P. 149 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14801-14900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2idlcpbl 14801	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝑅 ~QG 𝑆)    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
Theorem	qus2idrng 14802	The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 14804 analog). (Contributed by AV, 23-Feb-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑈 ∈ Rng)
Theorem	qus1 14803	The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈)))
Theorem	qusring 14804	If 𝑆 is a two-sided ideal in 𝑅, then 𝑈 = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
Theorem	qusrhm 14805*	If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (2Ideal‘𝑅)    &   ⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))
Theorem	qusmul2 14806	Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑄)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
Theorem	crngridl 14807	In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))
Theorem	crng2idl 14808	In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
Theorem	qusmulrng 14809	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 14810. Similar to qusmul2 14806. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
⊢ ∼ = (𝑅 ~QG 𝑆)    &   ⊢ 𝐻 = (𝑅 /s ∼ )    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝐻)    ⇒   ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ (2Ideal‘𝑅) ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ )
Theorem	quscrng 14810	The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   ⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ CRing)
7.6.4  Principal ideal rings. Divisibility in the integers
Theorem	rspsn 14811*	Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥})
7.7  The complex numbers as an algebraic extensible structure
7.7.1  Definition and basic properties
Syntax	cpsmet 14812	Extend class notation with the class of all pseudometric spaces.
class PsMet
Syntax	cxmet 14813	Extend class notation with the class of all extended metric spaces.
class ∞Met
Syntax	cmet 14814	Extend class notation with the class of all metrics.
class Met
Syntax	cbl 14815	Extend class notation with the metric space ball function.
class ball
Syntax	cfbas 14816	Extend class definition to include the class of filter bases.
class fBas
Syntax	cfg 14817	Extend class definition to include the filter generating function.
class filGen
Syntax	cmopn 14818	Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
Syntax	cmetu 14819	Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
Definition	df-psmet 14820*	Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
Definition	df-xmet 14821*	Define the set of all extended metrics on a given base set. The definition is similar to df-met 14822, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
⊢ ∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
Definition	df-met 14822*	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.)
⊢ Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤 ∈ 𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
Definition	df-bl 14823*	Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑧 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑧}))
Definition	df-mopn 14824	Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.)
⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
Definition	df-fbas 14825*	Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ fBas = (𝑤 ∈ V ↦ {𝑥 ∈ 𝒫 𝒫 𝑤 ∣ (𝑥 ≠ ∅ ∧ ∅ ∉ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑥 ∩ 𝒫 (𝑦 ∩ 𝑧)) ≠ ∅)})
Definition	df-fg 14826*	Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
⊢ filGen = (𝑤 ∈ V, 𝑥 ∈ (fBas‘𝑤) ↦ {𝑦 ∈ 𝒫 𝑤 ∣ (𝑥 ∩ 𝒫 𝑦) ≠ ∅})
Definition	df-metu 14827*	Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
Theorem	blfn 14828	The ball function has universal domain. (Contributed by Jim Kingdon, 24-Sep-2025.)
⊢ ball Fn V
Theorem	mopnset 14829	Getting a set by applying MetOpen. (Contributed by Jim Kingdon, 24-Sep-2025.)
⊢ (𝐷 ∈ 𝑉 → (MetOpen‘𝐷) ∈ V)
Theorem	cndsex 14830	The standard distance function on the complex numbers is a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
⊢ (abs ∘ − ) ∈ V
Theorem	cntopex 14831	The standard topology on the complex numbers is a set. (Contributed by Jim Kingdon, 25-Sep-2025.)
⊢ (MetOpen‘(abs ∘ − )) ∈ V
Theorem	metuex 14832	Applying metUnif yields a set. (Contributed by Jim Kingdon, 28-Sep-2025.)
⊢ (𝐴 ∈ 𝑉 → (metUnif‘𝐴) ∈ V)
Syntax	ccnfld 14833	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-cnfld 14834*	The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator. The contract of this set is defined entirely by cnfldex 14836, cnfldadd 14839, cnfldmul 14841, cnfldcj 14842, cnfldtset 14843, cnfldle 14844, cnfldds 14845, and cnfldbas 14837. We may add additional members to this in the future. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) Use maps-to notation for addition and multiplication. (Revised by GG, 31-Mar-2025.) (New usage is discouraged.)
⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦))⟩, ⟨(.r‘ndx), (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}))
Theorem	cnfldstr 14835	The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂfld Struct ⟨1, ;13⟩
Theorem	cnfldex 14836	The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂfld ∈ V
Theorem	cnfldbas 14837	The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ℂ = (Base‘ℂfld)
Theorem	mpocnfldadd 14838*	The addition operation of the field of complex numbers. Version of cnfldadd 14839 using maps-to notation, which does not require ax-addf 8265. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 + 𝑦)) = (+g‘ℂfld)
Theorem	cnfldadd 14839	The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
⊢ + = (+g‘ℂfld)
Theorem	mpocnfldmul 14840*	The multiplication operation of the field of complex numbers. Version of cnfldmul 14841 using maps-to notation, which does not require ax-mulf 8266. (Contributed by GG, 31-Mar-2025.)
⊢ (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) = (.r‘ℂfld)
Theorem	cnfldmul 14841	The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 27-Apr-2025.)
⊢ · = (.r‘ℂfld)
Theorem	cnfldcj 14842	The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.)
⊢ ∗ = (*𝑟‘ℂfld)
Theorem	cnfldtset 14843	The topology component of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by GG, 31-Mar-2025.)
⊢ (MetOpen‘(abs ∘ − )) = (TopSet‘ℂfld)
Theorem	cnfldle 14844	The ordering of the field of complex numbers. Note that this is not actually an ordering on ℂ, but we put it in the structure anyway because restricting to ℝ does not affect this component, so that (ℂfld ↾s ℝ) is an ordered field even though ℂfld itself is not. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14834. (Revised by GG, 31-Mar-2025.)
⊢ ≤ = (le‘ℂfld)
Theorem	cnfldds 14845	The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 14834. (Revised by GG, 31-Mar-2025.)
⊢ (abs ∘ − ) = (dist‘ℂfld)
Theorem	cncrng 14846	The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.)
⊢ ℂfld ∈ CRing
Theorem	cnring 14847	The complex numbers form a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ ℂfld ∈ Ring
Theorem	cnfld0 14848	Zero is the zero element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 0 = (0g‘ℂfld)
Theorem	cnfld1 14849	One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 1 = (1r‘ℂfld)
Theorem	cnfldneg 14850	The additive inverse in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ (𝑋 ∈ ℂ → ((invg‘ℂfld)‘𝑋) = -𝑋)
Theorem	cnfldplusf 14851	The functionalized addition operation of the field of complex numbers. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ + = (+𝑓‘ℂfld)
Theorem	cnfldsub 14852	The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ − = (-g‘ℂfld)
Theorem	cnfldmulg 14853	The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℂ) → (𝐴(.g‘ℂfld)𝐵) = (𝐴 · 𝐵))
Theorem	cnfldexp 14854	The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℕ0) → (𝐵(.g‘(mulGrp‘ℂfld))𝐴) = (𝐴↑𝐵))
Theorem	cnsubmlem 14855*	Lemma for nn0subm 14860 and friends. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ 0 ∈ 𝐴    ⇒   ⊢ 𝐴 ∈ (SubMnd‘ℂfld)
Theorem	cnsubglem 14856*	Lemma for cnsubrglem 14857 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴)    &   ⊢ 𝐵 ∈ 𝐴    ⇒   ⊢ 𝐴 ∈ (SubGrp‘ℂfld)
Theorem	cnsubrglem 14857*	Lemma for zsubrg 14858 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   ⊢ (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴)    &   ⊢ 1 ∈ 𝐴    &   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    ⇒   ⊢ 𝐴 ∈ (SubRing‘ℂfld)
Theorem	zsubrg 14858	The integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ ∈ (SubRing‘ℂfld)
Theorem	gzsubrg 14859	The gaussian integers form a subring of the complex numbers. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ ℤ[i] ∈ (SubRing‘ℂfld)
Theorem	nn0subm 14860	The nonnegative integers form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ ℕ0 ∈ (SubMnd‘ℂfld)
Theorem	rege0subm 14861	The nonnegative reals form a submonoid of the complex numbers. (Contributed by Mario Carneiro, 20-Jun-2015.)
⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
Theorem	zsssubrg 14862	The integers are a subset of any subring of the complex numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ (𝑅 ∈ (SubRing‘ℂfld) → ℤ ⊆ 𝑅)
Theorem	gsumfzfsumlem0 14863*	Lemma for gsumfzfsum 14865. The case where the sum is empty. (Contributed by Jim Kingdon, 9-Sep-2025.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 < 𝑀)    ⇒   ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
Theorem	gsumfzfsumlemm 14864*	Lemma for gsumfzfsum 14865. The case where the sum is inhabited. (Contributed by Jim Kingdon, 9-Sep-2025.)
