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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cnfldds 20101 | 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 20102 | The uniform structure component of the complex numbers. (Contributed by Thierry Arnoux, 17-Dec-2017.) |
⊢ (metUnif‘(abs ∘ − )) = (UnifSet‘ℂfld) | ||
Theorem | cnfldfun 20103 | The field of complex numbers is a function. (Contributed by AV, 14-Nov-2021.) |
⊢ Fun ℂfld | ||
Theorem | cnfldfunALT 20104 | Alternate proof of cnfldfun 20103 (much shorter proof, using cnfldstr 20093 and structn0fun 16487: in addition, it must be shown that ∅ ∉ ℂfld). (Contributed by AV, 18-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Fun ℂfld | ||
Theorem | xrsstr 20105 | The extended real structure is a structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ*𝑠 Struct 〈1, ;12〉 | ||
Theorem | xrsex 20106 | The extended real structure is a set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ*𝑠 ∈ V | ||
Theorem | xrsbas 20107 | The base set of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ℝ* = (Base‘ℝ*𝑠) | ||
Theorem | xrsadd 20108 | The addition operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ +𝑒 = (+g‘ℝ*𝑠) | ||
Theorem | xrsmul 20109 | The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ·e = (.r‘ℝ*𝑠) | ||
Theorem | xrstset 20110 | The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (ordTop‘ ≤ ) = (TopSet‘ℝ*𝑠) | ||
Theorem | xrsle 20111 | The ordering of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ ≤ = (le‘ℝ*𝑠) | ||
Theorem | cncrng 20112 | The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) |
⊢ ℂfld ∈ CRing | ||
Theorem | cnring 20113 | The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ ℂfld ∈ Ring | ||
Theorem | xrsmcmn 20114 | The "multiplicative group" of the extended reals is a commutative monoid (even though the "additive group" is not a semigroup, see xrsmgmdifsgrp 20128.) (Contributed by Mario Carneiro, 21-Aug-2015.) |
⊢ (mulGrp‘ℝ*𝑠) ∈ CMnd | ||
Theorem | cnfld0 20115 | Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 0 = (0g‘ℂfld) | ||
Theorem | cnfld1 20116 | One is the unit element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 1 = (1r‘ℂfld) | ||
Theorem | cnfldneg 20117 | 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 20118 | The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.) |
⊢ + = (+𝑓‘ℂfld) | ||
Theorem | cnfldsub 20119 | The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ − = (-g‘ℂfld) | ||
Theorem | cndrng 20120 | The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ ℂfld ∈ DivRing | ||
Theorem | cnflddiv 20121 | 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 20122 | 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 20123 | 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 20124 | 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 20125 | The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ ℂfld ∈ *-Ring | ||
Theorem | xrsmgm 20126 | The "additive group" of the extended reals is a magma. (Contributed by AV, 30-Jan-2020.) |
⊢ ℝ*𝑠 ∈ Mgm | ||
Theorem | xrsnsgrp 20127 | The "additive group" of the extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.) |
⊢ ℝ*𝑠 ∉ Smgrp | ||
Theorem | xrsmgmdifsgrp 20128 | 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 20114. (Contributed by AV, 30-Jan-2020.) |
⊢ ℝ*𝑠 ∈ (Mgm ∖ Smgrp) | ||
Theorem | xrs1mnd 20129 | The extended real numbers, restricted to ℝ* ∖ {-∞}, form an additive monoid - in contrast to the full structure, see xrsmgmdifsgrp 20128. (Contributed by Mario Carneiro, 27-Nov-2014.) |
⊢ 𝑅 = (ℝ*𝑠 ↾s (ℝ* ∖ {-∞})) ⇒ ⊢ 𝑅 ∈ Mnd | ||
Theorem | xrs10 20130 | 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 20131 | 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 20132 | 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 20133 | The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.) |
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | ||
Theorem | xrsds 20134* | 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 20135 | 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 20136 | 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 20137 | Lemma for xrsdsreclb 20138. (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 20138 | 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 20139* | Lemma for nn0subm 20146 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 20140* | Lemma for resubdrg 20297 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 20141* | Lemma for resubdrg 20297 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 20142* | Lemma for resubdrg 20297 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 20143 | 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 20144 | The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ ℤ ∈ (SubRing‘ℂfld) | ||
Theorem | gzsubrg 20145 | The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ ℤ[i] ∈ (SubRing‘ℂfld) | ||
