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	mulgghm2 21501*	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 21502*	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 21503*	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 21504	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 21505	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 20592: " 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 21522), but provide the prerequisites for ring2idlqusb 21353 to show that 𝑅 is a unital ring, and for ring2idlqus1 21362 to show that ⟨1, 1⟩ is its ring unity.
Theorem	pzriprnglem1 21506	Lemma 1 for pzriprng 21522: 𝑅 is a non-unital (actually a unital!) ring. (Contributed by AV, 17-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    ⇒   ⊢ 𝑅 ∈ Rng
Theorem	pzriprnglem2 21507	Lemma 2 for pzriprng 21522: 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 21508*	Lemma 3 for pzriprng 21522: 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 21509	Lemma 4 for pzriprng 21522: 𝐼 is a subgroup of 𝑅. (Contributed by AV, 18-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    ⇒   ⊢ 𝐼 ∈ (SubGrp‘𝑅)
Theorem	pzriprnglem5 21510	Lemma 5 for pzriprng 21522: 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 21511	Lemma 6 for pzriprng 21522: 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 21512	Lemma 7 for pzriprng 21522: 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 21513	Lemma 8 for pzriprng 21522: 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 21514	Lemma 9 for pzriprng 21522: 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 21515	Lemma 10 for pzriprng 21522: 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 21516*	Lemma 11 for pzriprng 21522: 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 21517	Lemma 12 for pzriprng 21522: 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 21518	Lemma 13 for pzriprng 21522: 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 21519	Lemma 14 for pzriprng 21522: 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 21520	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 21353). (Contributed by AV, 23-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (ℤring ×s ℤring) ∈ Ring
Theorem	pzriprng1ALT 21521	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 21362). (Contributed by AV, 24-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
Theorem	pzriprng 21522	The non-unital ring (ℤring ×s ℤring) is unital. Direct proof in contrast to pzriprngALT 21520. (Contributed by AV, 25-Mar-2025.)
⊢ (ℤring ×s ℤring) ∈ Ring
Theorem	pzriprng1 21523	The ring unity of the ring (ℤring ×s ℤring). Direct proof in contrast to pzriprng1ALT 21521. (Contributed by AV, 25-Mar-2025.)
⊢ (1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
10.8.3  Algebraic constructions based on the complex numbers
Syntax	czrh 21524	Map the rationals into a field, or the integers into a ring.
class ℤRHom
Syntax	czlm 21525	Augment an abelian group with vector space operations to turn it into a ℤ-module.
class ℤMod
Syntax	cchr 21526	Syntax for ring characteristic.
class chr
Syntax	czn 21527	The ring of integers modulo 𝑛.
class ℤ/nℤ
Definition	df-zrh 21528	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 19086). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟))
Definition	df-zlm 21529	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 21530	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 21531*	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 21532	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 21533*	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 21534	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 21535	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 21536	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 21537	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 21538	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 21539*	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 21540	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 21541	Lemma for zlmbas 21542 and zlmplusg 21543. (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	zlmbas 21542	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	zlmplusg 21543	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	zlmmulr 21544	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	zlmsca 21545	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 21546	Scalar multiplication operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ · = ( ·𝑠 ‘𝑊)
Theorem	zlmlmod 21547	The ℤ-module operation turns an arbitrary abelian group into a left module over ℤ. Also see zlmassa 21928. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod)
Theorem	chrval 21548	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 21549	Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝐶 = (chr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐶 ∈ ℕ0)
Theorem	chrid 21550	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 21551	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 21552	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 21553	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 21554	A generalization of Fermat's little theorem in a commutative ring 𝐹 of prime characteristic. See fermltl 16795. (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 21555	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 21556	The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))
