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Theorem List for Metamath Proof Explorer - 19601-19700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremxrsnsgrp 19601 The (additive group of the) extended reals is not a semigroup. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∉ SGrp

Theoremxrsmgmdifsgrp 19602 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 19588. (Contributed by AV, 30-Jan-2020.)
*𝑠 ∈ (Mgm ∖ SGrp)

Theoremxrs1mnd 19603 The extended real numbers, restricted to * ∖ {-∞}, form a monoid - in contrast to the full structure, see xrsmgmdifsgrp 19602. (Contributed by Mario Carneiro, 27-Nov-2014.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       𝑅 ∈ Mnd

Theoremxrs10 19604 The zero of the extended real number monoid. (Contributed by Mario Carneiro, 21-Aug-2015.)
𝑅 = (ℝ*𝑠s (ℝ* ∖ {-∞}))       0 = (0g𝑅)

Theoremxrs1cmn 19605 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 19606 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 19607 The nonnegative extended real numbers are a monoid. (Contributed by Mario Carneiro, 30-Aug-2015.)
(ℝ*𝑠s (0[,]+∞)) ∈ CMnd

Theoremxrsds 19608* 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 19609 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 19610 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 19611 Lemma for xrsdsreclb 19612. (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 19612 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 19613* Lemma for nn0subm 19620 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   0 ∈ 𝐴       𝐴 ∈ (SubMnd‘ℂfld)

Theoremcnsubglem 19614* Lemma for resubdrg 19773 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(𝑥𝐴𝑥 ∈ ℂ)    &   ((𝑥𝐴𝑦𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥𝐴 → -𝑥𝐴)    &   𝐵𝐴       𝐴 ∈ (SubGrp‘ℂfld)

Theoremcnsubrglem 19615* Lemma for resubdrg 19773 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 19616* Lemma for resubdrg 19773 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 19617 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 19618 The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ ∈ (SubRing‘ℂfld)

Theoremgzsubrg 19619 The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
ℤ[i] ∈ (SubRing‘ℂfld)

Theoremnn0subm 19620 The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
0 ∈ (SubMnd‘ℂfld)

Theoremrege0subm 19621 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
(0[,)+∞) ∈ (SubMnd‘ℂfld)

Theoremabsabv 19622 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 19623 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 19624 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 19625 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 19626 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 19627 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 19628* Lemma for rpmsubg 19629 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 19629 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 19630 Lemma for gzrngunit 19631. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑍 = (ℂflds ℤ[i])       (𝐴 ∈ (Unit‘𝑍) → 1 ≤ (abs‘𝐴))

Theoremgzrngunit 19631 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 19632* 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 19633* Relate a group sum on (ℂflds ℝ) to a finite sum on the reals. Cf. gsumfsum 19632. (Contributed by Thierry Arnoux, 7-Sep-2018.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ ℝ)       (𝜑 → ((ℂflds ℝ) Σg (𝑘𝐴𝐵)) = Σ𝑘𝐴 𝐵)

Theoremexpmhm 19634* 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 19635 The nonnegative integers form a semiring (commutative by subcmn 18065). (Contributed by Thierry Arnoux, 1-May-2018.)
(ℂflds0) ∈ SRing

Theoremrge0srg 19636 The nonnegative real numbers form a semiring (commutative by subcmn 18065). (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 19640 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 19656), and zringbas 19643 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as definition df-zring 19638 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 19637 Extend class notation with the (unital) ring of integers.
class ring

Definitiondf-zring 19638 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
ring = (ℂflds ℤ)

Theoremzringcrng 19639 The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
ring ∈ CRing

Theoremzringring 19640 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.)
ring ∈ Ring

Theoremzringabl 19641 The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
ring ∈ Abel

Theoremzringgrp 19642 The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
ring ∈ Grp

Theoremzringbas 19643 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)

Theoremzringplusg 19644 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
+ = (+g‘ℤring)

Theoremzringmulg 19645 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))

Theoremzringmulr 19646 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
· = (.r‘ℤring)

Theoremzring0 19647 The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
0 = (0g‘ℤring)

Theoremzring1 19648 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)

Theoremzringnzr 19649 The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
ring ∈ NzRing

Theoremdvdsrzring 19650 Ring divisibility in the ring of integers corresponds to ordinary divisibility in . (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
∥ = (∥r‘ℤring)

Theoremzringlpirlem1 19651 Lemma for zringlpir 19656. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
(𝜑𝐼 ∈ (LIdeal‘ℤring))    &   (𝜑𝐼 ≠ {0})       (𝜑 → (𝐼 ∩ ℕ) ≠ ∅)

Theoremzringlpirlem2 19652 Lemma for zringlpir 19656. 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.)
(𝜑𝐼 ∈ (LIdeal‘ℤring))    &   (𝜑𝐼 ≠ {0})    &   𝐺 = inf((𝐼 ∩ ℕ), ℝ, < )       (𝜑𝐺𝐼)

Theoremzringlpirlem3 19653 Lemma for zringlpir 19656. 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.)
(𝜑𝐼 ∈ (LIdeal‘ℤring))    &   (𝜑𝐼 ≠ {0})    &   𝐺 = inf((𝐼 ∩ ℕ), ℝ, < )    &   (𝜑𝑋𝐼)       (𝜑𝐺𝑋)

Theoremzringinvg 19654 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))

Theoremzringunit 19655 The units of 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))

Theoremzringlpir 19656 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.)
ring ∈ LPIR

Theoremzringndrg 19657 The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.)
ring ∉ DivRing

Theoremzringcyg 19658 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.)
ring ∈ CycGrp

Theoremzringmpg 19659 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)