Theorem | nn0subm 20146 | The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ ℕ0 ∈ (SubMnd‘ℂfld) | ||
Theorem | rege0subm 20147 | The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.) |
⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld) | ||
Theorem | absabv 20148 | 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) | ||
Theorem | zsssubrg 20149 | The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅) | ||
Theorem | qsssubdrg 20150 | The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ (ℂfld ↾s 𝑅) ∈ DivRing) → ℚ ⊆ 𝑅) | ||
Theorem | cnsubrg 20151 | There are no subrings of the complex numbers strictly between ℝ and ℂ. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ((𝑅 ∈ (SubRing‘ℂfld) ∧ ℝ ⊆ 𝑅) → 𝑅 ∈ {ℝ, ℂ}) | ||
Theorem | cnmgpabl 20152 | 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 | ||
Theorem | cnmgpid 20153 | 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})) ⇒ ⊢ (0g‘𝑀) = 1 | ||
Theorem | cnmsubglem 20154* | Lemma for rpmsubg 20155 and friends. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) & ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ) & ⊢ (𝑥 ∈ 𝐴 → 𝑥 ≠ 0) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴) & ⊢ 1 ∈ 𝐴 & ⊢ (𝑥 ∈ 𝐴 → (1 / 𝑥) ∈ 𝐴) ⇒ ⊢ 𝐴 ∈ (SubGrp‘𝑀) | ||
Theorem | rpmsubg 20155 | The positive reals form a multiplicative subgroup of the complex numbers. (Contributed by Mario Carneiro, 21-Jun-2015.) |
⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ⇒ ⊢ ℝ+ ∈ (SubGrp‘𝑀) | ||
Theorem | gzrngunitlem 20156 | Lemma for gzrngunit 20157. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑍 = (ℂfld ↾s ℤ[i]) ⇒ ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴)) | ||
Theorem | gzrngunit 20157 | The units on ℤ[i] are the gaussian integers with norm 1. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑍 = (ℂfld ↾s ℤ[i]) ⇒ ⊢ (𝐴 ∈ (Unit‘𝑍) ↔ (𝐴 ∈ ℤ[i] ∧ (abs‘𝐴) = 1)) | ||
Theorem | gsumfsum 20158* | Relate a group sum on ℂfld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ⇒ ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | regsumfsum 20159* | Relate a group sum on (ℂfld ↾s ℝ) to a finite sum on the reals. Cf. gsumfsum 20158. (Contributed by Thierry Arnoux, 7-Sep-2018.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((ℂfld ↾s ℝ) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) | ||
Theorem | expmhm 20160* | Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝑁 = (ℂfld ↾s ℕ0) & ⊢ 𝑀 = (mulGrp‘ℂfld) ⇒ ⊢ (𝐴 ∈ ℂ → (𝑥 ∈ ℕ0 ↦ (𝐴↑𝑥)) ∈ (𝑁 MndHom 𝑀)) | ||
Theorem | nn0srg 20161 | The nonnegative integers form a semiring (commutative by subcmn 18950). (Contributed by Thierry Arnoux, 1-May-2018.) |
⊢ (ℂfld ↾s ℕ0) ∈ SRing | ||
Theorem | rge0srg 20162 | The nonnegative real numbers form a semiring (commutative by subcmn 18950). (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ (ℂfld ↾s (0[,)+∞)) ∈ SRing | ||
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 (ℂfld ↾s ℤ), the field of complex numbers restricted to the integers. In zringring 20166 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 20182), and zringbas 20169 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition df-zring 20164 of the ring of integers. Remark: Instead of using the symbol "ZZrng" analogous to ℂfld 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) 20164). | ||
Syntax | zring 20163 | Extend class notation with the (unital) ring of integers. |
class ℤring | ||
Definition | df-zring 20164 | The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.) |
⊢ ℤring = (ℂfld ↾s ℤ) | ||
Theorem | zringcrng 20165 | The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.) |
⊢ ℤring ∈ CRing | ||
Theorem | zringring 20166 | The ring of integers is a ring. (Contributed by AV, 20-May-2019.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 13-Jun-2019.) |
⊢ ℤring ∈ Ring | ||
Theorem | zringabl 20167 | The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.) |
⊢ ℤring ∈ Abel | ||
Theorem | zringgrp 20168 | The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.) |
⊢ ℤring ∈ Grp | ||
Theorem | zringbas 20169 | The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.) |
⊢ ℤ = (Base‘ℤring) | ||
Theorem | zringplusg 20170 | The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.) |
⊢ + = (+g‘ℤring) | ||
Theorem | zringmulg 20171 | The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(.g‘ℤring)𝐵) = (𝐴 · 𝐵)) | ||
Theorem | zringmulr 20172 | The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) |
⊢ · = (.r‘ℤring) | ||
Theorem | zring0 20173 | The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) |
⊢ 0 = (0g‘ℤring) | ||
Theorem | zring1 20174 | The multiplicative neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.) |
⊢ 1 = (1r‘ℤring) | ||
Theorem | zringnzr 20175 | The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.) |
⊢ ℤring ∈ NzRing | ||
Theorem | dvdsrzring 20176 | Ring divisibility in the ring of integers corresponds to ordinary divisibility in ℤ. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) |
⊢ ∥ = (∥r‘ℤring) | ||
