P. 216 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfprm2 21501	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 21502	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 21503*	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 21504*	The powers of a group element give a homomorphism from ℤ to a group. The name 1 should not be taken as a constraint as it may be any group element. (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 21505*	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 21506*	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})
Theorem	irinitoringc 21507	The ring of integers is an initial object in the category of unital rings (within a universe containing the ring of integers). Example 7.2 (6) of [Adamek] p. 101 , and example in [Lang] p. 58. (Contributed by AV, 3-Apr-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → ℤring ∈ 𝑈)    &   ⊢ 𝐶 = (RingCat‘𝑈)    ⇒   ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶))
Theorem	nzerooringczr 21508	There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → ℤring ∈ 𝑈)    ⇒   ⊢ (𝜑 → (ZeroO‘𝐶) = ∅)
10.8.2.1  Example for a condition for a non-unital ring to be unital In this subsubsection, an example is given for a condition for a non-unital ring to be unital. This example is already mentioned in the comment for df-subrg 20586: " The subset (ℤ × {0}) of (ℤ × ℤ) (where multiplication is componentwise) contains the false identity ⟨1, 0⟩ which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring." The theorems in this subsubsection do not assume that 𝑅 = (ℤring ×s ℤring) is a ring (which can be proven directly very easily, see pzriprng 21525), but provide the prerequisites for ring2idlqusb 21337 to show that 𝑅 is a unital ring, and for ring2idlqus1 21346 to show that ⟨1, 1⟩ is its ring unity.
Theorem	pzriprnglem1 21509	Lemma 1 for pzriprng 21525: 𝑅 is a non-unital (actually a unital!) ring. (Contributed by AV, 17-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    ⇒   ⊢ 𝑅 ∈ Rng
Theorem	pzriprnglem2 21510	Lemma 2 for pzriprng 21525: The base set of 𝑅 is the cartesian product of the integers. (Contributed by AV, 17-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    ⇒   ⊢ (Base‘𝑅) = (ℤ × ℤ)
Theorem	pzriprnglem3 21511*	Lemma 3 for pzriprng 21525: An element of 𝐼 is an ordered pair. (Contributed by AV, 18-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    ⇒   ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑥 ∈ ℤ 𝑋 = ⟨𝑥, 0⟩)
Theorem	pzriprnglem4 21512	Lemma 4 for pzriprng 21525: 𝐼 is a subgroup of 𝑅. (Contributed by AV, 18-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    ⇒   ⊢ 𝐼 ∈ (SubGrp‘𝑅)
Theorem	pzriprnglem5 21513	Lemma 5 for pzriprng 21525: 𝐼 is a subring of the non-unital ring 𝑅. (Contributed by AV, 18-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    ⇒   ⊢ 𝐼 ∈ (SubRng‘𝑅)
Theorem	pzriprnglem6 21514	Lemma 6 for pzriprng 21525: 𝐽 has a ring unity. (Contributed by AV, 19-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    ⇒   ⊢ (𝑋 ∈ 𝐼 → ((⟨1, 0⟩(.r‘𝐽)𝑋) = 𝑋 ∧ (𝑋(.r‘𝐽)⟨1, 0⟩) = 𝑋))
Theorem	pzriprnglem7 21515	Lemma 7 for pzriprng 21525: 𝐽 is a unital ring. (Contributed by AV, 19-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    ⇒   ⊢ 𝐽 ∈ Ring
Theorem	pzriprnglem8 21516	Lemma 8 for pzriprng 21525: 𝐼 resp. 𝐽 is a two-sided ideal of the non-unital ring 𝑅. (Contributed by AV, 21-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    ⇒   ⊢ 𝐼 ∈ (2Ideal‘𝑅)
Theorem	pzriprnglem9 21517	Lemma 9 for pzriprng 21525: The ring unity of the ring 𝐽. (Contributed by AV, 22-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 1 = (1r‘𝐽)    ⇒   ⊢ 1 = ⟨1, 0⟩
Theorem	pzriprnglem10 21518	Lemma 10 for pzriprng 21525: The equivalence classes of 𝑅 modulo 𝐽. (Contributed by AV, 22-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    ⇒   ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → [⟨𝑋, 𝑌⟩] ∼ = (ℤ × {𝑌}))
Theorem	pzriprnglem11 21519*	Lemma 11 for pzriprng 21525: The base set of the quotient of 𝑅 and 𝐽. (Contributed by AV, 22-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ (Base‘𝑄) = ∪ 𝑟 ∈ ℤ {(ℤ × {𝑟})}
Theorem	pzriprnglem12 21520	Lemma 12 for pzriprng 21525: 𝑄 has a ring unity. (Contributed by AV, 23-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ (𝑋 ∈ (Base‘𝑄) → (((ℤ × {1})(.r‘𝑄)𝑋) = 𝑋 ∧ (𝑋(.r‘𝑄)(ℤ × {1})) = 𝑋))
Theorem	pzriprnglem13 21521	Lemma 13 for pzriprng 21525: 𝑄 is a unital ring. (Contributed by AV, 23-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ 𝑄 ∈ Ring
Theorem	pzriprnglem14 21522	Lemma 14 for pzriprng 21525: The ring unity of the ring 𝑄. (Contributed by AV, 23-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    &   ⊢ 𝐽 = (𝑅 ↾s 𝐼)    &   ⊢ 1 = (1r‘𝐽)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑄 = (𝑅 /s ∼ )    ⇒   ⊢ (1r‘𝑄) = (ℤ × {1})
