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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremevl1vsd 20501 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑𝑁𝐵)    &    = ( ·𝑠𝑃)    &    · = (.r𝑅)       (𝜑 → ((𝑁 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 𝑀))‘𝑌) = (𝑁 · 𝑉)))

Theoremevl1expd 20502 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &    = (.g‘(mulGrp‘𝑃))    &    = (.g‘(mulGrp‘𝑅))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑁 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 𝑀))‘𝑌) = (𝑁 𝑉)))

Theorempf1const 20503 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝐵 × {𝑋}) ∈ 𝑄)

Theorempf1id 20504 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ 𝑄)

Theorempf1subrg 20505 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       (𝑅 ∈ CRing → 𝑄 ∈ (SubRing‘(𝑅s 𝐵)))

Theorempf1rcl 20506 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)       (𝑋𝑄𝑅 ∈ CRing)

Theorempf1f 20507 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝐹𝑄𝐹:𝐵𝐵)

Theoremmpfpf1 20508* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐸 = ran (1o eval 𝑅)       (𝐹𝐸 → (𝐹 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)

Theorempf1mpf 20509* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐸 = ran (1o eval 𝑅)       (𝐹𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)

Theorempf1addcl 20510 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    + = (+g𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹f + 𝐺) ∈ 𝑄)

Theorempf1mulcl 20511 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    · = (.r𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹f · 𝐺) ∈ 𝑄)

Theorempf1ind 20512* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝑄 = ran (eval1𝑅)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)    &   (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))    &   (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))    &   (𝑥 = 𝑓 → (𝜓𝜏))    &   (𝑥 = 𝑔 → (𝜓𝜂))    &   (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))    &   (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))    &   (𝑥 = 𝐴 → (𝜓𝜌))    &   ((𝜑𝑓𝐵) → 𝜒)    &   (𝜑𝜃)    &   (𝜑𝐴𝑄)       (𝜑𝜌)

Theoremevl1gsumdlem 20513* Lemma for evl1gsumd 20514 (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 20514* 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 20515* 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 20481. (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 20516* 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 20517* 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 20518 Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 20515, the proof is shorter using evls1varpw 20484 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 20519 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 20520 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 20521 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 20522* 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)))))

10.10  The complex numbers as an algebraic extensible structure

10.10.1  Definition and basic properties

Syntaxcpsmet 20523 Extend class notation with the class of all pseudometric spaces.
class PsMet

Syntaxcxmet 20524 Extend class notation with the class of all extended metric spaces.
class ∞Met

Syntaxcmet 20525 Extend class notation with the class of all metrics.
class Met

Syntaxcbl 20526 Extend class notation with the metric space ball function.
class ball

Syntaxcfbas 20527 Extend class definition to include the class of filter bases.
class fBas

Syntaxcfg 20528 Extend class definition to include the filter generating function.
class filGen

Syntaxcmopn 20529 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen

Syntaxcmetu 20530 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif

Definitiondf-psmet 20531* 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 20532* Define the set of all extended metrics on a given base set. The definition is similar to df-met 20533, 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 20533* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 22925. 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 22947, metgt0 22963, metsym 22954, and mettri 22956. (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 20534* Define the metric space ball function. See blval 22990 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 20535 Define a function whose value is the family of open sets of a metric space. See elmopn 23046 for its main property. (Contributed by NM, 1-Sep-2006.)
MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))

Definitiondf-fbas 20536* 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 20537* 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 20538* 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 20539 Extend class notation with the field of complex numbers.
class fld

Definitiondf-cnfld 20540 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 20542, cnfldadd 20544, cnfldmul 20545, cnfldcj 20546, cnfldtset 20547, cnfldle 20548, cnfldds 20549, and cnfldbas 20543. 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 20541 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 20542 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 20543 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 20544 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 20545 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 20546 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 20547 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 20548 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 20549 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 20550 The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.)
(metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld)

Theoremcnfldfun 20551 The field of complex numbers is a function. (Contributed by AV, 14-Nov-2021.)
Fun ℂfld

TheoremcnfldfunALT 20552 Alternate proof of cnfldfun 20551 (much shorter proof, using cnfldstr 20541 and structn0fun 16489: 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 20553 The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 Struct ⟨1, 12⟩

Theoremxrsex 20554 The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.)
*𝑠 ∈ V

Theoremxrsbas 20555 The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
* = (Base‘ℝ*𝑠)

Theoremxrsadd 20556 The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
+𝑒 = (+g‘ℝ*𝑠)

Theoremxrsmul 20557 The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
·e = (.r‘ℝ*𝑠)

Theoremxrstset 20558 The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
(ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠)

Theoremxrsle 20559 The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.)
≤ = (le‘ℝ*𝑠)

Theoremcncrng 20560 The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
fld ∈ CRing

Theoremcnring 20561 The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ Ring

Theoremxrsmcmn 20562 The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 20576.) (Contributed by Mario Carneiro, 21-Aug-2015.)
(mulGrp‘ℝ*𝑠) ∈ CMnd

Theoremcnfld0 20563 Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
0 = (0g‘ℂfld)

Theoremcnfld1 20564 One is the unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r‘ℂfld)

Theoremcnfldneg 20565 The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
(𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋)

Theoremcnfldplusf 20566 The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
+ = (+𝑓‘ℂfld)

Theoremcnfldsub 20567 The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
− = (-g‘ℂfld)

Theoremcndrng 20568 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
fld ∈ DivRing

Theoremcnflddiv 20569 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 20570 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 20571 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 20572 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 20573 The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
fld ∈ *-Ring

Theoremxrsmgm 20574 The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∈ Mgm

Theoremxrsnsgrp 20575 The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∉ Smgrp

Theoremxrsmgmdifsgrp 20576 The "additive group" of the extended reals is a magma but not a semigroup, and therefore also not a monoid nor a group, in contrast to the "multiplicative group", see xrsmcmn 20562. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∈ (Mgm ∖ Smgrp)

Theoremxrs1mnd 20577 The extended real numbers, restricted to * ∖ {-∞}, form an additive monoid - in contrast to the full structure, see xrsmgmdifsgrp 20576. (Contributed by Mario Carneiro, 27-Nov-2014.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       𝑅 ∈ Mnd

Theoremxrs10 20578 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       0 = (0g𝑅)

Theoremxrs1cmn 20579 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 20580 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 20581 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
(ℝ*𝑠s (0[,]+∞)) ∈ CMnd

Theoremxrsds 20582* 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 20583 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 20584 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 20585 Lemma for xrsdsreclb 20586. (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 20586 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 20587* Lemma for nn0subm 20594 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   0 ∈ 𝐴       𝐴 ∈ (SubMnd‘ℂfld)

Theoremcnsubglem 20588* Lemma for resubdrg 20746 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥𝐴 → -𝑥𝐴)    &   𝐵𝐴       𝐴 ∈ (SubGrp‘ℂfld)

Theoremcnsubrglem 20589* Lemma for resubdrg 20746 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 20590* Lemma for resubdrg 20746 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 20591 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 20592 The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ ∈ (SubRing‘ℂfld)

Theoremgzsubrg 20593 The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ[i] ∈ (SubRing‘ℂfld)

Theoremnn0subm 20594 The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
0 ∈ (SubMnd‘ℂfld)

Theoremrege0subm 20595 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
(0[,)+∞) ∈ (SubMnd‘ℂfld)

Theoremabsabv 20596 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 20597 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 20598 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 20599 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 20600 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

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