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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremevl1gsumdlem 19701* Lemma for evl1gsumd 19702 (induction step). (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)       ((𝑚 ∈ Fin ∧ ¬ 𝑎𝑚𝜑) → ((∀𝑥𝑚 𝑀𝑈 → ((𝑂‘(𝑃 Σg (𝑥𝑚𝑀)))‘𝑌) = (𝑅 Σg (𝑥𝑚 ↦ ((𝑂𝑀)‘𝑌)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀𝑈 → ((𝑂‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂𝑀)‘𝑌))))))
 
Theoremevl1gsumd 19702* Polynomial evaluation builder for a finite group sum of polynomials. (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → ∀𝑥𝑁 𝑀𝑈)    &   (𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑂‘(𝑃 Σg (𝑥𝑁𝑀)))‘𝑌) = (𝑅 Σg (𝑥𝑁 ↦ ((𝑂𝑀)‘𝑌))))
 
Theoremevl1gsumadd 19703* Univariate polynomial evaluation maps (additive) group sums to group sums. Remark: the proof would be shorter if the theorem is proved directly instead of using evls1gsumadd 19670. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &    0 = (0g𝑊)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 0 )       (𝜑 → (𝑄‘(𝑊 Σg (𝑥𝑁𝑌))) = (𝑃 Σg (𝑥𝑁 ↦ (𝑄𝑌))))
 
Theoremevl1gsumaddval 19704* Value of a univariate polynomial evaluation mapping an additive group sum to a group sum of the evaluated variable. (Contributed by AV, 17-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑊 Σg (𝑥𝑁𝑌)))‘𝐶) = (𝑅 Σg (𝑥𝑁 ↦ ((𝑄𝑌)‘𝐶))))
 
Theoremevl1gsummul 19705* Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &    1 = (1r𝑊)    &   𝐺 = (mulGrp‘𝑊)    &   𝐻 = (mulGrp‘𝑃)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 1 )       (𝜑 → (𝑄‘(𝐺 Σg (𝑥𝑁𝑌))) = (𝐻 Σg (𝑥𝑁 ↦ (𝑄𝑌))))
 
Theoremevl1varpw 19706 Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 19703, the proof is shorter using evls1varpw 19672 instead of proving it directly. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑅s 𝐵)))(𝑄𝑋)))
 
Theoremevl1varpwval 19707 Value of a univariate polynomial evaluation mapping the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐶𝐵)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)       (𝜑 → ((𝑄‘(𝑁 𝑋))‘𝐶) = (𝑁𝐸𝐶))
 
Theoremevl1scvarpw 19708 Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &    × = ( ·𝑠𝑊)    &   (𝜑𝐴𝐵)    &   𝑆 = (𝑅s 𝐵)    &    = (.r𝑆)    &   𝑀 = (mulGrp‘𝑆)    &   𝐹 = (.g𝑀)       (𝜑 → (𝑄‘(𝐴 × (𝑁 𝑋))) = ((𝐵 × {𝐴}) (𝑁𝐹(𝑄𝑋))))
 
Theoremevl1scvarpwval 19709 Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &    × = ( ·𝑠𝑊)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &    · = (.r𝑅)       (𝜑 → ((𝑄‘(𝐴 × (𝑁 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶)))
 
Theoremevl1gsummon 19710* Value of a univariate polynomial evaluation mapping an additive group sum of a multiple of an exponentiation of a variable to a group sum of the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐵 = (Base‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &   𝐺 = (mulGrp‘𝑊)    &    = (.g𝐺)    &    × = ( ·𝑠𝑊)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑 → ∀𝑥𝑀 𝐴𝐾)    &   (𝜑𝑀 ⊆ ℕ0)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑 → ∀𝑥𝑀 𝑁 ∈ ℕ0)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑊 Σg (𝑥𝑀 ↦ (𝐴 × (𝑁 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))))
 
10.11  The complex numbers as an algebraic extensible structure
 
10.11.1  Definition and basic properties
 
Syntaxcpsmet 19711 Extend class notation with the class of all pseudometric spaces.
class PsMet
 
Syntaxcxmt 19712 Extend class notation with the class of all extended metric spaces.
class ∞Met
 
Syntaxcme 19713 Extend class notation with the class of all metrics.
class Met
 
Syntaxcbl 19714 Extend class notation with the metric space ball function.
class ball
 
Syntaxcfbas 19715 Extend class definition to include the class of filter bases.
class fBas
 
Syntaxcfg 19716 Extend class definition to include the filter generating function.
class filGen
 
Syntaxcmopn 19717 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
 
Syntaxcmetu 19718 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
 
Definitiondf-psmet 19719* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
 
Definitiondf-xmet 19720* Define the set of all extended metrics on a given base set. The definition is similar to df-met 19721, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
 
Definitiondf-met 19721* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 22107. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 22129, metgt0 22145, metsym 22136, and mettri 22138. (Contributed by NM, 25-Aug-2006.)
Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
 
Definitiondf-bl 19722* Define the metric space ball function. See blval 22172 for its value. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
 
Definitiondf-mopn 19723 Define a function whose value is the family of open sets of a metric space. See elmopn 22228 for its main property. (Contributed by NM, 1-Sep-2006.)
MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
 
Definitiondf-fbas 19724* Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑥 ∩ 𝒫 (𝑦𝑧)) ≠ ∅)})
 
Definitiondf-fg 19725* Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
 
Definitiondf-metu 19726* Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
 
Syntaxccnfld 19727 Extend class notation with the field of complex numbers.
class fld
 
Definitiondf-cnfld 19728 The field of complex numbers. Other number fields and rings can be constructed by applying the s restriction operator, for instance (ℂfld ↾ 𝔸) is the field of algebraic numbers.

The contract of this set is defined entirely by cnfldex 19730, cnfldadd 19732, cnfldmul 19733, cnfldcj 19734, cnfldtset 19735, cnfldle 19736, cnfldds 19737, and cnfldbas 19731. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (New usage is discouraged.)

fld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
 
Theoremcnfldstr 19729 The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
fld Struct ⟨1, 13⟩
 
Theoremcnfldex 19730 The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
fld ∈ V
 
Theoremcnfldbas 19731 The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
ℂ = (Base‘ℂfld)
 
Theoremcnfldadd 19732 The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
+ = (+g‘ℂfld)
 
Theoremcnfldmul 19733 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
· = (.r‘ℂfld)
 
Theoremcnfldcj 19734 The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.)
∗ = (*𝑟‘ℂfld)
 
Theoremcnfldtset 19735 The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
(MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
 
Theoremcnfldle 19736 The ordering of the field of complex numbers. (Note that this is not actually an ordering on , but we put it in the structure anyway because restricting to does not affect this component, so that (ℂflds ℝ) is an ordered field even though fld itself is not.) (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
≤ = (le‘ℂfld)
 
Theoremcnfldds 19737 The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
(abs ∘ − ) = (dist‘ℂfld)
 
Theoremcnfldunif 19738 The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)
 
Theoremcnfldfun 19739 The field of complex numbers is a function. (Contributed by AV, 14-Nov-2021.)
Fun ℂfld
 
TheoremcnfldfunALT 19740 Alternate proof of cnfldfun 19739 (much shorter proof, using cnfldstr 19729 and structn0fun 15850: in addition, it must be shown that ∅ ∉ ℂfld). (Contributed by AV, 18-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Fun ℂfld
 
Theoremxrsstr 19741 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 Struct ⟨1, 12⟩
 
Theoremxrsex 19742 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 ∈ V
 
Theoremxrsbas 19743 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
* = (Base‘ℝ*𝑠)
 
Theoremxrsadd 19744 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 = (+g‘ℝ*𝑠)
 
Theoremxrsmul 19745 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e = (.r‘ℝ*𝑠)
 
Theoremxrstset 19746 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
(ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)
 
Theoremxrsle 19747 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
≤ = (le‘ℝ*𝑠)
 
Theoremcncrng 19748 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
fld ∈ CRing
 
Theoremcnring 19749 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ Ring
 
Theoremxrsmcmn 19750 The multiplicative group of the extended reals forms a commutative monoid (even though the additive group is not, see xrsmgmdifsgrp 19764.) (Contributed by Mario Carneiro, 21-Aug-2015.)
(mulGrp‘ℝ*𝑠) ∈ CMnd
 
Theoremcnfld0 19751 The zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
0 = (0g‘ℂfld)
 
Theoremcnfld1 19752 The unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r‘ℂfld)
 
Theoremcnfldneg 19753 The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋)
 
Theoremcnfldplusf 19754 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
+ = (+𝑓‘ℂfld)
 
Theoremcnfldsub 19755 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
− = (-g‘ℂfld)
 
