 Home Metamath Proof ExplorerTheorem List (p. 292 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26948) Hilbert Space Explorer (26949-28473) Users' Mathboxes (28474-42413)

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

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)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42413
 Copyright terms: Public domain < Previous  Next >