Theorem	domnchr 21557	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 21558	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 21559	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 21560	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 21561	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 21562	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 21563	Lemma for znbas 21568. (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	znbas2 21564	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	znadd 21565	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	znmul 21566	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	znzrh 21567	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 21568	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 21569	ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
Theorem	znzrh2 21570*	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 21571	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 21572	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 21573	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 21574	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 21575	Special case of zndvds 21574 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 21576	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 21577	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 21578	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 21579	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 21580	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 21581	Lemma for zntos 21582. (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 21582	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 21583	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 21584	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 21585	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 21586	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 21587	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 21588	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 21589	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))
Theorem	znrrg 21590	The regular elements of ℤ/nℤ are exactly the units. (This theorem fails for 𝑁 = 0, where all nonzero integers are regular, but only ±1 are units.) (Contributed by Mario Carneiro, 18-Apr-2016.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝑈 = (Unit‘𝑌)    &   ⊢ 𝐸 = (RLReg‘𝑌)    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐸 = 𝑈)
Theorem	cygznlem1 21591*	Lemma for cygzn 21595. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   ⊢ (𝜑 → 𝐺 ∈ CycGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    ⇒   ⊢ ((𝜑 ∧ (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝐿‘𝐾) = (𝐿‘𝑀) ↔ (𝐾 · 𝑋) = (𝑀 · 𝑋)))
Theorem	cygznlem2a 21592*	Lemma for cygzn 21595. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   ⊢ (𝜑 → 𝐺 ∈ CycGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿‘𝑚), (𝑚 · 𝑋)⟩)    ⇒   ⊢ (𝜑 → 𝐹:(Base‘𝑌)⟶𝐵)
Theorem	cygznlem2 21593*	Lemma for cygzn 21595. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   ⊢ (𝜑 → 𝐺 ∈ CycGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿‘𝑚), (𝑚 · 𝑋)⟩)    ⇒   ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → (𝐹‘(𝐿‘𝑀)) = (𝑀 · 𝑋))
Theorem	cygznlem3 21594*	A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛ℤ. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ · = (.g‘𝐺)    &   ⊢ 𝐿 = (ℤRHom‘𝑌)    &   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ran (𝑛 ∈ ℤ ↦ (𝑛 · 𝑥)) = 𝐵}    &   ⊢ (𝜑 → 𝐺 ∈ CycGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ 𝐹 = ran (𝑚 ∈ ℤ ↦ ⟨(𝐿‘𝑚), (𝑚 · 𝑋)⟩)    ⇒   ⊢ (𝜑 → 𝐺 ≃𝑔 𝑌)
Theorem	cygzn 21595	A cyclic group with 𝑛 elements is isomorphic to ℤ / 𝑛ℤ, and an infinite cyclic group is isomorphic to ℤ / 0ℤ ≈ ℤ. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = if(𝐵 ∈ Fin, (♯‘𝐵), 0)    &   ⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝐺 ∈ CycGrp → 𝐺 ≃𝑔 𝑌)
Theorem	cygth 21596*	The "fundamental theorem of cyclic groups". Cyclic groups are exactly the additive groups ℤ / 𝑛ℤ, for 0 ≤ 𝑛 (where 𝑛 = 0 is the infinite cyclic group ℤ), up to isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝐺 ∈ CycGrp ↔ ∃𝑛 ∈ ℕ0 𝐺 ≃𝑔 (ℤ/nℤ‘𝑛))
Theorem	cyggic 21597	Cyclic groups are isomorphic precisely when they have the same order. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐶 = (Base‘𝐻)    ⇒   ⊢ ((𝐺 ∈ CycGrp ∧ 𝐻 ∈ CycGrp) → (𝐺 ≃𝑔 𝐻 ↔ 𝐵 ≈ 𝐶))
Theorem	frgpcyg 21598	A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 18-Apr-2021.)
⊢ 𝐺 = (freeGrp‘𝐼)    ⇒   ⊢ (𝐼 ≼ 1o ↔ 𝐺 ∈ CycGrp)
Theorem	freshmansdream 21599	For a prime number 𝑃, if 𝑋 and 𝑌 are members of a commutative ring 𝑅 of characteristic 𝑃, then ((𝑋 + 𝑌)↑𝑃) = ((𝑋↑𝑃) + (𝑌↑𝑃)). This theorem is sometimes referred to as "the freshman's dream" . (Contributed by Thierry Arnoux, 18-Sep-2023.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝑃 = (chr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝑌)) = ((𝑃 ↑ 𝑋) + (𝑃 ↑ 𝑌)))
Theorem	frobrhm 21600*	In a commutative ring with prime characteristic, the Frobenius function 𝐹 is a ring endomorphism, thus named the Frobenius endomorphism. (Contributed by Thierry Arnoux, 31-May-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (chr‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑃 ↑ 𝑥))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑅))