P. 146 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14501-14600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mulgass3 14501	An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.g‘𝑅)    &   ⊢ × = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
7.3.7  Divisibility
Syntax	cdsr 14502	Ring divisibility relation.
class ∥r
Syntax	cui 14503	Units in a ring.
class Unit
Syntax	cir 14504	Ring irreducibles.
class Irred
Definition	df-dvdsr 14505*	Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through (∥r‘(oppr‘𝑅)). (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)})
Definition	df-unit 14506	Define the set of units in a ring, that is, all elements with a left and right multiplicative inverse. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ Unit = (𝑤 ∈ V ↦ (◡((∥r‘𝑤) ∩ (∥r‘(oppr‘𝑤))) “ {(1r‘𝑤)}))
Definition	df-irred 14507*	Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑤)𝑦) ≠ 𝑧})
Theorem	reldvdsr 14508	The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ Rel ∥
Theorem	reldvdsrsrg 14509	The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
⊢ (𝑅 ∈ SRing → Rel (∥r‘𝑅))
Theorem	dvdsrvald 14510*	Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → ∥ = (∥r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → · = (.r‘𝑅))    ⇒   ⊢ (𝜑 → ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)})
Theorem	dvdsrd 14511*	Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → ∥ = (∥r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → · = (.r‘𝑅))    ⇒   ⊢ (𝜑 → (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)))
Theorem	dvdsr2d 14512*	Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → ∥ = (∥r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∥ 𝑌 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌))
Theorem	dvdsrmuld 14513	A left-multiple of 𝑋 is divisible by 𝑋. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → ∥ = (∥r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∥ (𝑌 · 𝑋))
Theorem	dvdsrcld 14514	Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → ∥ = (∥r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝑋 ∥ 𝑌)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐵)
Theorem	dvdsrex 14515	Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
⊢ (𝑅 ∈ SRing → (∥r‘𝑅) ∈ V)
Theorem	dvdsrcl2 14516	Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∥ 𝑌) → 𝑌 ∈ 𝐵)
Theorem	dvdsrid 14517	An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ 𝑋)
Theorem	dvdsrtr 14518	Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∥ 𝑍 ∧ 𝑍 ∥ 𝑋) → 𝑌 ∥ 𝑋)
Theorem	dvdsrmul1 14519	The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∥ 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍))
Theorem	dvdsrneg 14520	An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ (𝑁‘𝑋))
Theorem	dvdsr01 14521	In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ 0 )
Theorem	dvdsr02 14522	Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 ))
Theorem	isunitd 14523	Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)
⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 1 = (1r‘𝑅))    &   ⊢ (𝜑 → ∥ = (∥r‘𝑅))    &   ⊢ (𝜑 → 𝑆 = (oppr‘𝑅))    &   ⊢ (𝜑 → 𝐸 = (∥r‘𝑆))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 )))
Theorem	1unit 14524	The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 1 ∈ 𝑈)
Theorem	unitcld 14525	A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐵)
Theorem	unitssd 14526	The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    ⇒   ⊢ (𝜑 → 𝑈 ⊆ 𝐵)
Theorem	opprunitd 14527	Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝑆 = (oppr‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝑈 = (Unit‘𝑆))
Theorem	crngunit 14528	Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∥ 1 ))
Theorem	dvdsunit 14529	A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ∧ 𝑋 ∈ 𝑈) → 𝑌 ∈ 𝑈)
Theorem	unitmulcl 14530	The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ 𝑈)
Theorem	unitmulclb 14531	Reversal of unitmulcl 14530 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) ∈ 𝑈 ↔ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)))
Theorem	unitgrpbasd 14532	The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    ⇒   ⊢ (𝜑 → 𝑈 = (Base‘𝐺))
Theorem	unitgrp 14533	The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp)
Theorem	unitabl 14534	The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel)
Theorem	unitgrpid 14535	The identity of the group of units of a ring is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 1 = (0g‘𝐺))
Theorem	unitsubm 14536	The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀))
Syntax	cinvr 14537	Extend class notation with multiplicative inverse.
class invr
Definition	df-invr 14538	Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
Theorem	invrfvald 14539	Multiplicative inverse function for a ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈))    &   ⊢ (𝜑 → 𝐼 = (invr‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → 𝐼 = (invg‘𝐺))
Theorem	unitinvcl 14540	The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝑈)
Theorem	unitinvinv 14541	The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘(𝐼‘𝑋)) = 𝑋)
Theorem	ringinvcl 14542	The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝐵)
Theorem	unitlinv 14543	A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = 1 )
Theorem	unitrinv 14544	A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 )
Theorem	1rinv 14545	The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) = 1 )
Theorem	0unit 14546	The additive identity is a unit if and only if 1 = 0, i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( 0 ∈ 𝑈 ↔ 1 = 0 ))
Theorem	unitnegcl 14547	The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈)
Syntax	cdvr 14548	Extend class notation with ring division.