Theoremcndrng 19756 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ DivRing
 
Theoremcnflddiv 19757 The division operation in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
/ = (/r‘ℂfld)
 
Theoremcnfldinv 19758 The multiplicative inverse in the field of complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → ((invr‘ℂfld)‘𝑋) = (1 / 𝑋))
 
Theoremcnfldmulg 19759 The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℂ) → (𝐴(.g‘ℂfld)𝐵) = (𝐴 · 𝐵))
 
Theoremcnfldexp 19760 The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴𝐵))
 
Theoremcnsrng 19761 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
fld ∈ *-Ring
 
Theoremxrsmgm 19762 The (additive group of the) extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∈ Mgm
 
Theoremxrsnsgrp 19763 The (additive group of the) extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∉ SGrp
 
Theoremxrsmgmdifsgrp 19764 The (additive group of the) extended reals is a magma, but not a semigroup, and therefore also no monoid and no group, in contrast to the multiplicative group, see xrsmcmn 19750. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∈ (Mgm ∖ SGrp)
 
Theoremxrs1mnd 19765 The extended real numbers, restricted to * ∖ {-∞}, form a monoid - in contrast to the full structure, see xrsmgmdifsgrp 19764. (Contributed by Mario Carneiro, 27-Nov-2014.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       𝑅 ∈ Mnd
 
Theoremxrs10 19766 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       0 = (0g𝑅)
 
Theoremxrs1cmn 19767 The extended real numbers restricted to * ∖ {-∞} form a commutative monoid. They are not a group because 1 + +∞ = 2 + +∞ even though 1 ≠ 2. (Contributed by Mario Carneiro, 27-Nov-2014.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       𝑅 ∈ CMnd
 
Theoremxrge0subm 19768 The nonnegative extended real numbers are a submonoid of the nonnegative-infinite extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       (0[,]+∞) ∈ (SubMnd‘𝑅)
 
Theoremxrge0cmn 19769 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
(ℝ*𝑠s (0[,]+∞)) ∈ CMnd
 
Theoremxrsds 19770* The metric of the extended real number structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐷 = (dist‘ℝ*𝑠)       𝐷 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦)))
 
Theoremxrsdsval 19771 The metric of the extended real number structure. (Contributed by Mario Carneiro, 20-Aug-2015.)
𝐷 = (dist‘ℝ*𝑠)       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐷𝐵) = if(𝐴𝐵, (𝐵 +𝑒 -𝑒𝐴), (𝐴 +𝑒 -𝑒𝐵)))
 
Theoremxrsdsreval 19772 The metric of the extended real number structure coincides with the real number metric on the reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐷 = (dist‘ℝ*𝑠)       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴𝐵)))
 
Theoremxrsdsreclblem 19773 Lemma for xrsdsreclb 19774. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐷 = (dist‘ℝ*𝑠)       (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) ∧ 𝐴𝐵) → ((𝐵 +𝑒 -𝑒𝐴) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
 
Theoremxrsdsreclb 19774 The metric of the extended real number structure is only real when both arguments are real. (Contributed by Mario Carneiro, 3-Sep-2015.)
𝐷 = (dist‘ℝ*𝑠)       ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐴𝐵) → ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
 
Theoremcnsubmlem 19775* Lemma for nn0subm 19782 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   0 ∈ 𝐴       𝐴 ∈ (SubMnd‘ℂfld)
 
Theoremcnsubglem 19776* Lemma for resubdrg 19935 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥𝐴 → -𝑥𝐴)    &   𝐵𝐴       𝐴 ∈ (SubGrp‘ℂfld)
 
Theoremcnsubrglem 19777* Lemma for resubdrg 19935 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥𝐴 → -𝑥𝐴)    &   1 ∈ 𝐴    &   ((𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)       𝐴 ∈ (SubRing‘ℂfld)
 
Theoremcnsubdrglem 19778* Lemma for resubdrg 19935 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥𝐴 → -𝑥𝐴)    &   1 ∈ 𝐴    &   ((𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    &   ((𝑥𝐴𝑥 ≠ 0) → (1 / 𝑥) ∈ 𝐴)       (𝐴 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝐴) ∈ DivRing)
 
Theoremqsubdrg 19779 The rational numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
(ℚ ∈ (SubRing‘ℂfld) ∧ (ℂflds ℚ) ∈ DivRing)
 
