P. 211 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	domnlcanb 21001	Left-cancellation law for domains, biconditional version of domnlcan 21002. (Contributed by Thierry Arnoux, 8-Jun-2025.) Shorten this theorem and domnlcan 21002 overall. (Revised by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍))
Theorem	domnlcan 21002	Left-cancellation law for domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍))    ⇒   ⊢ (𝜑 → 𝑌 = 𝑍)
Theorem	domnrcanb 21003	Right-cancellation law for domains, biconditional version of domnrcan 21004. (Contributed by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑍) = (𝑌 · 𝑍) ↔ 𝑋 = 𝑌))
Theorem	domnrcan 21004	Right-cancellation law for domains. (Contributed by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	domneq0r 21005	Right multiplication by a nonzero element does not change zeroness in a domain. Compare rrgeq0 20982. (Contributed by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 ))
Theorem	isidom 21006	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
Theorem	idomdomd 21007	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Domn)
Theorem	idomcringd 21008	An integral domain is a commutative ring with unity. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 21009. (Proof shortened by SN, 14-May-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ CRing)
Theorem	idomringd 21009	An integral domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised 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 21010	Extend class notation with class of all division rings.
class DivRing
Syntax	cfield 21011	Class of fields.
class Field
Definition	df-drng 21012	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, 4-Apr-2009.) (Revised by NM, 18-Oct-2012.)
⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
Definition	df-field 21013	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ Field = (DivRing ∩ CRing)
Theorem	isdrng 21014	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 21015	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 21016	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 21017	A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
Theorem	drngringd 21018	A division ring is a ring. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Ring)
Theorem	drnggrpd 21019	A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Grp)
Theorem	drnggrp 21020	A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)
Theorem	ringinveu 21021	If a ring unit element 𝑋 admits both a left inverse 𝑌 and a right inverse 𝑍, they are equal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑌 · 𝑋) = 1 )    &   ⊢ (𝜑 → (𝑋 · 𝑍) = 1 )    ⇒   ⊢ (𝜑 → 𝑍 = 𝑌)
Theorem	isdrng4 21022*	A division ring is a ring in which 1 ≠ 0 and every nonzero element has a left and right inverse. (Contributed by Thierry Arnoux, 2-Mar-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝑅 ∈ DivRing ↔ ( 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 1 ∧ (𝑦 · 𝑥) = 1 ))))
Theorem	isfld 21023	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
Theorem	flddrngd 21024	A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by SN, 17-Jan-2025.)
⊢ (𝜑 → 𝑅 ∈ Field)    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	fldcrngd 21025	A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by SN, 23-Nov-2024.)
⊢ (𝜑 → 𝑅 ∈ Field)    ⇒   ⊢ (𝜑 → 𝑅 ∈ CRing)
Theorem	drngprops 21026	Properties of a division ring. (Contributed by NM, 4-Apr-2009.) (Revised by AV, 26-Aug-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing → (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
Theorem	isdrng2 21027	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.) (Proof shortened by AV, 12-Sep-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
Theorem	drngprop 21028	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 21029	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 21030	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 21031	A division ring's unity is different from its zero. (Contributed by FL, 14-Feb-2010.) (Revised by NM, 8-Sep-2011.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 )
Theorem	drngnzr 21032	A division ring is a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
Theorem	drngdomn 21033	A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Domn)
Theorem	isdrng3lem0 21034	Lemma for isdrng3 21037: The base set of a multipication group restricted to a subset of the original base set. (Contributed by AV, 22-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (Base‘((mulGrp‘𝑅) ↾s (𝐵 ∖ 𝑋))) = (𝐵 ∖ 𝑋)
Theorem	isdrng3lem1 21035*	Lemma for isdrng3 21037 (for the left to right implication). Formerly part of proof for isdrng3 21037. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 22-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp) → ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ (𝐵 ∖ { 0 })(𝑦 · 𝑥) = 1 )
Theorem	isdrng3lem2 21036*	Lemma for isdrng3 21037 (for the right to left implication). Formerly part of proof for isdrng3 21037. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 22-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ (𝐵 ∖ { 0 })(𝑦 · 𝑥) = 1 ) → ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ∈ Grp)
Theorem	isdrng3 21037*	A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 22-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ (𝐵 ∖ { 0 })(𝑦 · 𝑥) = 1 ))
Theorem	isdrng5 21038*	A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by AV, 23-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ 𝐵 (𝑦 · 𝑥) = 1 ))
Theorem	drngmcl 21039	The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.) (Revised 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 21040	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 21041	Closure of the multiplicative inverse in a division ring. (reccl 12008 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 21042	The multiplicative inverse in a division ring is nonzero. (recne0 12014 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 21043	Closure of the multiplicative inverse in a division ring. (reccld 12113 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 21044	Property of the multiplicative inverse in a division ring. (recid2 12016 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 21045	Property of the multiplicative inverse in a division ring. (recid 12015 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 21046	Property of the multiplicative inverse in a division ring. (recid2d 12116 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 21047	Property of the multiplicative inverse in a division ring. (recidd 12115 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 21048	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 21049	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 21050	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 21051	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 21052*	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 21053 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 21053*	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 21052 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 21054*	Obsolete version of isdrngd 21052 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 21055*	Obsolete version of isdrngrd 21053 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	zrdrng 21056	A zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Revised by AV, 22-Jul-2026.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ( 0 = 1 → ¬ 𝑅 ∈ DivRing)
Theorem	drngpropd 21057*	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 21058*	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 21059	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 21060*	Lemma for fidomndrng 21061. (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 21061	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 21062	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 21063	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 21064	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 20434, and semirings, see srg1zr 20471. (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 21065	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 20436, and semirings, see srgen1zr0 20472. (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 21066	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 21067*	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 21068*	According to df-subc 18016, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 18044 and subcss2 18047). 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 21069*	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 21070*	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 21071*	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 21072*	According to df-subc 18016, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 18044 and subcss2 18047). 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 21073	Syntax for subfields (sub-division-rings).
class SubDRing
Definition	df-sdrg 21074*	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 21085), 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 21075	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 21076	Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
Theorem	sdrgdrng 21077	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 21078	A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))
Theorem	sdrgid 21079	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 21080	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 21081	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 21082*	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 21083	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 21084	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 21085	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 21086*	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 21087	Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵))
Theorem	sdrgacs 21088	Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵))
Theorem	cntzsdrg 21089	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 21090*	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 21091	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 21092	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 21093	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 21094	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 21095	The set of absolute values on a ring.
class AbsVal
Definition	df-abv 21096*	Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 15432 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 21097*	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 21098*	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 21099*	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 21100	Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring)