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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | qsssubdrg 20201 | 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 20202 | 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 20203 | 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 20204 | 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 20205* | Lemma for rpmsubg 20206 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 20206 | 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 20207 | Lemma for gzrngunit 20208. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑍 = (ℂfld ↾s ℤ[i]) ⇒ ⊢ (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴)) | ||
Theorem | gzrngunit 20208 | 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 20209* | 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 20210* | Relate a group sum on (ℂfld ↾s ℝ) to a finite sum on the reals. Cf. gsumfsum 20209. (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 20211* | 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 20212 | The nonnegative integers form a semiring (commutative by subcmn 18628). (Contributed by Thierry Arnoux, 1-May-2018.) |
⊢ (ℂfld ↾s ℕ0) ∈ SRing | ||
Theorem | rge0srg 20213 | The nonnegative real numbers form a semiring (commutative by subcmn 18628). (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 20217 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 20233), and zringbas 20220 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition df-zring 20215 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)). | ||
Syntax | zring 20214 | Extend class notation with the (unital) ring of integers. |
class ℤring | ||
Definition | df-zring 20215 | The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.) |
⊢ ℤring = (ℂfld ↾s ℤ) | ||
Theorem | zringcrng 20216 | The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.) |
⊢ ℤring ∈ CRing | ||
Theorem | zringring 20217 | 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 20218 | The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.) |
⊢ ℤring ∈ Abel | ||
Theorem | zringgrp 20219 | The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.) |
⊢ ℤring ∈ Grp | ||
Theorem | zringbas 20220 | 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 20221 | 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 20222 | 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 20223 | 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 20224 | 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 20225 | 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 20226 | The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.) |
⊢ ℤring ∈ NzRing | ||
Theorem | dvdsrzring 20227 | 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 20228 | Lemma for zringlpir 20233. 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 20229 | Lemma for zringlpir 20233. 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 20230 | Lemma for zringlpir 20233. 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 20231 | 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 20232 | 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 20233 | 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 20234 | The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.) |
⊢ ℤring ∉ DivRing | ||
Theorem | zringcyg 20235 | The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.) |
⊢ ℤring ∈ CycGrp | ||
Theorem | zringmpg 20236 | 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 20237 | 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 20238 | 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 20239 | 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 20240* | 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 20241* | 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 20242* | 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 20243* | 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 20244 | Map the rationals into a field, or the integers into a ring. |
class ℤRHom | ||
Syntax | czlm 20245 | Augment an abelian group with vector space operations to turn it into a ℤ-module. |
class ℤMod | ||
Syntax | cchr 20246 | Syntax for ring characteristic. |
class chr | ||
Syntax | czn 20247 | The ring of integers modulo 𝑛. |
class ℤ/nℤ | ||
Definition | df-zrh 20248 | 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 17928). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟)) | ||
Definition | df-zlm 20249 | 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 20250 | 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 20251* | 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)〉)) | ||
Theorem | zrhval 20252 | Define the unique homomorphism from the integers to a ring or field. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ 𝐿 = ∪ (ℤring RingHom 𝑅) | ||
Theorem | zrhval2 20253* | Alternate value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐿 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))) | ||
Theorem | zrhmulg 20254 | Value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿‘𝑁) = (𝑁 · 1 )) | ||
Theorem | zrhrhmb 20255 | The ℤRHom homomorphism is the unique ring homomorphism from 𝑍. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐹 ∈ (ℤring RingHom 𝑅) ↔ 𝐹 = 𝐿)) | ||
Theorem | zrhrhm 20256 | The ℤRHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝐿 = (ℤRHom‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑅)) | ||
Theorem | zrh1 20257 | Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐿‘1) = 1 ) | ||
Theorem | zrh0 20258 | Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐿‘0) = 0 ) | ||
Theorem | zrhpropd 20259* | The ℤ ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (ℤRHom‘𝐾) = (ℤRHom‘𝐿)) | ||
Theorem | zlmval 20260 | 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.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), · 〉)) | ||
Theorem | zlmlem 20261 | Lemma for zlmbas 20262 and zlmplusg 20263. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < 5 ⇒ ⊢ (𝐸‘𝐺) = (𝐸‘𝑊) | ||
Theorem | zlmbas 20262 | Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ 𝐵 = (Base‘𝑊) | ||
Theorem | zlmplusg 20263 | Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ + = (+g‘𝐺) ⇒ ⊢ + = (+g‘𝑊) | ||
Theorem | zlmmulr 20264 | Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ · = (.r‘𝐺) ⇒ ⊢ · = (.r‘𝑊) | ||
Theorem | zlmsca 20265 | Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) |
⊢ 𝑊 = (ℤMod‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊)) | ||
Theorem | zlmvsca 20266 | Scalar multiplication operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ · = (.g‘𝐺) ⇒ ⊢ · = ( ·𝑠 ‘𝑊) | ||
Theorem | zlmlmod 20267 | The ℤ-module operation turns an arbitrary abelian group into a left module over ℤ. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) ⇒ ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod) | ||
Theorem | zlmassa 20268 | The ℤ-module operation turns a ring into an associative algebra over ℤ. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) ⇒ ⊢ (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg) | ||
Theorem | chrval 20269 | Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝑂 = (od‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐶 = (chr‘𝑅) ⇒ ⊢ (𝑂‘ 1 ) = 𝐶 | ||
Theorem | chrcl 20270 | Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.) |
⊢ 𝐶 = (chr‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐶 ∈ ℕ0) | ||
Theorem | chrid 20271 | The canonical ℤ ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.) |
⊢ 𝐶 = (chr‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐿‘𝐶) = 0 ) | ||
Theorem | chrdvds 20272 | The ℤ ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.) |
⊢ 𝐶 = (chr‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ 𝑁 ↔ (𝐿‘𝑁) = 0 )) | ||
Theorem | chrcong 20273 | If two integers are congruent relative to the ring characteristic, their images in the ring are the same. (Contributed by Mario Carneiro, 24-Sep-2015.) |
⊢ 𝐶 = (chr‘𝑅) & ⊢ 𝐿 = (ℤRHom‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ (𝑀 − 𝑁) ↔ (𝐿‘𝑀) = (𝐿‘𝑁))) | ||
Theorem | chrnzr 20274 | Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (chr‘𝑅) ≠ 1)) | ||
Theorem | chrrhm 20275 | The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅)) | ||
Theorem | domnchr 20276 | The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
⊢ (𝑅 ∈ Domn → ((chr‘𝑅) = 0 ∨ (chr‘𝑅) ∈ ℙ)) | ||
Theorem | znlidl 20277 | The set 𝑛ℤ is an ideal in ℤ. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) ⇒ ⊢ (𝑁 ∈ ℤ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring)) | ||
Theorem | zncrng2 20278 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) ⇒ ⊢ (𝑁 ∈ ℤ → 𝑈 ∈ CRing) | ||
Theorem | znval 20279 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ≤ 〉)) | ||
Theorem | znle 20280 | The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) | ||
Theorem | znval2 20281 | Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet 〈(le‘ndx), ≤ 〉)) | ||
Theorem | znbaslem 20282 | Lemma for znbas 20287. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 9-Sep-2021.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐸 = Slot 𝐾 & ⊢ 𝐾 ∈ ℕ & ⊢ 𝐾 < ;10 ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌)) | ||
Theorem | znbas2 20283 | The base set of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌)) | ||
Theorem | znadd 20284 | The additive structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌)) | ||
Theorem | znmul 20285 | The multiplicative structure of ℤ/nℤ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌)) | ||
Theorem | znzrh 20286 | The ℤ ring homomorphism of ℤ/nℤ is inherited from the quotient ring it is based on. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → (ℤRHom‘𝑈) = (ℤRHom‘𝑌)) | ||
Theorem | znbas 20287 | The base set of ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝑅 = (ℤring ~QG (𝑆‘{𝑁})) ⇒ ⊢ (𝑁 ∈ ℕ0 → (ℤ / 𝑅) = (Base‘𝑌)) | ||
Theorem | zncrng 20288 | ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing) | ||
Theorem | znzrh2 20289* | The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐿 = (𝑥 ∈ ℤ ↦ [𝑥] ∼ )) | ||
Theorem | znzrhval 20290 | The ℤ ring homomorphism maps elements to their equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑆 = (RSpan‘ℤring) & ⊢ ∼ = (ℤring ~QG (𝑆‘{𝑁})) & ⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴] ∼ ) | ||
Theorem | znzrhfo 20291 | The ℤ ring homomorphism is a surjection onto ℤ / 𝑛ℤ. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐿:ℤ–onto→𝐵) | ||
Theorem | zncyg 20292 | The group ℤ / 𝑛ℤ is cyclic for all 𝑛 (including 𝑛 = 0). (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CycGrp) | ||
Theorem | zndvds 20293 | Express equality of equivalence classes in ℤ / 𝑛ℤ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵))) | ||
Theorem | zndvds0 20294 | Special case of zndvds 20293 when one argument is zero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ 0 = (0g‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) = 0 ↔ 𝑁 ∥ 𝐴)) | ||
Theorem | znf1o 20295 | The function 𝐹 enumerates all equivalence classes in ℤ/nℤ for each 𝑛. When 𝑛 = 0, ℤ / 0ℤ = ℤ / {0} ≈ ℤ so we let 𝑊 = ℤ; otherwise 𝑊 = {0, ..., 𝑛 − 1} enumerates all the equivalence classes. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝑊–1-1-onto→𝐵) | ||
Theorem | zzngim 20296 | The ℤ ring homomorphism is an isomorphism for 𝑁 = 0. (We only show group isomorphism here, but ring isomorphism follows, since it is a bijective ring homomorphism.) (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘0) & ⊢ 𝐿 = (ℤRHom‘𝑌) ⇒ ⊢ 𝐿 ∈ (ℤring GrpIso 𝑌) | ||
Theorem | znle2 20297 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → ≤ = ((𝐹 ∘ ≤ ) ∘ ◡𝐹)) | ||
Theorem | znleval 20298 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → (𝐴 ≤ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵)))) | ||
Theorem | znleval2 20299 | The ordering of the ℤ/nℤ structure. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ≤ 𝐵 ↔ (◡𝐹‘𝐴) ≤ (◡𝐹‘𝐵))) | ||
Theorem | zntoslem 20300 | Lemma for zntos 20301. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑁) & ⊢ 𝐹 = ((ℤRHom‘𝑌) ↾ 𝑊) & ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁)) & ⊢ ≤ = (le‘𝑌) & ⊢ 𝑋 = (Base‘𝑌) ⇒ ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Toset) |
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