Theoremzsubrg 19780 The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ ∈ (SubRing‘ℂfld)
 
Theoremgzsubrg 19781 The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ[i] ∈ (SubRing‘ℂfld)
 
Theoremnn0subm 19782 The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
0 ∈ (SubMnd‘ℂfld)
 
Theoremrege0subm 19783 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
(0[,)+∞) ∈ (SubMnd‘ℂfld)
 
Theoremabsabv 19784 The regular absolute value function on the complex numbers is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 4-Dec-2014.)
abs ∈ (AbsVal‘ℂfld)
 
Theoremzsssubrg 19785 The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
 
Theoremqsssubdrg 19786 The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂflds 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅)
 
Theoremcnsubrg 19787 There are no subrings of the complex numbers strictly between and . (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝑅 ∈ (SubRing‘ℂfld) ∧ ℝ ⊆ 𝑅) → 𝑅 ∈ {ℝ, ℂ})
 
Theoremcnmgpabl 19788 The unit group of the complex numbers is an abelian group. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))       𝑀 ∈ Abel
 
Theoremcnmgpid 19789 The group identity element of nonzero complex number multiplication is one. (Contributed by Steve Rodriguez, 23-Feb-2007.) (Revised by AV, 26-Aug-2021.)
𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))       (0g𝑀) = 1
 
Theoremcnmsubglem 19790* Lemma for rpmsubg 19791 and friends. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))    &   (𝑥𝐴𝑥 ∈ ℂ)    &   (𝑥𝐴𝑥 ≠ 0)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    &   1 ∈ 𝐴    &   (𝑥𝐴 → (1 / 𝑥) ∈ 𝐴)       𝐴 ∈ (SubGrp‘𝑀)
 
Theoremrpmsubg 19791 The positive reals form a multiplicative subgroup of the complex numbers. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))       + ∈ (SubGrp‘𝑀)
 
Theoremgzrngunitlem 19792 Lemma for gzrngunit 19793. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑍 = (ℂflds ℤ[i])       (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))
 
Theoremgzrngunit 19793 The units on ℤ[i] are the gaussian integers with norm 1. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑍 = (ℂflds ℤ[i])       (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1))
 
Theoremgsumfsum 19794* Relate a group sum on fld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)       (𝜑 → (ℂfld Σg (𝑘𝐴𝐵)) = Σ𝑘𝐴 𝐵)
 
Theoremregsumfsum 19795* Relate a group sum on (ℂflds ℝ) to a finite sum on the reals. Cf. gsumfsum 19794. (Contributed by Thierry Arnoux, 7-Sep-2018.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((ℂflds ℝ) Σg (𝑘𝐴𝐵)) = Σ𝑘𝐴 𝐵)
 
Theoremexpmhm 19796* Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑁 = (ℂflds0)    &   𝑀 = (mulGrp‘ℂfld)       (𝐴 ∈ ℂ → (𝑥 ∈ ℕ0 ↦ (𝐴𝑥)) ∈ (𝑁 MndHom 𝑀))
 
Theoremnn0srg 19797 The nonnegative integers form a semiring (commutative by subcmn 18223). (Contributed by Thierry Arnoux, 1-May-2018.)
(ℂflds0) ∈ SRing
 
Theoremrge0srg 19798 The nonnegative real numbers form a semiring (commutative by subcmn 18223). (Contributed by Thierry Arnoux, 6-Sep-2018.)
(ℂflds (0[,)+∞)) ∈ SRing
 
10.11.2  Ring of integers

According to Wikipedia ("Integer", 25-May-2019, https://en.wikipedia.org/wiki/Integer) "The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of [unital] rings, characterizes the ring 𝑍." In set.mm, there was no explicit definition for the ring of integers until June 2019, but it was denoted by (ℂflds ℤ), the field of complex numbers restricted to the integers. In zringring 19802 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 19818), and zringbas 19805 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition df-zring 19800 of the ring of integers.

Remark: Instead of using the symbol "ZZrng" analogous to fld used for the field of complex numbers, we have chosen the version with an "i" to indicate that the ring of integers is a unital ring, see also Wikipedia ("Rng (algebra)", 9-Jun-2019, https://en.wikipedia.org/wiki/Rng_(algebra)).

 
Syntaxzring 19799 Extend class notation with the (unital) ring of integers.
class ring
 
Definitiondf-zring 19800 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
ring = (ℂflds ℤ)
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