P. 209 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20801-20900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	idomdomd 20801	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Domn)
Theorem	idomcringd 20802	An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20803. (Proof shortened by SN, 14-May-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ CRing)
Theorem	idomringd 20803	An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Ring)
10.4  Division rings and fields
10.4.1  Definition and basic properties
Syntax	cdr 20804	Extend class notation with class of all division rings.
class DivRing
Syntax	cfield 20805	Class of fields.
class Field
Definition	df-drng 20806	Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
Definition	df-field 20807	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ Field = (DivRing ∩ CRing)
Theorem	isdrng 20808	The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
Theorem	drngunit 20809	Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))
Theorem	drngui 20810	The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑅 ∈ DivRing    ⇒   ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)
Theorem	drngring 20811	A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
Theorem	drngringd 20812	A division ring is a ring. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Ring)
Theorem	drnggrpd 20813	A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Grp)
Theorem	drnggrp 20814	A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)
Theorem	isfld 20815	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
Theorem	flddrngd 20816	A field is a division ring. (Contributed by SN, 17-Jan-2025.)
⊢ (𝜑 → 𝑅 ∈ Field)    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	fldcrngd 20817	A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
⊢ (𝜑 → 𝑅 ∈ Field)    ⇒   ⊢ (𝜑 → 𝑅 ∈ CRing)
Theorem	isdrng2 20818	A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
Theorem	drngprop 20819	If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
⊢ (Base‘𝐾) = (Base‘𝐿)    &   ⊢ (+g‘𝐾) = (+g‘𝐿)    &   ⊢ (.r‘𝐾) = (.r‘𝐿)    ⇒   ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)
Theorem	drngmgp 20820	A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing → 𝐺 ∈ Grp)
Theorem	drngid 20821	A division ring's unity is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing → 1 = (0g‘𝐺))
Theorem	drngunz 20822	A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 )
Theorem	drngnzr 20823	A division ring is a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
Theorem	drngdomn 20824	A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Domn)
Theorem	drngmcl 20825	The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) (Proof shortened by SN, 25-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 }))
Theorem	drngid2 20826	Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼))
Theorem	drnginvrcl 20827	Closure of the multiplicative inverse in a division ring. (reccl 11867 analog). (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵)
Theorem	drnginvrn0 20828	The multiplicative inverse in a division ring is nonzero. (recne0 11873 analog). (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ≠ 0 )
Theorem	drnginvrcld 20829	Closure of the multiplicative inverse in a division ring. (reccld 11975 analog). (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵)
Theorem	drnginvrl 20830	Property of the multiplicative inverse in a division ring. (recid2 11875 analog). (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 )
Theorem	drnginvrr 20831	Property of the multiplicative inverse in a division ring. (recid 11874 analog). (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑋 · (𝐼‘𝑋)) = 1 )
Theorem	drnginvrld 20832	Property of the multiplicative inverse in a division ring. (recid2d 11978 analog). (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → ((𝐼‘𝑋) · 𝑋) = 1 )
Theorem	drnginvrrd 20833	Property of the multiplicative inverse in a division ring. (recidd 11977 analog). (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 · (𝐼‘𝑋)) = 1 )
Theorem	drngmul0or 20834	A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.) (Proof shortened by SN, 25-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 )))
Theorem	drngmulne0 20835	A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )))
Theorem	drngmuleq0 20836	An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 ))
Theorem	opprdrng 20837	The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)
Theorem	isdrngd 20838*	Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a left-inverse 𝐼(𝑥). See isdrngrd 20839 for the characterization using right-inverses. (Contributed by NM, 2-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 0 = (0g‘𝑅))    &   ⊢ (𝜑 → 1 = (1r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 )    &   ⊢ (𝜑 → 1 ≠ 0 )    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 )    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	isdrngrd 20839*	Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a right-inverse 𝐼(𝑥). See isdrngd 20838 for the characterization using left-inverses. (Contributed by NM, 10-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 0 = (0g‘𝑅))    &   ⊢ (𝜑 → 1 = (1r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 )    &   ⊢ (𝜑 → 1 ≠ 0 )    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 )    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	isdrngdOLD 20840*	Obsolete version of isdrngd 20838 as of 19-Feb-2025. (Contributed by NM, 2-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 0 = (0g‘𝑅))    &   ⊢ (𝜑 → 1 = (1r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 )    &   ⊢ (𝜑 → 1 ≠ 0 )    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 )    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 )    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	isdrngrdOLD 20841*	Obsolete version of isdrngrd 20839 as of 19-Feb-2025. (Contributed by NM, 10-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 0 = (0g‘𝑅))    &   ⊢ (𝜑 → 1 = (1r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 )    &   ⊢ (𝜑 → 1 ≠ 0 )    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 )    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 )    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	drngpropd 20842*	If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-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 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing))
Theorem	fldpropd 20843*	If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-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 → (𝐾 ∈ Field ↔ 𝐿 ∈ Field))
Theorem	fldidom 20844	A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.)
⊢ (𝑅 ∈ Field → 𝑅 ∈ IDomn)
Theorem	fidomndrnglem 20845*	Lemma for fidomndrng 20846. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ { 0 }))    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝐴))    ⇒   ⊢ (𝜑 → 𝐴 ∥ 1 )
Theorem	fidomndrng 20846	A finite domain is a division ring. Note that Wedderburn's little theorem (not proved) states that finite division rings are fields. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing))
Theorem	fiidomfld 20847	A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐵 ∈ Fin → (𝑅 ∈ IDomn ↔ 𝑅 ∈ Field))
Theorem	rng1nnzr 20848	The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    ⇒   ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ NzRing)
Theorem	ring1zr 20849	The only unital ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rng1zr 20251, and semirings, see srg1zr 20288. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	ringen1zr0 20850	The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). This holds already for nonunital rings, see rngen1zr0 20253, and semirings, see srgen1zr0 20289. (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 19-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
Theorem	rng1nfld 20851	The zero ring is not a field. (Contributed by AV, 29-Apr-2019.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    ⇒   ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∉ Field)
Theorem	issubdrg 20852*	Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴))
Theorem	drhmsubc 20853*	According to df-subc 17859, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17887 and subcss2 17890). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
Theorem	drngcat 20854*	The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)
Theorem	fldcat 20855*	The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    &   ⊢ 𝐷 = (𝑈 ∩ Field)    &   ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat)
Theorem	fldc 20856*	The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    &   ⊢ 𝐷 = (𝑈 ∩ Field)    &   ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)
Theorem	fldhmsubc 20857*	According to df-subc 17859, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17887 and subcss2 17890). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    &   ⊢ 𝐷 = (𝑈 ∩ Field)    &   ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽)))
10.4.2  Sub-division rings
Syntax	csdrg 20858	Syntax for subfields (sub-division-rings).
class SubDRing
Definition	df-sdrg 20859*	Define the function associating with a ring the set of its sub-division-rings. A sub-division-ring of a ring is a subset of its base set which is a division ring when equipped with the induced structure (sum, multiplication, zero, and unity). If a ring is commutative (resp., a field), then its sub-division-rings are commutative (resp., are fields) (fldsdrgfld 20870), so we do not make a specific definition for subfields. (Contributed by Stefan O'Rear, 3-Oct-2015.) TODO: extend this definition to a function with domain V or at least Ring and not only DivRing.
⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})
Theorem	issdrg 20860	Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
Theorem	sdrgrcl 20861	Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
Theorem	sdrgdrng 20862	A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)
Theorem	sdrgsubrg 20863	A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))
Theorem	sdrgid 20864	Every division ring is a division subring of itself. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝑅))
Theorem	sdrgss 20865	A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵)
Theorem	sdrgbas 20866	Base set of a sub-division-ring structure. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 = (Base‘𝑆))
Theorem	issdrg2 20867*	Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆))
Theorem	sdrgunit 20868	A unit of a sub-division-ring is a nonzero element of the subring. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑆)    ⇒   ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 )))
Theorem	imadrhmcl 20869	The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.)
⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆))    &   ⊢ 0 = (0g‘𝑁)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑀))    &   ⊢ (𝜑 → ran 𝐹 ≠ { 0 })    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	fldsdrgfld 20870	A sub-division-ring of a field is itself a field, so it is a subfield. We can therefore use SubDRing to express subfields. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ ((𝐹 ∈ Field ∧ 𝐴 ∈ (SubDRing‘𝐹)) → (𝐹 ↾s 𝐴) ∈ Field)
Theorem	acsfn1p 20871*	Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))
Theorem	subrgacs 20872	Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵))
Theorem	sdrgacs 20873	Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵))
Theorem	cntzsdrg 20874	Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubDRing‘𝑅))
Theorem	subdrgint 20875*	The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ⊆ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑅 ↾s 𝑠) ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝐿 ∈ DivRing)
Theorem	sdrgint 20876	The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ⊆ (SubDRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubDRing‘𝑅))
Theorem	primefld 20877	The smallest sub division ring of a division ring, here named 𝑃, is a field, called the Prime Field of 𝑅. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅))    ⇒   ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Field)
Theorem	primefld0cl 20878	The prime field contains the zero element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 0 ∈ ∩ (SubDRing‘𝑅))
Theorem	primefld1cl 20879	The prime field contains the unity element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 1 ∈ ∩ (SubDRing‘𝑅))
10.4.3  Absolute value (abstract algebra)
Syntax	cabv 20880	The set of absolute values on a ring.
class AbsVal
Definition	df-abv 20881*	Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 15277 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
Theorem	abvfval 20882*	Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
Theorem	isabv 20883*	Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))
Theorem	isabvd 20884*	Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → + = (+g‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 0 = (0g‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹:𝐵⟶ℝ)    &   ⊢ (𝜑 → (𝐹‘ 0 ) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 0 < (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐴)
Theorem	abvrcl 20885	Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring)
Theorem	abvfge0 20886	An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶(0[,)+∞))
Theorem	abvf 20887	An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶ℝ)
Theorem	abvcl 20888	An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ ℝ)
Theorem	abvge0 20889	The absolute value of a number is greater than or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 0 ≤ (𝐹‘𝑋))
Theorem	abveq0 20890	The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 0 ))
Theorem	abvne0 20891	The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0)
Theorem	abvgt0 20892	The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 < (𝐹‘𝑋))
Theorem	abvmul 20893	An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌)))
Theorem	abvtri 20894	An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌)))
Theorem	abv0 20895	The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0)
Theorem	abv1z 20896	The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = 1)
Theorem	abv1 20897	The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) → (𝐹‘ 1 ) = 1)
Theorem	abvneg 20898	The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁‘𝑋)) = (𝐹‘𝑋))
Theorem	abvsubtri 20899	An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 − 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌)))
Theorem	abvrec 20900	The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝐼‘𝑋)) = (1 / (𝐹‘𝑋)))