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
Theorem	gsumfzfsum 14865*	Relate a group sum on ℂfld to a finite sum on the complex numbers. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ (𝑀...𝑁) ↦ 𝐵)) = Σ𝑘 ∈ (𝑀...𝑁)𝐵)
Theorem	cnfldui 14866	The invertible complex numbers are exactly those apart from zero. This is recapb 8965 but expressed in terms of ℂfld. (Contributed by Jim Kingdon, 11-Sep-2025.)
⊢ {𝑧 ∈ ℂ ∣ 𝑧 # 0} = (Unit‘ℂfld)
7.7.2  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 (ℂfld ↾s ℤ), the field of complex numbers restricted to the integers. In zringring 14870 it is shown that this restriction is a ring, and zringbas 14873 shows that its base set is the integers. As of June 2019, there is an abbreviation of this expression as Definition df-zring 14868 of the ring of integers. Remark: Instead of using the symbol "ZZrng" analogous to ℂfld 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) 14868).
Syntax	czring 14867	Extend class notation with the (unital) ring of integers.
class ℤring
Definition	df-zring 14868	The (unital) ring of integers. (Contributed by Alexander van der Vekens, 9-Jun-2019.)
⊢ ℤring = (ℂfld ↾s ℤ)
Theorem	zringcrng 14869	The ring of integers is a commutative ring. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ CRing
Theorem	zringring 14870	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.)
⊢ ℤring ∈ Ring
Theorem	zringabl 14871	The ring of integers is an (additive) abelian group. (Contributed by AV, 13-Jun-2019.)
⊢ ℤring ∈ Abel
Theorem	zringgrp 14872	The ring of integers is an (additive) group. (Contributed by AV, 10-Jun-2019.)
⊢ ℤring ∈ Grp
Theorem	zringbas 14873	The integers are the base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
⊢ ℤ = (Base‘ℤring)
Theorem	zringplusg 14874	The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ + = (+g‘ℤring)
Theorem	zringmulg 14875	The multiplication (group power) operation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.) (Revised by AV, 9-Jun-2019.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(.g‘ℤring)𝐵) = (𝐴 · 𝐵))
Theorem	zringmulr 14876	The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ · = (.r‘ℤring)
Theorem	zring0 14877	The zero element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ 0 = (0g‘ℤring)
Theorem	zring1 14878	The unity element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.) (Revised by AV, 9-Jun-2019.)
⊢ 1 = (1r‘ℤring)
Theorem	zringnzr 14879	The ring of integers is a nonzero ring. (Contributed by AV, 18-Apr-2020.)
⊢ ℤring ∈ NzRing
Theorem	dvdsrzring 14880	Ring divisibility in the ring of integers corresponds to ordinary divisibility in ℤ. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by AV, 9-Jun-2019.)
⊢ ∥ = (∥r‘ℤring)
Theorem	zringinvg 14881	The additive inverse of an element of the ring of integers. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ (𝐴 ∈ ℤ → -𝐴 = ((invg‘ℤring)‘𝐴))
Theorem	zringsubgval 14882	Subtraction in the ring of integers. (Contributed by AV, 3-Aug-2019.)
⊢ − = (-g‘ℤring)    ⇒   ⊢ ((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑋 − 𝑌) = (𝑋 − 𝑌))
Theorem	zringmpg 14883	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	expghmap 14884*	Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)
⊢ 𝑀 = (mulGrp‘ℂfld)    &   ⊢ 𝑈 = (𝑀 ↾s {𝑧 ∈ ℂ ∣ 𝑧 # 0})    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (𝑥 ∈ ℤ ↦ (𝐴↑𝑥)) ∈ (ℤring GrpHom 𝑈))
Theorem	mulgghm2 14885*	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 14886*	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 14887*	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})
7.7.3  Algebraic constructions based on the complex numbers
Syntax	czrh 14888	Map the rationals into a field, or the integers into a ring.
class ℤRHom
Syntax	czlm 14889	Augment an abelian group with vector space operations to turn it into a ℤ-module.
class ℤMod
Syntax	czn 14890	The ring of integers modulo 𝑛.
class ℤ/nℤ
Definition	df-zrh 14891	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 13876). (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)
⊢ ℤRHom = (𝑟 ∈ V ↦ ∪ (ℤring RingHom 𝑟))
Definition	df-zlm 14892	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-zn 14893*	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 14894	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	zrhvalg 14895	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)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐿 = ∪ (ℤring RingHom 𝑅))
Theorem	zrhval2 14896*	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 14897	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	zrhex 14898	Set existence for ℤRHom. (Contributed by Jim Kingdon, 19-May-2025.)
⊢ 𝐿 = (ℤRHom‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐿 ∈ V)
Theorem	zrhrhmb 14899	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 14900	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))