Theorem	pzriprngALT 21523	The non-unital ring (ℤring ×s ℤring) is unital because it has the two-sided ideal (ℤ × {0}), which is unital, and the quotient of the ring and the ideal is also unital (using ring2idlqusb 21337). (Contributed by AV, 23-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (ℤring ×s ℤring) ∈ Ring
Theorem	pzriprng1ALT 21524	The ring unity of the ring (ℤring ×s ℤring) constructed from the ring unity of the two-sided ideal (ℤ × {0}) and the ring unity of the quotient of the ring and the ideal (using ring2idlqus1 21346). (Contributed by AV, 24-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
Theorem	pzriprng 21525	The non-unital ring (ℤring ×s ℤring) is unital. Direct proof in contrast to pzriprngALT 21523. (Contributed by AV, 25-Mar-2025.)
⊢ (ℤring ×s ℤring) ∈ Ring
Theorem	pzriprng1 21526	The ring unity of the ring (ℤring ×s ℤring). Direct proof in contrast to pzriprng1ALT 21524. (Contributed by AV, 25-Mar-2025.)
⊢ (1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
10.8.3  Algebraic constructions based on the complex numbers
Syntax	czrh 21527	Map the rationals into a field, or the integers into a ring.
class ℤRHom
Syntax	czlm 21528	Augment an abelian group with vector space operations to turn it into a ℤ-module.
class ℤMod
Syntax	cchr 21529	Syntax for ring characteristic.
class chr
Syntax	czn 21530	The ring of integers modulo 𝑛.
class ℤ/nℤ
Definition	df-zrh 21531	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 19098). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟))
Definition	df-zlm 21532	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 21533	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 21534*	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 21535	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 21536*	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 21537	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 21538	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 21539	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 21540	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 21541	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 21542*	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 21543	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 21544	Lemma for zlmbas 21546 and zlmplusg 21548. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx)    &   ⊢ (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)    ⇒   ⊢ (𝐸‘𝐺) = (𝐸‘𝑊)
Theorem	zlmlemOLD 21545	Obsolete version of zlmlem 21544 as of 3-Nov-2024. Lemma for zlmbas 21546 and zlmplusg 21548. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐸 = Slot 𝑁    &   ⊢ 𝑁 ∈ ℕ    &   ⊢ 𝑁 < 5    ⇒   ⊢ (𝐸‘𝐺) = (𝐸‘𝑊)
Theorem	zlmbas 21546	Base set of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝐵 = (Base‘𝑊)
Theorem	zlmbasOLD 21547	Obsolete version of zlmbas 21546 as of 3-Nov-2024. Base set of a ℤ -module. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ 𝐵 = (Base‘𝑊)
Theorem	zlmplusg 21548	Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ + = (+g‘𝑊)
Theorem	zlmplusgOLD 21549	Obsolete version of zlmbas 21546 as of 3-Nov-2024. Group operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ + = (+g‘𝑊)
Theorem	zlmmulr 21550	Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ · = (.r‘𝐺)    ⇒   ⊢ · = (.r‘𝑊)
Theorem	zlmmulrOLD 21551	Obsolete version of zlmbas 21546 as of 3-Nov-2024. Ring operation of a ℤ-module (if present). (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ · = (.r‘𝐺)    ⇒   ⊢ · = (.r‘𝑊)
Theorem	zlmsca 21552	Scalar ring of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 12-Jun-2019.) (Proof shortened by AV, 2-Nov-2024.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑉 → ℤring = (Scalar‘𝑊))
Theorem	zlmvsca 21553	Scalar multiplication operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ · = ( ·𝑠 ‘𝑊)
Theorem	zlmlmod 21554	The ℤ-module operation turns an arbitrary abelian group into a left module over ℤ. Also see zlmassa 21940. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod)
Theorem	chrval 21555	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 21556	Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝐶 = (chr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐶 ∈ ℕ0)
Theorem	chrid 21557	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 21558	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 21559	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	dvdschrmulg 21560	In a ring, any multiple of the characteristics annihilates all elements. (Contributed by Thierry Arnoux, 6-Sep-2016.)
