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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnsubrglem 21001* Lemma for resubdrg 21167 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ β„‚)    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ + 𝑦) ∈ 𝐴)    &   (π‘₯ ∈ 𝐴 β†’ -π‘₯ ∈ 𝐴)    &   1 ∈ 𝐴    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ Β· 𝑦) ∈ 𝐴)    β‡’   π΄ ∈ (SubRingβ€˜β„‚fld)
 
Theoremcnsubdrglem 21002* Lemma for resubdrg 21167 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
(π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ β„‚)    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ + 𝑦) ∈ 𝐴)    &   (π‘₯ ∈ 𝐴 β†’ -π‘₯ ∈ 𝐴)    &   1 ∈ 𝐴    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ Β· 𝑦) ∈ 𝐴)    &   ((π‘₯ ∈ 𝐴 ∧ π‘₯ β‰  0) β†’ (1 / π‘₯) ∈ 𝐴)    β‡’   (𝐴 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝐴) ∈ DivRing)
 
Theoremqsubdrg 21003 The rational numbers form a division subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
(β„š ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs β„š) ∈ DivRing)
 
Theoremzsubrg 21004 The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
β„€ ∈ (SubRingβ€˜β„‚fld)
 
Theoremgzsubrg 21005 The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
β„€[i] ∈ (SubRingβ€˜β„‚fld)
 
Theoremnn0subm 21006 The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
β„•0 ∈ (SubMndβ€˜β„‚fld)
 
Theoremrege0subm 21007 The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
(0[,)+∞) ∈ (SubMndβ€˜β„‚fld)
 
Theoremabsabv 21008 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 ∈ (AbsValβ€˜β„‚fld)
 
Theoremzsssubrg 21009 The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝑅 ∈ (SubRingβ€˜β„‚fld) β†’ β„€ βŠ† 𝑅)
 
Theoremqsssubdrg 21010 The rational numbers are a subset of any subfield of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ (β„‚fld β†Ύs 𝑅) ∈ DivRing) β†’ β„š βŠ† 𝑅)
 
Theoremcnsubrg 21011 There are no subrings of the complex numbers strictly between ℝ and β„‚. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝑅 ∈ (SubRingβ€˜β„‚fld) ∧ ℝ βŠ† 𝑅) β†’ 𝑅 ∈ {ℝ, β„‚})
 
Theoremcnmgpabl 21012 The unit group of the complex numbers is an abelian group. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))    β‡’   π‘€ ∈ Abel
 
Theoremcnmgpid 21013 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 = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))    β‡’   (0gβ€˜π‘€) = 1
 
Theoremcnmsubglem 21014* Lemma for rpmsubg 21015 and friends. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))    &   (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ β„‚)    &   (π‘₯ ∈ 𝐴 β†’ π‘₯ β‰  0)    &   ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) β†’ (π‘₯ Β· 𝑦) ∈ 𝐴)    &   1 ∈ 𝐴    &   (π‘₯ ∈ 𝐴 β†’ (1 / π‘₯) ∈ 𝐴)    β‡’   π΄ ∈ (SubGrpβ€˜π‘€)
 
Theoremrpmsubg 21015 The positive reals form a multiplicative subgroup of the complex numbers. (Contributed by Mario Carneiro, 21-Jun-2015.)
𝑀 = ((mulGrpβ€˜β„‚fld) β†Ύs (β„‚ βˆ– {0}))    β‡’   β„+ ∈ (SubGrpβ€˜π‘€)
 
Theoremgzrngunitlem 21016 Lemma for gzrngunit 21017. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑍 = (β„‚fld β†Ύs β„€[i])    β‡’   (𝐴 ∈ (Unitβ€˜π‘) β†’ 1 ≀ (absβ€˜π΄))
 
Theoremgzrngunit 21017 The units on β„€[i] are the gaussian integers with norm 1. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑍 = (β„‚fld β†Ύs β„€[i])    β‡’   (𝐴 ∈ (Unitβ€˜π‘) ↔ (𝐴 ∈ β„€[i] ∧ (absβ€˜π΄) = 1))
 
Theoremgsumfsum 21018* Relate a group sum on β„‚fld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)    β‡’   (πœ‘ β†’ (β„‚fld Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝐡)) = Ξ£π‘˜ ∈ 𝐴 𝐡)
 
Theoremregsumfsum 21019* Relate a group sum on (β„‚fld β†Ύs ℝ) to a finite sum on the reals. Cf. gsumfsum 21018. (Contributed by Thierry Arnoux, 7-Sep-2018.)
(πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ ((β„‚fld β†Ύs ℝ) Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝐡)) = Ξ£π‘˜ ∈ 𝐴 𝐡)
 
Theoremexpmhm 21020* Exponentiation is a monoid homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑁 = (β„‚fld β†Ύs β„•0)    &   π‘€ = (mulGrpβ€˜β„‚fld)    β‡’   (𝐴 ∈ β„‚ β†’ (π‘₯ ∈ β„•0 ↦ (𝐴↑π‘₯)) ∈ (𝑁 MndHom 𝑀))
 
Theoremnn0srg 21021 The nonnegative integers form a semiring (commutative by subcmn 19707). (Contributed by Thierry Arnoux, 1-May-2018.)
(β„‚fld β†Ύs β„•0) ∈ SRing
 
Theoremrge0srg 21022 The nonnegative real numbers form a semiring (commutative by subcmn 19707). (Contributed by Thierry Arnoux, 6-Sep-2018.)
(β„‚fld β†Ύs (0[,)+∞)) ∈ SRing
 
10.8.2  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 (β„‚fld β†Ύs β„€), the field of complex numbers restricted to the integers. In zringring 21026 it is shown that this restriction is a ring (it is actually a principal ideal ring as shown in zringlpir 21043), and zringbas 21029 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 21024 of the ring of integers.

Remark: Instead of using the symbol "ZZrng" analogous to β„‚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) 21024).

 
Syntaxczring 21023 Extend class notation with the (unital) ring of integers.
class β„€ring
 
Definitiondf-zring 21024 The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
β„€ring = (β„‚fld β†Ύs β„€)
 
Theoremzringcrng 21025 The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
β„€ring ∈ CRing
 
Theoremzringring 21026 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.)
β„€ring ∈ Ring
 
Theoremzringabl 21027 The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
β„€ring ∈ Abel
 
Theoremzringgrp 21028 The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
β„€ring ∈ Grp
 
Theoremzringbas 21029 The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
β„€ = (Baseβ€˜β„€ring)
 
Theoremzringplusg 21030 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
+ = (+gβ€˜β„€ring)
 
Theoremzringsub 21031 The subtraction of elements in the ring of integers. (Contributed by AV, 24-Mar-2025.)
βˆ’ = (-gβ€˜β„€ring)    β‡’   ((𝑋 ∈ β„€ ∧ π‘Œ ∈ β„€) β†’ (𝑋 βˆ’ π‘Œ) = (𝑋 βˆ’ π‘Œ))
 
Theoremzringmulg 21032 The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ (𝐴(.gβ€˜β„€ring)𝐡) = (𝐴 Β· 𝐡))
 
Theoremzringmulr 21033 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
Β· = (.rβ€˜β„€ring)
 
Theoremzring0 21034 The zero element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
0 = (0gβ€˜β„€ring)
 
Theoremzring1 21035 The unity element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
1 = (1rβ€˜β„€ring)
 
Theoremzringnzr 21036 The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
β„€ring ∈ NzRing
 
Theoremdvdsrzring 21037 Ring divisibility in the ring of integers corresponds to ordinary divisibility in β„€. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
βˆ₯ = (βˆ₯rβ€˜β„€ring)
 
Theoremzringlpirlem1 21038 Lemma for zringlpir 21043. A nonzero ideal of integers contains some positive integers. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
(πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜β„€ring))    &   (πœ‘ β†’ 𝐼 β‰  {0})    β‡’   (πœ‘ β†’ (𝐼 ∩ β„•) β‰  βˆ…)
 
Theoremzringlpirlem2 21039 Lemma for zringlpir 21043. 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.)
(πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜β„€ring))    &   (πœ‘ β†’ 𝐼 β‰  {0})    &   πΊ = inf((𝐼 ∩ β„•), ℝ, < )    β‡’   (πœ‘ β†’ 𝐺 ∈ 𝐼)
 
