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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-idom 20901 | An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ IDomn = (CRing ∩ Domn) | ||
Definition | df-pid 20902 | A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ PID = (IDomn ∩ LPIR) | ||
Theorem | rrgval 20903* | Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} | ||
Theorem | isrrg 20904* | Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) | ||
Theorem | rrgeq0i 20905 | Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )) | ||
Theorem | rrgeq0 20906 | Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 )) | ||
Theorem | rrgsupp 20907 | Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.) |
⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐸) & ⊢ (𝜑 → 𝑌:𝐼⟶𝐵) ⇒ ⊢ (𝜑 → (((𝐼 × {𝑋}) ∘f · 𝑌) supp 0 ) = (𝑌 supp 0 )) | ||
Theorem | rrgss 20908 | Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ 𝐸 ⊆ 𝐵 | ||
Theorem | unitrrg 20909 | Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ 𝐸) | ||
Theorem | isdomn 20910* | Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))) | ||
Theorem | domnnzr 20911 | A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.) |
⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | ||
Theorem | domnring 20912 | A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.) |
⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | ||
Theorem | domneq0 20913 | In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) | ||
Theorem | domnmuln0 20914 | In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 ) | ||
Theorem | isdomn2 20915 | A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸)) | ||
Theorem | domnrrg 20916 | In a domain, any nonzero element is a nonzero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸) | ||
Theorem | isdomn5 20917* | The right conjunct in the right hand side of the equivalence of isdomn 20910 is logically equivalent to a less symmetric version where one of the variables is restricted to be nonzero. (Contributed by SN, 16-Sep-2024.) |
⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) | ||
Theorem | isdomn4 20918* | A ring is a domain iff it is nonzero and the cancellation law for multiplication holds. (Contributed by SN, 15-Sep-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎 · 𝑏) = (𝑎 · 𝑐) → 𝑏 = 𝑐))) | ||
Theorem | opprdomn 20919 | The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn) | ||
Theorem | abvn0b 20920 | Another characterization of domains, hinted at in abvtriv 20449: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ 𝐴 = (AbsVal‘𝑅) ⇒ ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅)) | ||
Theorem | drngdomn 20921 | A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Domn) | ||
Theorem | isidom 20922 | An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn)) | ||
Theorem | fldidom 20923 | A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.) |
⊢ (𝑅 ∈ Field → 𝑅 ∈ IDomn) | ||
Theorem | fldidomOLD 20924 | Obsolete version of fldidom 20923 as of 11-Nov-2024. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑅 ∈ Field → 𝑅 ∈ IDomn) | ||
Theorem | fidomndrnglem 20925* | Lemma for fidomndrng 20926. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ { 0 })) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝐴)) ⇒ ⊢ (𝜑 → 𝐴 ∥ 1 ) | ||
Theorem | fidomndrng 20926 | A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing)) | ||
Theorem | fiidomfld 20927 | A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝐵 ∈ Fin → (𝑅 ∈ IDomn ↔ 𝑅 ∈ Field)) | ||
Syntax | cpsmet 20928 | Extend class notation with the class of all pseudometric spaces. |
class PsMet | ||
Syntax | cxmet 20929 | Extend class notation with the class of all extended metric spaces. |
class ∞Met | ||
Syntax | cmet 20930 | Extend class notation with the class of all metrics. |
class Met | ||
Syntax | cbl 20931 | Extend class notation with the metric space ball function. |
class ball | ||
Syntax | cfbas 20932 | Extend class definition to include the class of filter bases. |
class fBas | ||
Syntax | cfg 20933 | Extend class definition to include the filter generating function. |
class filGen | ||
Syntax | cmopn 20934 | Extend class notation with a function mapping each metric space to the family of its open sets. |
class MetOpen | ||
Syntax | cmetu 20935 | Extend class notation with the function mapping metrics to the uniform structure generated by that metric. |
class metUnif | ||
Definition | df-psmet 20936* | 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)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) | ||
Definition | df-xmet 20937* | Define the set of all extended metrics on a given base set. The definition is similar to df-met 20938, 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)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))}) | ||
Definition | df-met 20938* | Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 23827. 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 23849, metgt0 23865, metsym 23856, and mettri 23858. (Contributed by NM, 25-Aug-2006.) |
⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))}) | ||
Definition | df-bl 20939* | Define the metric space ball function. See blval 23892 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})) | ||
Definition | df-mopn 20940 | Define a function whose value is the family of open sets of a metric space. See elmopn 23948 for its main property. (Contributed by NM, 1-Sep-2006.) |
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | ||
Definition | df-fbas 20941* | 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 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)}) | ||
Definition | df-fg 20942* | 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) ≠ ∅}) | ||
Definition | df-metu 20943* | 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))))) | ||
Syntax | ccnfld 20944 | Extend class notation with the field of complex numbers. |
class ℂfld | ||
Definition | df-cnfld 20945 |
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 20947, cnfldadd 20949, cnfldmul 20950, cnfldcj 20951, cnfldtset 20952, cnfldle 20953, cnfldds 20954, and cnfldbas 20948. 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.) |
⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) | ||
Theorem | cnfldstr 20946 | The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) |
⊢ ℂfld Struct ⟨1, ;13⟩ | ||
Theorem | cnfldex 20947 | 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.) |
⊢ ℂfld ∈ V | ||
Theorem | cnfldbas 20948 | 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) | ||
Theorem | cnfldadd 20949 | 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) | ||
Theorem | cnfldmul 20950 | 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) | ||
Theorem | cnfldcj 20951 | 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) | ||
Theorem | cnfldtset 20952 | 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) | ||
Theorem | cnfldle 20953 | The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld 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) | ||
Theorem | cnfldds 20954 | 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) | ||
Theorem | cnfldunif 20955 | The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | ||
Theorem | cnfldfun 20956 | The field of complex numbers is a function. The proof is much shorter than the proof of cnfldfunALT 20957 by using cnfldstr 20946 and structn0fun 17084: in addition, it must be shown that ∅ ∉ ℂfld. (Contributed by AV, 18-Nov-2021.) |
⊢ Fun ℂfld | ||
Theorem | cnfldfunALT 20957 | The field of complex numbers is a function. Alternate proof of cnfldfun 20956 not requiring that the index set of the components is ordered, but using quadratically many inequalities for the indices. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 11-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Fun ℂfld | ||
Theorem | cnfldfunALTOLD 20958 | Obsolete proof of cnfldfunALT 20957 as of 10-Nov-2024. The field of complex numbers is a function. (Contributed by AV, 14-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Fun ℂfld | ||
Theorem | xrsstr 20959 | The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ*𝑠 Struct ⟨1, ;12⟩ | ||
Theorem | xrsex 20960 | The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ*𝑠 ∈ V | ||
Theorem | xrsbas 20961 | The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ* = (Base‘ℝ*𝑠) | ||
Theorem | xrsadd 20962 | The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ +𝑒 = (+g‘ℝ*𝑠) | ||
Theorem | xrsmul 20963 | The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ·e = (.r‘ℝ*𝑠) | ||
Theorem | xrstset 20964 | The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠) | ||
Theorem | xrsle 20965 | The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ≤ = (le‘ℝ*𝑠) | ||
Theorem | cncrng 20966 | The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) |
⊢ ℂfld ∈ CRing | ||
Theorem | cnring 20967 | The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ ℂfld ∈ Ring | ||
Theorem | xrsmcmn 20968 | The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 20982.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd | ||