⊢ 𝐶 = (chr‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∥ 𝑁 ∧ 𝐴 ∈ 𝐵) → (𝑁 · 𝐴) = 0 )
Theorem	fermltlchr 21561	A generalization of Fermat's little theorem in a commutative ring 𝐹 of prime characteristic. See fermltl 16817. (Contributed by Thierry Arnoux, 9-Jan-2024.)
⊢ 𝑃 = (chr‘𝐹)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ ↑ = (.g‘(mulGrp‘𝐹))    &   ⊢ 𝐴 = ((ℤRHom‘𝐹)‘𝐸)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝐸 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ CRing)    ⇒   ⊢ (𝜑 → (𝑃 ↑ 𝐴) = 𝐴)
Theorem	chrnzr 21562	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 21563	The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))
Theorem	domnchr 21564	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 21565	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 21566	Making a commutative ring as a quotient of ℤ and 𝑛ℤ. (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 21567	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 21568	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 21569	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 21570	Lemma for znbas 21579. (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 = (RSpan‘ℤring)    &   ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (le‘ndx)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝐸‘𝑈) = (𝐸‘𝑌))
Theorem	znbaslemOLD 21571	Obsolete version of znbaslem 21570 as of 3-Nov-2024. Lemma for znbas 21579. (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 = (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 21572	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.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑆 = (RSpan‘ℤring)    &   ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌))
Theorem	znbas2OLD 21573	Obsolete version of znbas2 21572 as of 3-Nov-2024. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (RSpan‘ℤring)    &   ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (Base‘𝑈) = (Base‘𝑌))
Theorem	znadd 21574	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.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑆 = (RSpan‘ℤring)    &   ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌))
Theorem	znaddOLD 21575	Obsolete version of znadd 21574 as of 3-Nov-2024. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (RSpan‘ℤring)    &   ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (+g‘𝑈) = (+g‘𝑌))
Theorem	znmul 21576	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.) (Revised by AV, 3-Nov-2024.)
⊢ 𝑆 = (RSpan‘ℤring)    &   ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌))
Theorem	znmulOLD 21577	Obsolete version of znadd 21574 as of 3-Nov-2024. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑆 = (RSpan‘ℤring)    &   ⊢ 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (.r‘𝑈) = (.r‘𝑌))
Theorem	znzrh 21578	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 21579	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 21580	ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
Theorem	znzrh2 21581*	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 21582	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 21583	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 21584	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 21585	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 21586	Special case of zndvds 21585 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 21587	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 21588	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 21589	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 21590	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 21591	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 21592	Lemma for zntos 21593. (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)
Theorem	zntos 21593	The ℤ/nℤ structure is a totally ordered set. (The order is not respected by the operations, except in the case 𝑁 = 0 when it coincides with the ordering on ℤ.) (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ Toset)
Theorem	znhash 21594	The ℤ/nℤ structure has 𝑛 elements. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ (𝑁 ∈ ℕ → (♯‘𝐵) = 𝑁)
Theorem	znfi 21595	The ℤ/nℤ structure is a finite ring. (Contributed by Mario Carneiro, 2-May-2016.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
Theorem	znfld 21596	The ℤ/nℤ structure is a finite field when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℙ → 𝑌 ∈ Field)
Theorem	znidomb 21597	The ℤ/nℤ structure is a domain (and hence a field) precisely when 𝑛 is prime. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ → (𝑌 ∈ IDomn ↔ 𝑁 ∈ ℙ))
Theorem	znchr 21598	Cyclic rings are defined by their characteristic. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (chr‘𝑌) = 𝑁)
Theorem	znunit 21599	The units of ℤ/nℤ are the integers coprime to the base. (Contributed by Mario Carneiro, 18-Apr-2016.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝑈 = (Unit‘𝑌)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴 ∈ ℤ) → ((𝐿‘𝐴) ∈ 𝑈 ↔ (𝐴 gcd 𝑁) = 1))
Theorem	znunithash 21600	The size of the unit group of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝑈 = (Unit‘𝑌)    ⇒   ⊢ (𝑁 ∈ ℕ → (♯‘𝑈) = (ϕ‘𝑁))