Theoremzringlpirlem3 21040 Lemma for zringlpir 21043. 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.)
(πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜β„€ring))    &   (πœ‘ β†’ 𝐼 β‰  {0})    &   πΊ = inf((𝐼 ∩ β„•), ℝ, < )    &   (πœ‘ β†’ 𝑋 ∈ 𝐼)    β‡’   (πœ‘ β†’ 𝐺 βˆ₯ 𝑋)
 
Theoremzringinvg 21041 The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
(𝐴 ∈ β„€ β†’ -𝐴 = ((invgβ€˜β„€ring)β€˜π΄))
 
Theoremzringunit 21042 The units of β„€ 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 ∈ (Unitβ€˜β„€ring) ↔ (𝐴 ∈ β„€ ∧ (absβ€˜π΄) = 1))
 
Theoremzringlpir 21043 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.)
β„€ring ∈ LPIR
 
Theoremzringndrg 21044 The integers are not a division ring, and therefore not a field. (Contributed by AV, 22-Oct-2021.)
β„€ring βˆ‰ DivRing
 
Theoremzringcyg 21045 The integers are a cyclic group. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 9-Jun-2019.)
β„€ring ∈ CycGrp
 
Theoremzringsubgval 21046 Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
βˆ’ = (-gβ€˜β„€ring)    β‡’   ((𝑋 ∈ β„€ ∧ π‘Œ ∈ β„€) β†’ (𝑋 βˆ’ π‘Œ) = (𝑋 βˆ’ π‘Œ))
 
Theoremzringmpg 21047 The multiplicative group of the ring of integers is the restriction of the multiplicative group of the complex numbers to the integers. (Contributed by AV, 15-Jun-2019.)
((mulGrpβ€˜β„‚fld) β†Ύs β„€) = (mulGrpβ€˜β„€ring)
 
Theoremprmirredlem 21048 A positive integer is irreducible over β„€ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
𝐼 = (Irredβ€˜β„€ring)    β‡’   (𝐴 ∈ β„• β†’ (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ β„™))
 
Theoremdfprm2 21049 The positive irreducible elements of β„€ are the prime numbers. This is an alternative way to define β„™. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
𝐼 = (Irredβ€˜β„€ring)    β‡’   β„™ = (β„• ∩ 𝐼)
 
Theoremprmirred 21050 The irreducible elements of β„€ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
𝐼 = (Irredβ€˜β„€ring)    β‡’   (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ β„€ ∧ (absβ€˜π΄) ∈ β„™))
 
Theoremexpghm 21051* Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.)
𝑀 = (mulGrpβ€˜β„‚fld)    &   π‘ˆ = (𝑀 β†Ύs (β„‚ βˆ– {0}))    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐴 β‰  0) β†’ (π‘₯ ∈ β„€ ↦ (𝐴↑π‘₯)) ∈ (β„€ring GrpHom π‘ˆ))
 
Theoremmulgghm2 21052* The powers of a group element give a homomorphism from β„€ to a group. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
Β· = (.gβ€˜π‘…)    &   πΉ = (𝑛 ∈ β„€ ↦ (𝑛 Β· 1 ))    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Grp ∧ 1 ∈ 𝐡) β†’ 𝐹 ∈ (β„€ring GrpHom 𝑅))
 
Theoremmulgrhm 21053* The powers of the element 1 give a ring homomorphism from β„€ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
Β· = (.gβ€˜π‘…)    &   πΉ = (𝑛 ∈ β„€ ↦ (𝑛 Β· 1 ))    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐹 ∈ (β„€ring RingHom 𝑅))
 
Theoremmulgrhm2 21054* The powers of the element 1 give the unique ring homomorphism from β„€ to a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 12-Jun-2019.)
Β· = (.gβ€˜π‘…)    &   πΉ = (𝑛 ∈ β„€ ↦ (𝑛 Β· 1 ))    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (β„€ring RingHom 𝑅) = {𝐹})
 
10.8.3  Algebraic constructions based on the complex numbers
 
Syntaxczrh 21055 Map the rationals into a field, or the integers into a ring.
class β„€RHom
 
