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Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqqhvval 30001 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 30002 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 30003 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 30004 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 30005 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 30006 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 30007 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 30008 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 30009 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 30010 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))

20.3.13.4  Canonical embedding of the real numbers into a complete ordered field

Syntaxcrrh 30011 Map the real numbers into a complete field.
class ℝHom

Syntaxcrrext 30012 Extend class notation with the class of extension fields of .
class ℝExt

Definitiondf-rrh 30013 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 30014 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 30015 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 30016 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 30017 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 30018 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 30018 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 30019 An extension of is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)

Theoremrrextdrg 30020 An extension of is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)

Theoremrrextnlm 30021 The norm of an extension of is absolutely homogeneous. (Contributed by Thierry Arnoux, 2-May-2018.)
𝑍 = (ℤMod‘𝑅)       (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)

Theoremrrextchr 30022 The ring characteristic of an extension of is zero. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → (chr‘𝑅) = 0)

Theoremrrextcusp 30023 An extension of is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)

Theoremrrexttps 30024 An extension of is a topological space. (Contributed by Thierry Arnoux, 7-Sep-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)

Theoremrrexthaus 30025 The topology of an extension of is Hausdorff. (Contributed by Thierry Arnoux, 7-Sep-2018.)
𝐾 = (TopOpen‘𝑅)       (𝑅 ∈ ℝExt → 𝐾 ∈ Haus)

Theoremrrextust 30026 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 30027 The field of the real numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt

Theoremcnrrext 30028 The field of the complex numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt

Theoremqqtopn 30029 The topology of the field of the rational numbers. (Contributed by Thierry Arnoux, 29-Aug-2020.)
((TopOpen‘ℝfld) ↾t ℚ) = (TopOpen‘(ℂflds ℚ))

Theoremrrhfe 30030 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 30031 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 30032 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 30033 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))

20.3.13.5  Embedding from the extended real numbers into a complete lattice

Syntaxcxrh 30034 Map the extended real numbers into a complete lattice.
class *Hom

Definitiondf-xrh 30035* 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 30036* 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)))))

20.3.13.6  Canonical embeddings into the ordered field of the real numbers

Theoremzrhre 30037 The ℤRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℤRHom‘ℝfld) = ( I ↾ ℤ)

Theoremqqhre 30038 The ℚHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℚHom‘ℝfld) = ( I ↾ ℚ)

Theoremrrhre 30039 The ℝHom homomorphism for the real numbers structure is the identity. (Contributed by Thierry Arnoux, 22-Oct-2017.)
(ℝHom‘ℝfld) = ( I ↾ ℝ)

20.3.13.7  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 30040 The class of n-manifold topologies.
class ManTop

Definitiondf-mntop 30041* 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 30042 Manifold is a relation. (Contributed by Thierry Arnoux, 28-Dec-2019.)
Rel ManTop

Theoremismntoplly 30043 Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)
((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2nd𝜔 ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))

Theoremismntop 30044* 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))))))

20.3.14  Real and complex functions

20.3.14.1  Integer powers - misc. additions

Theoremnexple 30045 A lower bound for an exponentiation. (Contributed by Thierry Arnoux, 19-Aug-2017.)
((𝐴 ∈ ℕ0𝐵 ∈ ℝ ∧ 2 ≤ 𝐵) → 𝐴 ≤ (𝐵𝐴))

20.3.14.2  Indicator Functions

Syntaxcind 30046 Extend class notation with the indicator function generator.
class 𝟭

Definitiondf-ind 30047* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))))

Theoremindv 30048* 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 30049* 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 30050 Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0})))

Theoremindf 30051 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 30052 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋𝐴, 1, 0))

Theoremind1 30053 Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝐴) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 1)

Theoremind0 30054 Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋 ∈ (𝑂𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0)

Theoremind1a 30055 Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑋) = 1 ↔ 𝑋𝐴))

Theoremindpi1 30056 Preimage of the singleton {1} by the indicator function. See i1f1lem 23437. (Contributed by Thierry Arnoux, 21-Aug-2017.)
((𝑂𝑉𝐴𝑂) → (((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)

Theoremindsum 30057* 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 30058* 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 30059* 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 30060 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 30061 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 30062* 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})

20.3.14.3  Extended sum

Syntaxcesum 30063 Extend class notation to include infinite summations.
class Σ*𝑘𝐴𝐵

Definitiondf-esum 30064 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 30065 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Σ*𝑘𝐴𝐵 ∈ V

Theoremesumcl 30066* 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 30067 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

Theoremesumeq12dva 30068* 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 30069* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)

Theoremesumeq1 30070* Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)

Theoremesumeq1d 30071 Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)

Theoremesumeq2 30072* Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
(∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremesumeq2d 30073 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
𝑘𝜑    &   (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremesumeq2dv 30074* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremesumeq2sdv 30075* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)

Theoremnfesum1 30076 Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝐴       𝑘Σ*𝑘𝐴𝐵

Theoremnfesum2 30077* Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ*𝑘𝐴𝐵

Theoremcbvesum 30078* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)    &   𝑘𝐴    &   𝑗𝐴    &   𝑘𝐵    &   𝑗𝐶       Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶

Theoremcbvesumv 30079* Change bound variable in an extended sum. (Contributed by Thierry Arnoux, 19-Jun-2017.)
(𝑗 = 𝑘𝐵 = 𝐶)       Σ*𝑗𝐴𝐵 = Σ*𝑘𝐴𝐶

Theoremesumid 30080 Identify the extended sum as any limit points of the infinite sum. (Contributed by Thierry Arnoux, 9-May-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐶 ∈ ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)))       (𝜑 → Σ*𝑘𝐴𝐵 = 𝐶)

Theoremesumgsum 30081 A finite extended sum is the group sum over the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 24-Apr-2020.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴𝐵 = ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝐴𝐵)))

Theoremesumval 30082* Develop the value of the extended sum. (Contributed by Thierry Arnoux, 4-Jan-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝑥𝐵)) = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ 𝐶), ℝ*, < ))

Theoremesumel 30083* The extended sum is a limit point of the corresponding infinite group sum. (Contributed by Thierry Arnoux, 24-Mar-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴𝐵 ∈ ((ℝ*𝑠s (0[,]+∞)) tsums (𝑘𝐴𝐵)))

Theoremesumnul 30084 Extended sum over the empty set. (Contributed by Thierry Arnoux, 19-Feb-2017.)
Σ*𝑥 ∈ ∅𝐴 = 0

Theoremesum0 30085* Extended sum of zero. (Contributed by Thierry Arnoux, 3-Mar-2017.)
𝑘𝐴       (𝐴𝑉 → Σ*𝑘𝐴0 = 0)

Theoremesumf1o 30086* Re-index an extended sum using a bijection. (Contributed by Thierry Arnoux, 6-Apr-2017.)
𝑛𝜑    &   𝑛𝐵    &   𝑘𝐷    &   𝑛𝐴    &   𝑛𝐶    &   𝑛𝐹    &   (𝑘 = 𝐺𝐵 = 𝐷)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐶1-1-onto𝐴)    &   ((𝜑𝑛𝐶) → (𝐹𝑛) = 𝐺)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑛𝐶𝐷)

Theoremesumc 30087* Convert from the collection form to the class-variable form of a sum. (Contributed by Thierry Arnoux, 10-May-2017.)
𝑘𝐷    &   𝑘𝜑    &   𝑘𝐴    &   (𝑦 = 𝐶𝐷 = 𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑 → Fun (𝑘𝐴𝐶))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶𝑊)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘𝐴 𝑧 = 𝐶}𝐷)

Theoremesumrnmpt 30088* Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 27-May-2020.)
𝑘𝐴    &   (𝑦 = 𝐵𝐶 = 𝐷)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐷 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (𝑊 ∖ {∅}))    &   (𝜑Disj 𝑘𝐴 𝐵)       (𝜑 → Σ*𝑦 ∈ ran (𝑘𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐷)

Theoremesumsplit 30089 Split an extended sum into two parts. (Contributed by Thierry Arnoux, 9-May-2017.)
𝑘𝜑    &   𝑘𝐴    &   𝑘𝐵    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝐴𝐵) = ∅)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐵) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘 ∈ (𝐴𝐵)𝐶 = (Σ*𝑘𝐴𝐶 +𝑒 Σ*𝑘𝐵𝐶))

Theoremesummono 30090* Extended sum is monotonic. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝜑    &   (𝜑𝐶𝑉)    &   ((𝜑𝑘𝐶) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝐴𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 ≤ Σ*𝑘𝐶𝐵)

Theoremesumpad 30091* Extend an extended sum by padding outside with zeroes. (Contributed by Thierry Arnoux, 31-May-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐵) → 𝐶 = 0)       (𝜑 → Σ*𝑘 ∈ (𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐶)

Theoremesumpad2 30092* Remove zeroes from an extended sum. (Contributed by Thierry Arnoux, 5-Jun-2020.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐵) → 𝐶 = 0)       (𝜑 → Σ*𝑘 ∈ (𝐴𝐵)𝐶 = Σ*𝑘𝐴𝐶)

(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘𝐴𝐵 +𝑒 Σ*𝑘𝐴𝐶))

Theoremesumle 30094* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
(𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 ≤ Σ*𝑘𝐴𝐶)

Theoremgsumesum 30095* Relate a group sum on (ℝ*𝑠s (0[,]+∞)) to a finite extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)
𝑘𝜑    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))       (𝜑 → ((ℝ*𝑠s (0[,]+∞)) Σg (𝑘𝐴𝐵)) = Σ*𝑘𝐴𝐵)

Theoremesumlub 30096* The extended sum is the lowest upper bound for the partial sums. (Contributed by Thierry Arnoux, 19-Oct-2017.) (Proof shortened by AV, 12-Dec-2019.)
𝑘𝜑    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   (𝜑𝑋 ∈ ℝ*)    &   (𝜑𝑋 < Σ*𝑘𝐴𝐵)       (𝜑 → ∃𝑎 ∈ (𝒫 𝐴 ∩ Fin)𝑋 < Σ*𝑘𝑎𝐵)

𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘𝐴(𝐵 +𝑒 𝐶) = (Σ*𝑘𝐴𝐵 +𝑒 Σ*𝑘𝐴𝐶))

Theoremesumlef 30098* If all of the terms of an extended sums compare, so do the sums. (Contributed by Thierry Arnoux, 8-Jun-2017.)
𝑘𝜑    &   𝑘𝐴    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐶 ∈ (0[,]+∞))    &   ((𝜑𝑘𝐴) → 𝐵𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 ≤ Σ*𝑘𝐴𝐶)

Theoremesumcst 30099* The extended sum of a constant. (Contributed by Thierry Arnoux, 3-Mar-2017.) (Revised by Thierry Arnoux, 5-Jul-2017.)
𝑘𝐴    &   𝑘𝐵       ((𝐴𝑉𝐵 ∈ (0[,]+∞)) → Σ*𝑘𝐴𝐵 = ((#‘𝐴) ·e 𝐵))

Theoremesumsnf 30100* The extended sum of a singleton is the term. (Contributed by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 2-May-2020.)
𝑘𝐵    &   ((𝜑𝑘 = 𝑀) → 𝐴 = 𝐵)    &   (𝜑𝑀𝑉)    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → Σ*𝑘 ∈ {𝑀}𝐴 = 𝐵)

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