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Theorem List for Metamath Proof Explorer - 31101-31200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.3.14  Topology and algebraic structures
 
20.3.14.1  The norm on the ring of the integer numbers
 
Theoremzringnm 31101 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 31102 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 31103 Zero of a -module. (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &    0 = (0g𝐺)        0 = (0g𝑊)
 
Theoremzlm1 31104 Unit of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &    1 = (1r𝐺)        1 = (1r𝑊)
 
Theoremzlmds 31105 Distance in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝐷 = (dist‘𝐺)       (𝐺𝑉𝐷 = (dist‘𝑊))
 
Theoremzlmtset 31106 Topology in a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝐽 = (TopSet‘𝐺)       (𝐺𝑉𝐽 = (TopSet‘𝑊))
 
Theoremzlmnm 31107 Norm of a -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
𝑊 = (ℤMod‘𝐺)    &   𝑁 = (norm‘𝐺)       (𝐺𝑉𝑁 = (norm‘𝑊))
 
Theoremzhmnrg 31108 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 31109 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 31110 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 31111 The -module of is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.)
(ℤMod‘ℂfld) ∈ NrmMod
 
Theoremrezh 31112 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 31113 Map the rationals into a field.
class ℚHom
 
Definitiondf-qqh 31114* 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 31115* 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 31116 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 31117 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 31118 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 31119 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 31120 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 31121 Lemma for qqhval2 31123. (Contributed by Thierry Arnoux, 29-Oct-2017.)
(𝑀 ∈ (ℤ ∖ {0}) → (𝑀 ∈ ℕ ∨ -𝑀 ∈ ℕ))
 
Theoremqqhval2lem 31122 Lemma for qqhval2 31123. (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 31123* 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 31124 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 31125 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 31126 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 31127 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 31128 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 31129 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 31130 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 31131 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 31132 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 31133 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 31134 Map the real numbers into a complete field.
class ℝHom
 
Syntaxcrrext 31135 Extend class notation with the class of extension fields of .
class ℝExt
 
Definitiondf-rrh 31136 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 31137 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 31138 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 31139 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 31140 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 31141 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 31141 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 31142 An extension of is a normed ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing)
 
Theoremrrextdrg 31143 An extension of is a division ring. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ DivRing)
 
Theoremrrextnlm 31144 The norm of an extension of is absolutely homogeneous. (Contributed by Thierry Arnoux, 2-May-2018.)
𝑍 = (ℤMod‘𝑅)       (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod)
 
Theoremrrextchr 31145 The ring characteristic of an extension of is zero. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → (chr‘𝑅) = 0)
 
Theoremrrextcusp 31146 An extension of is a complete uniform space. (Contributed by Thierry Arnoux, 2-May-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ CUnifSp)
 
Theoremrrexttps 31147 An extension of is a topological space. (Contributed by Thierry Arnoux, 7-Sep-2018.)
(𝑅 ∈ ℝExt → 𝑅 ∈ TopSp)
 
Theoremrrexthaus 31148 The topology of an extension of is Hausdorff. (Contributed by Thierry Arnoux, 7-Sep-2018.)
𝐾 = (TopOpen‘𝑅)       (𝑅 ∈ ℝExt → 𝐾 ∈ Haus)
 
Theoremrrextust 31149 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 31150 The field of the real numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt
 
Theoremcnrrext 31151 The field of the complex numbers is an extension of the real numbers. (Contributed by Thierry Arnoux, 2-May-2018.)
fld ∈ ℝExt
 
Theoremqqtopn 31152 The topology of the field of the rational numbers. (Contributed by Thierry Arnoux, 29-Aug-2020.)
((TopOpen‘ℝfld) ↾t ℚ) = (TopOpen‘(ℂflds ℚ))
 
Theoremrrhfe 31153 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 31154 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 31155 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 31156 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 31157 Map the extended real numbers into a complete lattice.
class *Hom
 
Definitiondf-xrh 31158* 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 31159* 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 31160 The ℤRHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℤRHom‘ℝfld) = ( I ↾ ℤ)
 
Theoremqqhre 31161 The ℚHom homomorphism for the real number structure is the identity. (Contributed by Thierry Arnoux, 31-Oct-2017.)
(ℚHom‘ℝfld) = ( I ↾ ℚ)
 
Theoremrrhre 31162 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 31163 The class of n-manifold topologies.
class ManTop
 
Definitiondf-mntop 31164* 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 31165 Manifold is a relation. (Contributed by Thierry Arnoux, 28-Dec-2019.)
Rel ManTop
 
Theoremismntoplly 31166 Property of being a manifold. (Contributed by Thierry Arnoux, 28-Dec-2019.)
((𝑁 ∈ ℕ0𝐽𝑉) → (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndω ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpen‘(𝔼hil𝑁))] ≃ )))
 
Theoremismntop 31167* 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 31168 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 31169 Extend class notation with the indicator function generator.
class 𝟭
 
Definitiondf-ind 31170* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
𝟭 = (𝑜 ∈ V ↦ (𝑎 ∈ 𝒫 𝑜 ↦ (𝑥𝑜 ↦ if(𝑥𝑎, 1, 0))))
 
Theoremindv 31171* 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 31172* 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 31173 Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
((𝑂𝑉𝐴𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝐴 × {1}) ∪ ((𝑂𝐴) × {0})))
 
Theoremindf 31174 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 31175 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝑂) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = if(𝑋𝐴, 1, 0))
 
Theoremind1 31176 Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝐴) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 1)
 
Theoremind0 31177 Value of the indicator function where it is 0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋 ∈ (𝑂𝐴)) → (((𝟭‘𝑂)‘𝐴)‘𝑋) = 0)
 
Theoremind1a 31178 Value of the indicator function where it is 1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
((𝑂𝑉𝐴𝑂𝑋𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑋) = 1 ↔ 𝑋𝐴))
 
Theoremindpi1 31179 Preimage of the singleton {1} by the indicator function. See i1f1lem 24219. (Contributed by Thierry Arnoux, 21-Aug-2017.)
((𝑂𝑉𝐴𝑂) → (((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴)
 
Theoremindsum 31180* 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 31181* 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 31182* 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 31183 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 31184 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 31185* 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 31186 Extend class notation to include infinite summations.
class Σ*𝑘𝐴𝐵
 
Definitiondf-esum 31187 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 31188 An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)
Σ*𝑘𝐴𝐵 ∈ V
 
Theoremesumcl 31189* 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 31190 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 26-Mar-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑘𝐴) → 𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
 
Theoremesumeq12dva 31191* 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 31192* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐷)
 
Theoremesumeq1 31193* Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 18-Feb-2017.)
(𝐴 = 𝐵 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
 
Theoremesumeq1d 31194 Equality theorem for an extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝜑    &   (𝜑𝐴 = 𝐵)       (𝜑 → Σ*𝑘𝐴𝐶 = Σ*𝑘𝐵𝐶)
 
Theoremesumeq2 31195* Equality theorem for extended sum. (Contributed by Thierry Arnoux, 24-Dec-2016.)
(∀𝑘𝐴 𝐵 = 𝐶 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremesumeq2d 31196 Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016.)
𝑘𝜑    &   (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremesumeq2dv 31197* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 2-Jan-2017.)
((𝜑𝑘𝐴) → 𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremesumeq2sdv 31198* Equality deduction for extended sum. (Contributed by Thierry Arnoux, 25-Dec-2016.)
(𝜑𝐵 = 𝐶)       (𝜑 → Σ*𝑘𝐴𝐵 = Σ*𝑘𝐴𝐶)
 
Theoremnfesum1 31199 Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.)
𝑘𝐴       𝑘Σ*𝑘𝐴𝐵
 
Theoremnfesum2 31200* Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ*𝑘𝐴𝐵
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