Theorem | cnfld0 20969 | Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 0 = (0g‘ℂfld) | ||
Theorem | cnfld1 20970 | One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 1 = (1r‘ℂfld) | ||
Theorem | cnfldneg 20971 | The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ (𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋) | ||
Theorem | cnfldplusf 20972 | The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ + = (+𝑓‘ℂfld) | ||
Theorem | cnfldsub 20973 | The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ − = (-g‘ℂfld) | ||
Theorem | cndrng 20974 | The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ ℂfld ∈ DivRing | ||
Theorem | cnflddiv 20975 | 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) | ||
Theorem | cnfldinv 20976 | 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)) | ||
Theorem | cnfldmulg 20977 | 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)) | ||
Theorem | cnfldexp 20978 | 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) = (𝐴↑𝐵)) | ||
Theorem | cnsrng 20979 | The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ℂfld ∈ *-Ring | ||
Theorem | xrsmgm 20980 | The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.) |
⊢ ℝ*𝑠 ∈ Mgm | ||
Theorem | xrsnsgrp 20981 | The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.) |
⊢ ℝ*𝑠 ∉ Smgrp | ||
Theorem | xrsmgmdifsgrp 20982 | 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 20968. (Contributed by AV, 30-Jan-2020.) |
⊢ ℝ*𝑠 ∈ (Mgm ∖ Smgrp) | ||
Theorem | xrs1mnd 20983 | The extended real numbers, restricted to ℝ* ∖ {-∞}, form an additive monoid - in contrast to the full structure, see xrsmgmdifsgrp 20982. (Contributed by Mario Carneiro, 27-Nov-2014.) |
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ⇒ ⊢ 𝑅 ∈ Mnd | ||
Theorem | xrs10 20984 | The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ⇒ ⊢ 0 = (0g‘𝑅) | ||
Theorem | xrs1cmn 20985 | The extended real numbers restricted to ℝ* ∖ {-∞} 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*𝑠 ↾s (ℝ* ∖ {-∞})) ⇒ ⊢ 𝑅 ∈ CMnd | ||
Theorem | xrge0subm 20986 | The nonnegative extended real numbers are a submonoid of the nonnegative-infinite extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ⇒ ⊢ (0[,]+∞) ∈ (SubMnd‘𝑅) | ||
Theorem | xrge0cmn 20987 | The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | ||
Theorem | xrsds 20988* | 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(𝑥 ≤ 𝑦, (𝑦 +𝑒 -𝑒𝑥), (𝑥 +𝑒 -𝑒𝑦))) | ||
Theorem | xrsdsval 20989 | 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(𝐴 ≤ 𝐵, (𝐵 +𝑒 -𝑒𝐴), (𝐴 +𝑒 -𝑒𝐵))) | ||
Theorem | xrsdsreval 20990 | 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‘(𝐴 − 𝐵))) | ||
Theorem | xrsdsreclblem 20991 | Lemma for xrsdsreclb 20992. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ 𝐷 = (dist‘ℝ*𝑠) ⇒ ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≠ 𝐵) ∧ 𝐴 ≤ 𝐵) → ((𝐵 +𝑒 -𝑒𝐴) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))) | ||
Theorem | xrsdsreclb 20992 | 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* ∧ 𝐴 ≠ 𝐵) → ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))) | ||
Theorem | cnsubmlem 20993* | Lemma for nn0subm 21000 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) & ⊢ 0 ∈ 𝐴 ⇒ ⊢ 𝐴 ∈ (SubMnd‘ℂfld) | ||
Theorem | cnsubglem 20994* | Lemma for resubdrg 21161 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) & ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) & ⊢ 𝐵 ∈ 𝐴 ⇒ ⊢ 𝐴 ∈ (SubGrp‘ℂfld) | ||
Theorem | cnsubrglem 20995* | Lemma for resubdrg 21161 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) & ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) & ⊢ 1 ∈ 𝐴 & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) ⇒ ⊢ 𝐴 ∈ (SubRing‘ℂfld) | ||
Theorem | cnsubdrglem 20996* | Lemma for resubdrg 21161 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴) & ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴) & ⊢ 1 ∈ 𝐴 & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 0) → (1 / 𝑥) ∈ 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝐴) ∈ DivRing) | ||
Theorem | qsubdrg 20997 | The rational numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ (ℚ ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s ℚ) ∈ DivRing) | ||
Theorem | zsubrg 20998 | The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ ℤ ∈ (SubRing‘ℂfld) | ||
Theorem | gzsubrg 20999 | The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ ℤ[i] ∈ (SubRing‘ℂfld) | ||
Theorem | nn0subm 21000 | The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ ℕ0 ∈ (SubMnd‘ℂfld) |
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