class /r
Definition	df-dvr 14549*	Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦))))
Theorem	dvrfvald 14550*	Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝐼 = (invr‘𝑅))    &   ⊢ (𝜑 → / = (/r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ SRing)    ⇒   ⊢ (𝜑 → / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))))
Theorem	dvrvald 14551	Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝐼 = (invr‘𝑅))    &   ⊢ (𝜑 → / = (/r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌)))
Theorem	dvrcl 14552	Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) ∈ 𝐵)
Theorem	unitdvcl 14553	The units are closed under division. (Contributed by Mario Carneiro, 2-Jul-2014.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) ∈ 𝑈)
Theorem	dvrid 14554	A ring element divided by itself is the ring unity. (dividap 9040 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 / 𝑋) = 1 )
Theorem	dvr1 14555	A ring element divided by the ring unity is itself. (div1 9042 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 / 1 ) = 𝑋)
Theorem	dvrass 14556	An associative law for division. (divassap 9029 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑌) / 𝑍) = (𝑋 · (𝑌 / 𝑍)))
Theorem	dvrcan1 14557	A cancellation law for division. (divcanap1 9020 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) · 𝑌) = 𝑋)
Theorem	dvrcan3 14558	A cancellation law for division. (divcanap3 9037 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) / 𝑌) = 𝑋)
Theorem	dvreq1 14559	Equality in terms of ratio equal to ring unity. (diveqap1 9044 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) = 1 ↔ 𝑋 = 𝑌))
Theorem	dvrdir 14560	Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 + 𝑌) / 𝑍) = ((𝑋 / 𝑍) + (𝑌 / 𝑍)))
Theorem	rdivmuldivd 14561	Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑋 / 𝑌) · (𝑍 / 𝑊)) = ((𝑋 · 𝑍) / (𝑌 · 𝑊)))
Theorem	ringinvdv 14562	Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = ( 1 / 𝑋))
Theorem	rngidpropdg 14563*	The ring unity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿))
Theorem	dvdsrpropdg 14564*	The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ SRing)    &   ⊢ (𝜑 → 𝐿 ∈ SRing)    ⇒   ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿))
Theorem	unitpropdg 14565*	The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ Ring)    &   ⊢ (𝜑 → 𝐿 ∈ Ring)    ⇒   ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
Theorem	invrpropdg 14566*	The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐾 ∈ Ring)    &   ⊢ (𝜑 → 𝐿 ∈ Ring)    ⇒   ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿))
7.3.8  Ring homomorphisms
Syntax	crh 14567	Extend class notation with the ring homomorphisms.
class RingHom
Syntax	crs 14568	Extend class notation with the ring isomorphisms.
class RingIso
Definition	df-rhm 14569*	Define the set of ring homomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑𝑚 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))})
Definition	df-rim 14570*	Define the set of ring isomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)})
Theorem	dfrhm2 14571*	The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))
Theorem	rhmrcl1 14572	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
Theorem	rhmrcl2 14573	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
Theorem	rhmex 14574	Set existence for ring homomorphism. (Contributed by Jim Kingdon, 16-May-2025.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 RingHom 𝑆) ∈ V)
Theorem	isrhm 14575	A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁))))
Theorem	rhmmhm 14576	A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑀 MndHom 𝑁))
Theorem	rimrcl 14577	Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Theorem	isrim0 14578	A ring isomorphism is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
Theorem	rhmghm 14579	A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
Theorem	rhmf 14580	A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵⟶𝐶)
Theorem	rhmmul 14581	A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
Theorem	isrhm2d 14582*	Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	isrhmd 14583*	Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ⨣ = (+g‘𝑆)    &   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	rhm1 14584	Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (1r‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 1 ) = 𝑁)
Theorem	rhmf1o 14585	A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))
Theorem	isrim 14586	An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))
Theorem	rimf1o 14587	An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	rimrhm 14588	A ring isomorphism is a homomorphism. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	rhmfn 14589	The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
⊢ RingHom Fn (Ring × Ring)
Theorem	rhmval 14590	The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))
Theorem	rhmco 14591	The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ ((𝐹 ∈ (𝑇 RingHom 𝑈) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RingHom 𝑈))
Theorem	rhmdvdsr 14592	A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ / = (∥r‘𝑆)    ⇒   ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) / (𝐹‘𝐵))
Theorem	rhmopp 14593	A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))
Theorem	elrhmunit 14594	Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))
Theorem	rhmunitinv 14595	Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴)))
7.3.9  Nonzero rings and zero rings
Syntax	cnzr 14596	The class of nonzero rings.
class NzRing
Definition	df-nzr 14597	A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
Theorem	isnzr 14598	Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))
Theorem	nzrnz 14599	One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )
Theorem	nzrring 14600	A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)