremrrexthaus 31256 The topology of an extension of is Hausdorff. (Contributed by Thierry Arnoux, 7-Sep-2018.)
𝐾 = (TopOpen‘𝑅)       (𝑅 ∈ ℝExt → 𝐾 ∈ Haus)

Theoremrrextust 31257 The uniformity of an extension of 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))

Theoremrerrext 31258 The field of the real numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt

Theoremcnrrext 31259 The field of the complex numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt

Theoremqqtopn 31260 The topology of the field of the rational numbers. (Contributed by Thierry Arnoux, 29-Aug-2020.)
((TopOpen‘ℝfld) ↾t ℚ) = (TopOpen‘(ℂflds ℚ))

Theoremrrhfe 31261 If 𝑅 is an extension of , then the canonical homomorphism of into 𝑅 is a function. (Contributed by Thierry Arnoux, 2-May-2018.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ ℝExt → (ℝHom‘𝑅):ℝ⟶𝐵)

Theoremrrhcne 31262 If 𝑅 is an extension of , then the canonical homomorphism of into 𝑅 is continuous. (Contributed by Thierry Arnoux, 2-May-2018.)
𝐽 = (topGen‘ran (,))    &   𝐾 = (TopOpen‘𝑅)       (𝑅 ∈ ℝExt → (ℝHom‘𝑅) ∈ (𝐽 Cn 𝐾))

Theoremrrhqima 31263 The ℝHom homomorphism leaves rational numbers unchanged. (Contributed by Thierry Arnoux, 27-Mar-2018.)
((𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ) → ((ℝHom‘𝑅)‘𝑄) = ((ℚHom‘𝑅)‘𝑄))

Theoremrrh0 31264 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) = (0g𝑅))

20.3.14.5  Embedding from the extended real numbers into a complete lattice

Syntaxcxrh 31265 Map the extended real numbers into a complete lattice.
class *Hom

Definitiondf-xrh 31266* Define an embedding from the extended real number into a complete lattice. (Contributed by Thierry Arnoux, 19-Feb-2018.)
*Hom = (𝑟 ∈ V ↦ (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑟)‘𝑥), if(𝑥 = +∞, ((lub‘𝑟)‘((ℝHom‘𝑟) “ ℝ)), ((glb‘𝑟)‘((ℝHom‘𝑟) “ ℝ))))))

Theoremxrhval 31267* 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)       (𝑅𝑉 → (ℝ*Hom‘𝑅) = (𝑥 ∈ ℝ* ↦ if(𝑥 ∈ ℝ, ((ℝHom‘𝑅)‘𝑥), if(𝑥 = +∞, (𝑈𝐵), (𝐿𝐵)))))

20.3.14.6  Canonical embeddings into the ordered field of the real numbers

Theoremzrhre 31268 The ℤRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℤRHom‘ℝfld) = ( I ↾ ℤ)

Theoremqqhre 31269 The ℚHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℚHom‘ℝfld) = ( I ↾ ℚ)

Theoremrrhre 31270 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.

Syntaxcmntop 31271 The class of n-manifold topologies.
class ManTop

Definitiondf-mntop 31272* Define the class of N-manifold topologies, as 2nd 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‘(𝔼hil𝑛))] ≃ ))}

Theoremrelmntop 31273 Manifold is a relation. (Contributed by Thierry Arnoux, 28-Dec-2019.)
Rel ManTop

Theoremismntoplly 31274 Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)
((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))

Theoremismntop 31275* Property of being a manifold. (Contributed by Thierry Arnoux, 5-Jan-2020.)
((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦𝑢 ∧ (𝐽t 𝑢) ≃ (TopOpen‘(𝔼hil𝑁))))))

20.3.15  Real and complex functions

20.3.15.1  Integer powers - misc. additions

Theoremnexple 31276 A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))

20.3.15.2  Indicator Functions

Syntaxcind 31277 Extend class notation with the indicator function generator.
class 𝟭

Definitiondf-ind 31278* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))))

Theoremindv 31279* Value of the indicator function generator with domain 𝑂. (Contributed by Thierry Arnoux, 23-Aug-2017.)
(𝑂𝑉 → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥𝑂 ↦ if(𝑥𝑎, 1, 0))))

Theoremindval 31280* Value of the indicator function generator for a set 𝐴 and a domain 𝑂. (Contributed by Thierry Arnoux, 2-Feb-2017.)
((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = (𝑥𝑂 ↦ if(𝑥𝐴, 1, 0)))

Theoremindval2 31281 Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0})))

Theoremindf 31282 An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1})

Theoremindfval 31283 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋𝐴, 1, 0))

Theoremind1 31284 Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝐴) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 1)

Theoremind0 31285 Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋 ∈ (𝑂𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0)

Theoremind1a 31286 Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑋) = 1 ↔ 𝑋𝐴))

Theoremindpi1 31287 Preimage of the singleton {1} by the indicator function. See i1f1lem 24272. (Contributed by Thierry Arnoux, 21-Aug-2017.)
((𝑂𝑉𝐴𝑂) → (((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)

Theoremindsum 31288* Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
(𝜑𝑂 ∈ Fin)    &   (𝜑𝐴𝑂)    &   ((𝜑𝑥𝑂) → 𝐵 ∈ ℂ)       (𝜑 → Σ𝑥𝑂 ((((𝟭‘𝑂)‘𝐴)‘𝑥) · 𝐵) = Σ𝑥𝐴 𝐵)

Theoremindsumin 31289* Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 11-Dec-2021.)
(𝜑𝑂𝑉)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐴𝑂)    &   (𝜑𝐵𝑂)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ ℂ)       (𝜑 → Σ𝑘𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴𝐵)𝐶)

Theoremprodindf 31290* The product of indicators is one if and only if all values are in the set. (Contributed by Thierry Arnoux, 11-Dec-2021.)
(𝜑𝑂𝑉)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝑂)    &   (𝜑𝐹:𝐴𝑂)       (𝜑 → ∏𝑘𝐴 (((𝟭‘𝑂)‘𝐵)‘(𝐹𝑘)) = if(ran 𝐹𝐵, 1, 0))

Theoremindf1o 31291 The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
(𝑂𝑉 → (𝟭‘𝑂):𝒫 𝑂1-1-onto→({0, 1} ↑m 𝑂))

Theoremindpreima 31292 A function with range {0, 1} as an indicator of the preimage of {1}. (Contributed by Thierry Arnoux, 23-Aug-2017.)
((𝑂𝑉𝐹:𝑂⟶{0, 1}) → 𝐹 = ((𝟭‘𝑂)‘(𝐹 “ {1})))

Theoremindf1ofs 31293* The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
(𝑂𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑m 𝑂) ∣ (𝑓 “ {1}) ∈ Fin})

20.3.15.3  Extended sum

Syntaxcesum 31294 Extend class notation to include infinite summations.
class Σ*𝑘𝐴𝐵

Definitiondf-esum 31295 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*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵))

Theoremesumex 31296 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Σ*𝑘𝐴𝐵 ∈ V

Theoremesumcl 31297* Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.)
𝑘𝐴       ((𝐴𝑉 ∧ ∀𝑘𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘𝐴𝐵 ∈ (0[,]+∞))

Theoremesumeq12dvaf 31298 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

Theoremesumeq12dva 31299* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.) (Revised by Thierry Arnoux, 29-Jun-2017.)
(𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

Theoremesumeq12d 31300* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

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