Theorem | zringlpirlem1 20177 | Lemma for zringlpir 20182. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) |
⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) & ⊢ (𝜑 → 𝐼 ≠ {0}) ⇒ ⊢ (𝜑 → (𝐼 ∩ ℕ) ≠ ∅) | ||
Theorem | zringlpirlem2 20178 | Lemma for zringlpir 20182. A nonzero ideal of integers contains the least positive element. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Revised by AV, 27-Sep-2020.) |
⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) & ⊢ (𝜑 → 𝐼 ≠ {0}) & ⊢ 𝐺 = inf((𝐼 ∩ ℕ), ℝ, < ) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐼) | ||
Theorem | zringlpirlem3 20179 | Lemma for zringlpir 20182. All elements of a nonzero ideal of integers are divided by the least one. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 27-Sep-2020.) |
⊢ (𝜑 → 𝐼 ∈ (LIdeal‘ℤring)) & ⊢ (𝜑 → 𝐼 ≠ {0}) & ⊢ 𝐺 = inf((𝐼 ∩ ℕ), ℝ, < ) & ⊢ (𝜑 → 𝑋 ∈ 𝐼) ⇒ ⊢ (𝜑 → 𝐺 ∥ 𝑋) | ||
Theorem | zringinvg 20180 | The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
⊢ (𝐴 ∈ ℤ → -𝐴 = ((invg‘ℤring)‘𝐴)) | ||
Theorem | zringunit 20181 | The units of ℤ are the integers with norm 1, i.e. 1 and -1. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.) |
⊢ (𝐴 ∈ (Unit‘ℤring) ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) = 1)) | ||
Theorem | zringlpir 20182 | The integers are a principal ideal ring. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.) (Proof shortened by AV, 27-Sep-2020.) |
⊢ ℤring ∈ LPIR | ||
Theorem | zringndrg 20183 | The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.) |
⊢ ℤring ∉ DivRing | ||
Theorem | zringcyg 20184 | The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.) |
⊢ ℤring ∈ CycGrp | ||
Theorem | zringmpg 20185 | The multiplication group of the ring of integers is the restriction of the multiplication group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019.) |
⊢ ((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring) | ||
Theorem | prmirredlem 20186 | A positive integer is irreducible over ℤ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.) |
⊢ 𝐼 = (Irred‘ℤring) ⇒ ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ)) | ||
Theorem | dfprm2 20187 | The positive irreducible elements of ℤ are the prime numbers. This is an alternative way to define ℙ. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.) |
⊢ 𝐼 = (Irred‘ℤring) ⇒ ⊢ ℙ = (ℕ ∩ 𝐼) | ||
Theorem | prmirred 20188 | The irreducible elements of ℤ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.) |
⊢ 𝐼 = (Irred‘ℤring) ⇒ ⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ)) | ||
Theorem | expghm 20189* | Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) |
⊢ 𝑀 = (mulGrp‘ℂfld) & ⊢ 𝑈 = (𝑀 ↾s (ℂ ∖ {0})) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈)) | ||
Theorem | mulgghm2 20190* | The powers of a group element give a homomorphism from ℤ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ · = (.g‘𝑅) & ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Grp ∧ 1 ∈ 𝐵) → 𝐹 ∈ (ℤring GrpHom 𝑅)) | ||
Theorem | mulgrhm 20191* | The powers of the element 1 give a ring homomorphism from ℤ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ · = (.g‘𝑅) & ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐹 ∈ (ℤring RingHom 𝑅)) | ||
Theorem | mulgrhm2 20192* | The powers of the element 1 give the unique ring homomorphism from ℤ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ · = (.g‘𝑅) & ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (ℤring RingHom 𝑅) = {𝐹}) | ||
Syntax | czrh 20193 | Map the rationals into a field, or the integers into a ring. |
class ℤRHom | ||
Syntax | czlm 20194 | Augment an abelian group with vector space operations to turn it into a ℤ-module. |
class ℤMod | ||
Syntax | cchr 20195 | Syntax for ring characteristic. |
class chr | ||
Syntax | czn 20196 | The ring of integers modulo 𝑛. |
class ℤ/nℤ | ||
Definition | df-zrh 20197 | Define the unique homomorphism from the integers into a ring. This encodes the usual notation of 𝑛 = 1r + 1r + ... + 1r for integers (see also df-mulg 18217). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) | ||
Definition | df-zlm 20198 | Augment an abelian group with vector space operations to turn it into a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ ℤMod = (𝑔 ∈ V ↦ ((𝑔 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝑔)〉)) | ||
Definition | df-chr 20199 | The characteristic of a ring is the smallest positive integer which is equal to 0 when interpreted in the ring, or 0 if there is no such positive integer. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ chr = (𝑔 ∈ V ↦ ((od‘𝑔)‘(1r‘𝑔))) | ||
Definition | df-zn 20200* | Define the ring of integers mod 𝑛. This is literally the quotient ring of ℤ by the ideal 𝑛ℤ, but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ ℤ/nℤ = (𝑛 ∈ ℕ0 ↦ ⦋ℤring / 𝑧⦌⦋(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠⦌(𝑠 sSet 〈(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)〉)) |
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