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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrmulc1cn 29101* The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝐽 = (ordTop‘ ≤ )    &   𝐹 = (𝑥 ∈ ℝ* ↦ (𝑥 ·e 𝐶))    &   (𝜑𝐶 ∈ ℝ+)       (𝜑𝐹 ∈ (𝐽 Cn 𝐽))
 
Theoremfmcncfil 29102 The image of a Cauchy filter by a continuous filter map is a Cauchy filter. (Contributed by Thierry Arnoux, 12-Nov-2017.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = (MetOpen‘𝐸)       (((𝐷 ∈ (CMet‘𝑋) ∧ 𝐸 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝐵 ∈ (CauFil‘𝐷)) → ((𝑌 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐸))
 
20.3.10.13  Topology of the extended nonnegative real numbers ordered monoid
 
Theoremxrge0hmph 29103 The extended nonnegative reals are homeomorphic to the closed unit interval. (Contributed by Thierry Arnoux, 24-Mar-2017.)
II ≃ ((ordTop‘ ≤ ) ↾t (0[,]+∞))
 
Theoremxrge0iifcnv 29104* Define a bijection from [0, 1] to [0, +∞]. (Contributed by Thierry Arnoux, 29-Mar-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))       (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ 𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 0, (exp‘-𝑦))))
 
Theoremxrge0iifcv 29105* The defined function's value in the real. (Contributed by Thierry Arnoux, 1-Apr-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))       (𝑋 ∈ (0(,]1) → (𝐹𝑋) = -(log‘𝑋))
 
Theoremxrge0iifiso 29106* The defined bijection from the closed unit interval and the extended nonnegative reals is an order isomorphism. (Contributed by Thierry Arnoux, 31-Mar-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))       𝐹 Isom < , < ((0[,]1), (0[,]+∞))
 
Theoremxrge0iifhmeo 29107* Expose a homeomorphism from the closed unit interval and the extended nonnegative reals. (Contributed by Thierry Arnoux, 1-Apr-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       𝐹 ∈ (IIHomeo𝐽)
 
Theoremxrge0iifhom 29108* The defined function from the closed unit interval and the extended nonnegative reals is also a monoid homomorphism. (Contributed by Thierry Arnoux, 5-Apr-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) +𝑒 (𝐹𝑌)))
 
Theoremxrge0iif1 29109* Condition for the defined function, -(log‘𝑥) to be a monoid homomorphism. (Contributed by Thierry Arnoux, 20-Jun-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       (𝐹‘1) = 0
 
Theoremxrge0iifmhm 29110* The defined function from the closed unit interval and the extended nonnegative reals is a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))       𝐹 ∈ (((mulGrp‘ℂfld) ↾s (0[,]1)) MndHom (ℝ*𝑠s (0[,]+∞)))
 
Theoremxrge0pluscn 29111* The addition operation of the extended nonnegative real numbers monoid is continuous. (Contributed by Thierry Arnoux, 24-Mar-2017.)
𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 0, +∞, -(log‘𝑥)))    &   𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    &    + = ( +𝑒 ↾ ((0[,]+∞) × (0[,]+∞)))        + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
 
Theoremxrge0mulc1cn 29112* The operation multiplying a nonnegative real numbers by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.)
𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞))    &   𝐹 = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·e 𝐶))    &   (𝜑𝐶 ∈ (0[,)+∞))       (𝜑𝐹 ∈ (𝐽 Cn 𝐽))
 
Theoremxrge0tps 29113 The extended nonnegative real numbers monoid forms a topological space. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopSp
 
Theoremxrge0topn 29114 The topology of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 20-Jun-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞))
 
Theoremxrge0haus 29115 The topology of the extended nonnegative real numbers is Hausdorff. (Contributed by Thierry Arnoux, 26-Jul-2017.)
(TopOpen‘(ℝ*𝑠s (0[,]+∞))) ∈ Haus
 
Theoremxrge0tmdOLD 29116 The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopMnd
 
Theoremxrge0tmd 29117 The extended nonnegative real numbers monoid is a topological monoid. (Contributed by Thierry Arnoux, 26-Mar-2017.) (Proof Shortened by Thierry Arnoux, 21-Jun-2017.)
(ℝ*𝑠s (0[,]+∞)) ∈ TopMnd
 
20.3.10.14  Limits - misc additions
 
Theoremlmlim 29118 Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.)
𝐽 ∈ (TopOn‘𝑌)    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝑃𝑋)    &   (𝐽t 𝑋) = ((TopOpen‘ℂfld) ↾t 𝑋)    &   𝑋 ⊆ ℂ       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹𝑃))
 
Theoremlmlimxrge0 29119 Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝑃𝑋)    &   𝑋 ⊆ (0[,)+∞)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃𝐹𝑃))
 
Theoremrge0scvg 29120 Implication of convergence for a nonnegative series. This could be used to shorten prmreclem6 15345. (Contributed by Thierry Arnoux, 28-Jul-2017.)
((𝐹:ℕ⟶(0[,)+∞) ∧ seq1( + , 𝐹) ∈ dom ⇝ ) → sup(ran seq1( + , 𝐹), ℝ, < ) ∈ ℝ)
 
Theoremfsumcvg4 29121 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 29122* A neighborhood of +∞ contains an unbounded interval based at a real number. See pnfnei 20735. (Contributed by Thierry Arnoux, 31-Jul-2017.)
𝐽 = (TopOpen‘(ℝ*𝑠s (0[,]+∞)))       ((𝐴𝐽 ∧ +∞ ∈ 𝐴) → ∃𝑥 ∈ ℝ (𝑥(,]+∞) ⊆ 𝐴)
 
Theoremlmxrge0 29123* 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 29124* 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 29125* 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)+∞)
 
20.3.10.15  Univariate polynomials
 
Theorempl1cn 29126 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))
 
20.3.11  Uniform Stuctures and Spaces
 
20.3.11.1  Hausdorff uniform completion
 
Syntaxchcmp 29127 Extend class notation with the Hausdorff uniform completion relation.
class HCmp
 
Definitiondf-hcmp 29128* 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 unicity 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))}
 
20.3.12  Topology and algebraic structures
 
20.3.12.1  The norm on the ring of the integer numbers
 
Theoremzringnm 29129 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 29130 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))
 
20.3.12.2  Topological ` ZZ ` -modules
 
Theoremzlm0 29131 Zero of a -module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &    0 = (0g𝐺)        0 = (0g𝑊)
 
Theoremzlm1 29132 Unit of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &    1 = (1r𝐺)        1 = (1r𝑊)
 
Theoremzlmds 29133 Distance in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝐷 = (dist‘𝐺)       (𝐺𝑉𝐷 = (dist‘𝑊))
 
Theoremzlmtset 29134 Topology in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝐽 = (TopSet‘𝐺)       (𝐺𝑉𝐽 = (TopSet‘𝑊))
 
Theoremzlmnm 29135 Norm of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝑁 = (norm‘𝐺)       (𝐺𝑉𝑁 = (norm‘𝑊))
 
Theoremzhmnrg 29136 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 29137 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 29138 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 29139 The -module of is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
(ℤMod‘ℂfld) ∈ NrmMod
 
Theoremrezh 29140 The -module of is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(ℤMod‘ℝfld) ∈ NrmMod
 
20.3.12.3  Canonical embedding of the field of the rational numbers into a division ring
 
Syntaxcqqh 29141 Map the rationals into a field.
class ℚHom
 
Definitiondf-qqh 29142* 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 29143* 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 29144 The kernel of the homomorphism from the integers to a ring, if it is injective. (Contributed by Thierry Arnoux, 26-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐿:ℤ–1-1𝐵 ↔ (𝐿 “ { 0 }) = {0}))
 
Theoremzrhchr 29145 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 29146 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 29147 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 29148 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 29149 Lemma for qqhval2 29151. (Contributed by Thierry Arnoux, 29-Oct-2017.)
(𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))
 
Theoremqqhval2lem 29150 Lemma for qqhval2 29151. (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 29151* 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 29152 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 29153 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 29154 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 29155 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 29156 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 29157 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 29158 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 29159 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 29160 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 29161 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))
 
20.3.12.4  Canonical embedding of the real numbers into a complete ordered field
 
Syntaxcrrh 29162 Map the real numbers into a complete field.
class ℝHom
 
Syntaxcrrext 29163 Extend class notation with the class of extension fields of .
class ℝExt
 
Definitiondf-rrh 29164 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 29165 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 29166 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 29167 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 29168 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 29169 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 29169 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 29170 An extension of is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
 
Theoremrrextdrg 29171 An extension of is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)
 
Theoremrrextnlm 29172 The norm of an extension of is absolutely homogeneous. (Contributed by Thierry Arnoux, 2-May-2018.)
𝑍 = (ℤMod‘𝑅)       (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)
 
Theoremrrextchr 29173 The ring characteristic of an extension of is zero. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → (chr‘𝑅) = 0)
 
Theoremrrextcusp 29174 An extension of is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)
 
Theoremrrexttps 29175 An extension of is a topological space. (Contributed by Thierry Arnoux, 7-Sep-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)
 
Theoremrrexthaus 29176 The topology of an extension of is Hausdorff. (Contributed by Thierry Arnoux, 7-Sep-2018.)
𝐾 = (TopOpen‘𝑅)       (𝑅 ∈ ℝExt → 𝐾 ∈ Haus)
 
Theoremrrextust 29177 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 29178 The field of the real numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt
 
Theoremcnrrext 29179 The field of the complex numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt
 
Theoremqqtopn 29180 The topology of the field of the rational numbers. (Contributed by Thierry Arnoux, 29-Aug-2020.)
((TopOpen‘ℝfld) ↾t ℚ) = (TopOpen‘(ℂflds ℚ))
 
Theoremrrhfe 29181 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 29182 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 29183 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 29184 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))
 
20.3.12.5  Embedding from the extended real numbers into a complete lattice
 
Syntaxcxrh 29185 Map the extended real numbers into a complete lattice.
class *Hom
 
Definitiondf-xrh 29186* 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 29187* 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)))))
 
20.3.12.6  Canonical embeddings into the ordered field of the real numbers
 
Theoremzrhre 29188 The ℤRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℤRHom‘ℝfld) = ( I ↾ ℤ)
 
Theoremqqhre 29189 The ℚHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℚHom‘ℝfld) = ( I ↾ ℚ)
 
Theoremrrhre 29190 The ℝHom homomorphism for the real numbers structure is the identity. (Contributed by Thierry Arnoux, 22-Oct-2017.)
(ℝHom‘ℝfld) = ( I ↾ ℝ)
 
20.3.12.7  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 characterise topological manifold, and then to extends 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 29191 The class of n-manifold topologies.
class ManTop
 
Definitiondf-mntop 29192* 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 29193 Manifold is a relation. (Contributed by Thierry Arnoux, 28-Dec-2019.)
Rel ManTop
 
Theoremismntoplly 29194 Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)
((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
 
Theoremismntop 29195* 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))))))
 
20.3.13  Real and complex functions
 
20.3.13.1  Integer powers - misc. additions
 
Theoremnexple 29196 A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))
 
20.3.13.2  Indicator Functions
 
Syntaxcind 29197 Extend class notation with the indicator function generator.
class 𝟭
 
Definitiondf-ind 29198* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))))
 
Theoremindv 29199* 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 29200* 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)))
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