Theoremprmirredlem 19660 A positive integer is irreducible over iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
𝐼 = (Irred‘ℤring)       (𝐴 ∈ ℕ → (𝐴𝐼𝐴 ∈ ℙ))

Theoremdfprm2 19661 The positive irreducible elements of are the prime numbers. This is an alternative way to define . (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
𝐼 = (Irred‘ℤring)       ℙ = (ℕ ∩ 𝐼)

Theoremprmirred 19662 The irreducible elements of are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
𝐼 = (Irred‘ℤring)       (𝐴𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ))

Theoremexpghm 19663* 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 = (𝑀s (ℂ ∖ {0}))       ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝑥 ∈ ℤ ↦ (𝐴𝑥)) ∈ (ℤring GrpHom 𝑈))

Theoremmulgghm2 19664* The powers of a group element give a homomorphism from to a group. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
· = (.g𝑅)    &   𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Grp ∧ 1𝐵) → 𝐹 ∈ (ℤring GrpHom 𝑅))

Theoremmulgrhm 19665* The powers of the element 1 give a ring homomorphism from to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
· = (.g𝑅)    &   𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 𝐹 ∈ (ℤring RingHom 𝑅))

Theoremmulgrhm2 19666* The powers of the element 1 give the unique ring homomorphism from to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
· = (.g𝑅)    &   𝐹 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 ))    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (ℤring RingHom 𝑅) = {𝐹})

10.11.3  Algebraic constructions based on the complex numbers

Syntaxczrh 19667 Map the rationals into a field, or the integers into a ring.
class ℤRHom

Syntaxczlm 19668 Augment an abelian group with vector space operations to turn it into a -module.
class ℤMod

Syntaxcchr 19669 Syntax for ring characteristic.
class chr

Syntaxczn 19670 The ring of integers modulo 𝑛.
class ℤ/n

Definitiondf-zrh 19671 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 17364). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
ℤRHom = (𝑟 ∈ V ↦ (ℤring RingHom 𝑟))

Definitiondf-zlm 19672 Augment an abelian group with vector space operations to turn it into a -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), (.g𝑔)⟩))

Definitiondf-chr 19673 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)‘(1r𝑔)))

Definitiondf-zn 19674* Define the ring of integers mod 𝑛. This is literally the quotient ring of by the ideal 𝑛, but we augment it with a total order. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
ℤ/nℤ = (𝑛 ∈ ℕ0ring / 𝑧(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩))

Theoremzrhval 19675 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)

Theoremzrhval2 19676* Alternate value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐿 = (ℤRHom‘𝑅)    &    · = (.g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 𝐿 = (𝑛 ∈ ℤ ↦ (𝑛 · 1 )))

Theoremzrhmulg 19677 Value of the ℤRHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐿 = (ℤRHom‘𝑅)    &    · = (.g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐿𝑁) = (𝑁 · 1 ))

Theoremzrhrhmb 19678 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))

Theoremzrhrhm 19679 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))

Theoremzrh1 19680 Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (ℤRHom‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐿‘1) = 1 )

Theoremzrh0 19681 Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐿‘0) = 0 )

Theoremzrhpropd 19682* The ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (ℤRHom‘𝐾) = (ℤRHom‘𝐿))

Theoremzlmval 19683 Augment an abelian group with vector space operations to turn it into a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
𝑊 = (ℤMod‘𝐺)    &    · = (.g𝐺)       (𝐺𝑉𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), · ⟩))

Theoremzlmlem 19684 Lemma for zlmbas 19685 and zlmplusg 19686. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 5       (𝐸𝐺) = (𝐸𝑊)

Theoremzlmbas 19685 Base set of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝐵 = (Base‘𝐺)       𝐵 = (Base‘𝑊)

Theoremzlmplusg 19686 Group operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &    + = (+g𝐺)        + = (+g𝑊)

Theoremzlmmulr 19687 Ring operation of a -module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &    · = (.r𝐺)        · = (.r𝑊)

Theoremzlmsca 19688 Scalar ring of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
𝑊 = (ℤMod‘𝐺)       (𝐺𝑉 → ℤring = (Scalar‘𝑊))

Theoremzlmvsca 19689 Scalar multiplication operation of a -module. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &    · = (.g𝐺)        · = ( ·𝑠𝑊)

Theoremzlmlmod 19690 The -module operation turns an arbitrary abelian group into a left module over . (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)       (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod)

Theoremzlmassa 19691 The -module operation turns a ring into an associative algebra over . (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑊 = (ℤMod‘𝐺)       (𝐺 ∈ Ring ↔ 𝑊 ∈ AssAlg)

Theoremchrval 19692 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝑂 = (od‘𝑅)    &    1 = (1r𝑅)    &   𝐶 = (chr‘𝑅)       (𝑂1 ) = 𝐶

Theoremchrcl 19693 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐶 = (chr‘𝑅)       (𝑅 ∈ Ring → 𝐶 ∈ ℕ0)

Theoremchrid 19694 The canonical ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐶 = (chr‘𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐿𝐶) = 0 )

Theoremchrdvds 19695 The ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐶 = (chr‘𝑅)    &   𝐿 = (ℤRHom‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ ℤ) → (𝐶𝑁 ↔ (𝐿𝑁) = 0 ))

Theoremchrcong 19696 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 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐶 ∥ (𝑀𝑁) ↔ (𝐿𝑀) = (𝐿𝑁)))

Theoremchrnzr 19697 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ (chr‘𝑅) ≠ 1))

Theoremchrrhm 19698 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))

Theoremdomnchr 19699 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))

Theoremznlidl 19700 The set 𝑛 is an ideal in . (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
𝑆 = (RSpan‘ℤring)       (𝑁 ∈ ℤ → (𝑆‘{𝑁}) ∈ (LIdeal‘ℤring))

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