P. 217 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	zringsubgval 21601	Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
⊢ − = (-g‘ℤring)    ⇒   ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 − 𝑌) = (𝑋 − 𝑌))
Theorem	zringmpg 21602	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.)
⊢ ((mulGrp‘ℂfld) ↾s ℤ) = (mulGrp‘ℤring)
Theorem	prmirredlem 21603	A positive integer is irreducible over ℤ iff it is a prime number. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)
⊢ 𝐼 = (Irred‘ℤring)    ⇒   ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))
Theorem	dfprm2 21604	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 21605	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 21606*	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 21607*	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 21608*	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 21609*	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 21610	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 21611	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 20656: " 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 21628), but provide the prerequisites for ring2idlqusb 21431 to show that 𝑅 is a unital ring, and for ring2idlqus1 21440 to show that ⟨1, 1⟩ is its ring unity.
Theorem	pzriprnglem1 21612	Lemma 1 for pzriprng 21628: 𝑅 is a non-unital (actually a unital!) ring. (Contributed by AV, 17-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    ⇒   ⊢ 𝑅 ∈ Rng
Theorem	pzriprnglem2 21613	Lemma 2 for pzriprng 21628: 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 21614*	Lemma 3 for pzriprng 21628: 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 21615	Lemma 4 for pzriprng 21628: 𝐼 is a subgroup of 𝑅. (Contributed by AV, 18-Mar-2025.)
⊢ 𝑅 = (ℤring ×s ℤring)    &   ⊢ 𝐼 = (ℤ × {0})    ⇒   ⊢ 𝐼 ∈ (SubGrp‘𝑅)
Theorem	pzriprnglem5 21616	Lemma 5 for pzriprng 21628: 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 21617	Lemma 6 for pzriprng 21628: 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 21618	Lemma 7 for pzriprng 21628: 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 21619	Lemma 8 for pzriprng 21628: 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 21620	Lemma 9 for pzriprng 21628: 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 21621	Lemma 10 for pzriprng 21628: 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 21622*	Lemma 11 for pzriprng 21628: 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 21623	Lemma 12 for pzriprng 21628: 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 21624	Lemma 13 for pzriprng 21628: 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 21625	Lemma 14 for pzriprng 21628: 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 21626	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 21431). (Contributed by AV, 23-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (ℤring ×s ℤring) ∈ Ring
Theorem	pzriprng1ALT 21627	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 21440). (Contributed by AV, 24-Mar-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
Theorem	pzriprng 21628	The non-unital ring (ℤring ×s ℤring) is unital. Direct proof in contrast to pzriprngALT 21626. (Contributed by AV, 25-Mar-2025.)
⊢ (ℤring ×s ℤring) ∈ Ring
Theorem	pzriprng1 21629	The ring unity of the ring (ℤring ×s ℤring). Direct proof in contrast to pzriprng1ALT 21627. (Contributed by AV, 25-Mar-2025.)
⊢ (1r‘(ℤring ×s ℤring)) = ⟨1, 1⟩
10.8.3  Algebraic constructions based on the complex numbers
Syntax	czrh 21630	Map the rationals into a field, or the integers into a ring.
class ℤRHom
Syntax	czlm 21631	Augment an abelian group with vector space operations to turn it into a ℤ-module.
class ℤMod
Syntax	cchr 21632	Syntax for ring characteristic.
class chr
Syntax	czn 21633	The ring of integers modulo 𝑛.
class ℤ/nℤ
Definition	df-zrh 21634	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 19135). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟))
Definition	df-zlm 21635	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 21636	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 21637*	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 21638	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 21639*	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 21640	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 21641	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 21642	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 21643	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 21644	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 21645*	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 21646	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 21647	Lemma for zlmbas 21648 and zlmplusg 21649. (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 21648	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 21649	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 21650	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 21651	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 21652	Scalar multiplication operation of a ℤ-module. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ · = (.g‘𝐺)    ⇒   ⊢ · = ( ·𝑠 ‘𝑊)
Theorem	zlmlmod 21653	The ℤ-module operation turns an arbitrary abelian group into a left module over ℤ. Also see zlmassa 22034. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ LMod)
Theorem	chrval 21654	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 21655	Closure of the characteristic. (Contributed by Mario Carneiro, 23-Sep-2015.)
⊢ 𝐶 = (chr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐶 ∈ ℕ0)
Theorem	chrid 21656	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 21657	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 21658	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 21659	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 21660	A generalization of Fermat's little theorem in a commutative ring 𝐹 of prime characteristic. See fermltl 16844. (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 21661	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 21662	The characteristic restriction on ring homomorphisms. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (chr‘𝑆) ∥ (chr‘𝑅))
Theorem	domnchr 21663	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 21664	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 21665	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 21666	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 21667	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 21668	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 21669	Lemma for znbas 21674. (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 21670	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 21671	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 21672	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 21673	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 21674	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 21675	ℤ/nℤ is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝑌 ∈ CRing)
Theorem	znzrh2 21676*	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 21677	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 21678	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 21679	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 21680	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 21681	Special case of zndvds 21680 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 21682	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 21683	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 21684	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 21685	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 21686	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 21687	Lemma for zntos 21688. (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 21688	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 21689	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 21690	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 21691	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 21692	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 21693	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 21694	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 21695	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 21696	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 21697*	Lemma for cygzn 21701. (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 21698*	Lemma for cygzn 21701. (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 21699*	Lemma for cygzn 21701. (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 21700*	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)