Syntaxczlm 21056 Augment an abelian group with vector space operations to turn it into a β„€-module.
class β„€Mod
 
Syntaxcchr 21057 Syntax for ring characteristic.
class chr
 
Syntaxczn 21058 The ring of integers modulo 𝑛.
class β„€/nβ„€
 
Definitiondf-zrh 21059 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 18953). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
β„€RHom = (π‘Ÿ ∈ V ↦ βˆͺ (β„€ring RingHom π‘Ÿ))
 
Definitiondf-zlm 21060 Augment an abelian group with vector space operations to turn it into a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
β„€Mod = (𝑔 ∈ V ↦ ((𝑔 sSet ⟨(Scalarβ€˜ndx), β„€ring⟩) sSet ⟨( ·𝑠 β€˜ndx), (.gβ€˜π‘”)⟩))
 
Definitiondf-chr 21061 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 ↦ ((odβ€˜π‘”)β€˜(1rβ€˜π‘”)))
 
Definitiondf-zn 21062* Define the ring of integers mod 𝑛. This is literally the quotient ring of β„€ 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.)
β„€/nβ„€ = (𝑛 ∈ β„•0 ↦ ⦋℀ring / π‘§β¦Œβ¦‹(𝑧 /s (𝑧 ~QG ((RSpanβ€˜π‘§)β€˜{𝑛}))) / π‘ β¦Œ(𝑠 sSet ⟨(leβ€˜ndx), ⦋((β„€RHomβ€˜π‘ ) β†Ύ if(𝑛 = 0, β„€, (0..^𝑛))) / π‘“β¦Œ((𝑓 ∘ ≀ ) ∘ ◑𝑓)⟩))
 
Theoremzrhval 21063 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 = (β„€RHomβ€˜π‘…)    β‡’   πΏ = βˆͺ (β„€ring RingHom 𝑅)
 
Theoremzrhval2 21064* Alternate value of the β„€RHom homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐿 = (β„€RHomβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐿 = (𝑛 ∈ β„€ ↦ (𝑛 Β· 1 )))
 
Theoremzrhmulg 21065 Value of the β„€RHom homomorphism. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐿 = (β„€RHomβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑁 ∈ β„€) β†’ (πΏβ€˜π‘) = (𝑁 Β· 1 ))
 
Theoremzrhrhmb 21066 The β„€RHom homomorphism is the unique ring homomorphism from 𝑍. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 12-Jun-2019.)
𝐿 = (β„€RHomβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (𝐹 ∈ (β„€ring RingHom 𝑅) ↔ 𝐹 = 𝐿))
 
Theoremzrhrhm 21067 The β„€RHom homomorphism is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) (Revised by AV, 12-Jun-2019.)
𝐿 = (β„€RHomβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐿 ∈ (β„€ring RingHom 𝑅))
 
Theoremzrh1 21068 Interpretation of 1 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (β„€RHomβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (πΏβ€˜1) = 1 )
 
Theoremzrh0 21069 Interpretation of 0 in a ring. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐿 = (β„€RHomβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (πΏβ€˜0) = 0 )
 
Theoremzrhpropd 21070* The β„€ ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (β„€RHomβ€˜πΎ) = (β„€RHomβ€˜πΏ))
 
Theoremzlmval 21071 Augment an abelian group with vector space operations to turn it into a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.)
π‘Š = (β„€Modβ€˜πΊ)    &    Β· = (.gβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ π‘Š = ((𝐺 sSet ⟨(Scalarβ€˜ndx), β„€ring⟩) sSet ⟨( ·𝑠 β€˜ndx), Β· ⟩))
 
Theoremzlmlem 21072 Lemma for zlmbas 21074 and zlmplusg 21076. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &   πΈ = Slot (πΈβ€˜ndx)    &   (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)    &   (πΈβ€˜ndx) β‰  ( ·𝑠 β€˜ndx)    β‡’   (πΈβ€˜πΊ) = (πΈβ€˜π‘Š)
 
TheoremzlmlemOLD 21073 Obsolete version of zlmlem 21072 as of 3-Nov-2024. Lemma for zlmbas 21074 and zlmplusg 21076. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
π‘Š = (β„€Modβ€˜πΊ)    &   πΈ = Slot 𝑁    &   π‘ ∈ β„•    &   π‘ < 5    β‡’   (πΈβ€˜πΊ) = (πΈβ€˜π‘Š)
 
Theoremzlmbas 21074 Base set of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &   π΅ = (Baseβ€˜πΊ)    β‡’   π΅ = (Baseβ€˜π‘Š)
 
TheoremzlmbasOLD 21075 Obsolete version of zlmbas 21074 as of 3-Nov-2024. Base set of a β„€ -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
π‘Š = (β„€Modβ€˜πΊ)    &   π΅ = (Baseβ€˜πΊ)    β‡’   π΅ = (Baseβ€˜π‘Š)
 
Theoremzlmplusg 21076 Group operation of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’    + = (+gβ€˜π‘Š)
 
TheoremzlmplusgOLD 21077 Obsolete version of zlmbas 21074 as of 3-Nov-2024. Group operation of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
π‘Š = (β„€Modβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’    + = (+gβ€˜π‘Š)
 
Theoremzlmmulr 21078 Ring operation of a β„€-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    &    Β· = (.rβ€˜πΊ)    β‡’    Β· = (.rβ€˜π‘Š)
 
TheoremzlmmulrOLD 21079 Obsolete version of zlmbas 21074 as of 3-Nov-2024. Ring operation of a β„€-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
π‘Š = (β„€Modβ€˜πΊ)    &    Β· = (.rβ€˜πΊ)    β‡’    Β· = (.rβ€˜π‘Š)
 
Theoremzlmsca 21080 Scalar ring of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
π‘Š = (β„€Modβ€˜πΊ)    β‡’   (𝐺 ∈ 𝑉 β†’ β„€ring = (Scalarβ€˜π‘Š))
 
Theoremzlmvsca 21081 Scalar multiplication operation of a β„€-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
π‘Š = (β„€Modβ€˜πΊ)    &    Β· = (.gβ€˜πΊ)    β‡’    Β· = ( ·𝑠 β€˜π‘Š)
 
Theoremzlmlmod 21082 The β„€-module operation turns an arbitrary abelian group into a left module over β„€. Also see zlmassa 21462. (Contributed by Mario Carneiro, 2-Oct-2015.)
π‘Š = (β„€Modβ€˜πΊ)    β‡’   (𝐺 ∈ Abel ↔ π‘Š ∈ LMod)
 
Theoremchrval 21083 Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝑂 = (odβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΆ = (chrβ€˜π‘…)    β‡’   (π‘‚β€˜ 1 ) = 𝐢
 
Theoremchrcl 21084 Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐢 = (chrβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐢 ∈ β„•0)
 
Theoremchrid 21085 The canonical β„€ ring homomorphism applied to a ring's characteristic is zero. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐢 = (chrβ€˜π‘…)    &   πΏ = (β„€RHomβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (πΏβ€˜πΆ) = 0 )
 
Theoremchrdvds 21086 The β„€ ring homomorphism is zero only at multiples of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝐢 = (chrβ€˜π‘…)    &   πΏ = (β„€RHomβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑁 ∈ β„€) β†’ (𝐢 βˆ₯ 𝑁 ↔ (πΏβ€˜π‘) = 0 ))
 
Theoremchrcong 21087 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.)
𝐢 = (chrβ€˜π‘…)    &   πΏ = (β„€RHomβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐢 βˆ₯ (𝑀 βˆ’ 𝑁) ↔ (πΏβ€˜π‘€) = (πΏβ€˜π‘)))
 
Theoremchrnzr 21088 Nonzero rings are precisely those with characteristic not 1. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝑅 ∈ Ring β†’ (𝑅 ∈ NzRing ↔ (chrβ€˜π‘…) β‰  1))
 
Theoremchrrhm 21089 The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) β†’ (chrβ€˜π‘†) βˆ₯ (chrβ€˜π‘…))
 
Theoremdomnchr 21090 The characteristic of a domain can only be zero or a prime. (Contributed by Stefan O'Rear, 6-Sep-2015.)
(𝑅 ∈ Domn β†’ ((chrβ€˜π‘…) = 0 ∨ (chrβ€˜π‘…) ∈ β„™))
 
Theoremznlidl 21091 The set 𝑛℀ is an ideal in β„€. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
𝑆 = (RSpanβ€˜β„€ring)    β‡’   (𝑁 ∈ β„€ β†’ (π‘†β€˜{𝑁}) ∈ (LIdealβ€˜β„€ring))
 
Theoremzncrng2 21092 The value of the β„€/nβ„€ structure. It is defined as the quotient ring β„€ / 𝑛℀, with an "artificial" ordering added to make it a Toset. (In other words, β„€/nβ„€ 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 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    β‡’   (𝑁 ∈ β„€ β†’ π‘ˆ ∈ CRing)
 
Theoremznval 21093 The value of the β„€/nβ„€ structure. It is defined as the quotient ring β„€ / 𝑛℀, with an "artificial" ordering added to make it a Toset. (In other words, β„€/nβ„€ 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 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    &   πΉ = ((β„€RHomβ€˜π‘ˆ) β†Ύ π‘Š)    &   π‘Š = if(𝑁 = 0, β„€, (0..^𝑁))    &    ≀ = ((𝐹 ∘ ≀ ) ∘ ◑𝐹)    β‡’   (𝑁 ∈ β„•0 β†’ π‘Œ = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
 
Theoremznle 21094 The value of the β„€/nβ„€ structure. It is defined as the quotient ring β„€ / 𝑛℀, with an "artificial" ordering added to make it a Toset. (In other words, β„€/nβ„€ 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 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    &   πΉ = ((β„€RHomβ€˜π‘ˆ) β†Ύ π‘Š)    &   π‘Š = if(𝑁 = 0, β„€, (0..^𝑁))    &    ≀ = (leβ€˜π‘Œ)    β‡’   (𝑁 ∈ β„•0 β†’ ≀ = ((𝐹 ∘ ≀ ) ∘ ◑𝐹))
 
Theoremznval2 21095 Self-referential expression for the β„€/nβ„€ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
𝑆 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    &    ≀ = (leβ€˜π‘Œ)    β‡’   (𝑁 ∈ β„•0 β†’ π‘Œ = (π‘ˆ sSet ⟨(leβ€˜ndx), ≀ ⟩))
 
Theoremznbaslem 21096 Lemma for znbas 21105. (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.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    &   πΈ = Slot (πΈβ€˜ndx)    &   (πΈβ€˜ndx) β‰  (leβ€˜ndx)    β‡’   (𝑁 ∈ β„•0 β†’ (πΈβ€˜π‘ˆ) = (πΈβ€˜π‘Œ))
 
TheoremznbaslemOLD 21097 Obsolete version of znbaslem 21096 as of 3-Nov-2024. Lemma for znbas 21105. (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.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑆 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    &   πΈ = Slot 𝐾    &   πΎ ∈ β„•    &   πΎ < 10    β‡’   (𝑁 ∈ β„•0 β†’ (πΈβ€˜π‘ˆ) = (πΈβ€˜π‘Œ))
 
Theoremznbas2 21098 The base set of β„€/nβ„€ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    β‡’   (𝑁 ∈ β„•0 β†’ (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘Œ))
 
Theoremznbas2OLD 21099 Obsolete version of znbas2 21098 as of 3-Nov-2024. The base set of β„€/nβ„€ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑆 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    β‡’   (𝑁 ∈ β„•0 β†’ (Baseβ€˜π‘ˆ) = (Baseβ€˜π‘Œ))
 
Theoremznadd 21100 The additive structure of β„€/nβ„€ is the same as the quotient ring it is based on. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 13-Jun-2019.) (Revised by AV, 3-Nov-2024.)
𝑆 = (RSpanβ€˜β„€ring)    &   π‘ˆ = (β„€ring /s (β„€ring ~QG (π‘†β€˜{𝑁})))    &   π‘Œ = (β„€/nβ„€β€˜π‘)    β‡’   (𝑁 ∈ β„•0 β†’ (+gβ€˜π‘ˆ) = (+gβ€˜π‘Œ))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47936
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