P. 188 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isirred2 18701*	Expand out the class difference from isirred 18699. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
Theorem	opprirred 18702	Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (oppr‘𝑅)    &   ⊢ 𝐼 = (Irred‘𝑅)    ⇒   ⊢ 𝐼 = (Irred‘𝑆)
Theorem	irredn0 18703	The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ≠ 0 )
Theorem	irredcl 18704	An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵)
Theorem	irrednu 18705	An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈)
Theorem	irredn1 18706	The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ≠ 1 )
Theorem	irredrmul 18707	The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ 𝐼)
Theorem	irredlmul 18708	The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼) → (𝑋 · 𝑌) ∈ 𝐼)
Theorem	irredmul 18709	If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
Theorem	irredneg 18710	The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → (𝑁‘𝑋) ∈ 𝐼)
Theorem	irrednegb 18711	An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐼 = (Irred‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝐼 ↔ (𝑁‘𝑋) ∈ 𝐼))
10.4.6  Ring homomorphisms
Syntax	crh 18712	Extend class notation with the ring homomorphisms.
class RingHom
Syntax	crs 18713	Extend class notation with the ring isomorphisms.
class RingIso
Syntax	cric 18714	Extend class notation with the ring isomorphism relation.
class ≃𝑟
Definition	df-rnghom 18715*	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-rngiso 18716*	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 18717*	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))))
Definition	df-ric 18718	Define the ring isomorphism relation, analogous to df-gic 17702: Two (unital) rings are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic rings share all global ring properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by AV, 24-Dec-2019.)
⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1𝑜))
Theorem	rhmrcl1 18719	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
Theorem	rhmrcl2 18720	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
Theorem	isrhm 18721	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 18722	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	isrim0 18723	An isomorphism of rings is a homomorphism whose converse is also a homomorphism . (Contributed by AV, 22-Oct-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅))))
Theorem	rimrcl 18724	Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Theorem	rhmghm 18725	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 18726	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 18727	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 18728*	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 18729*	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 18730	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	idrhm 18731	The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅))
Theorem	rhmf1o 18732	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 18733	An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)))
Theorem	rimf1o 18734	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 18735	An isomorphism of rings is a homomorphism. (Contributed by AV, 22-Oct-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	rimgim 18736	An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))
Theorem	rhmco 18737	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	pwsco1rhm 18738*	Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐴)    &   ⊢ 𝑍 = (𝑅 ↑s 𝐵)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 RingHom 𝑌))
Theorem	pwsco2rhm 18739*	Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐴)    &   ⊢ 𝑍 = (𝑆 ↑s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 RingHom 𝑍))
Theorem	f1rhm0to0 18740	If a ring homomorphism 𝐹 is injective, it maps the zero of one ring (and only the zero) to the zero of the other ring. (Contributed by AV, 24-Oct-2019.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 ))
Theorem	f1rhm0to0ALT 18741	Alternate proof for f1rhm0to0 18740. Using ghmf1 17689 does not make the proof shorter and requires disjoint variable restrictions! (Contributed by AV, 24-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 ))
Theorem	rim0to0 18742	A ring isomorphism maps the zero of one ring (and only the zero) to the zero of the other ring. (Contributed by AV, 24-Oct-2019.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 ))
Theorem	kerf1hrm 18743	A ring homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑁 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁}))
Theorem	brric 18744	The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Theorem	brric2 18745*	The relation "is isomorphic to" for (unital) rings. This theorem corresponds to the definition df-risc 33782 of the ring isomorphism relation in JM's mathbox. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆)))
Theorem	ricgic 18746	If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑅 ≃𝑔 𝑆)
10.5  Division rings and fields
10.5.1  Definition and basic properties
Syntax	cdr 18747	Extend class notation with class of all division rings.
class DivRing
Syntax	cfield 18748	Class of fields.
class Field
Definition	df-drng 18749	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 18750	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ Field = (DivRing ∩ CRing)
Theorem	isdrng 18751	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 18752	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 18753	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 18754	A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
Theorem	drnggrp 18755	A division ring is a group. (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)
Theorem	isfld 18756	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
Theorem	isdrng2 18757	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 18758	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 18759	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	drngmcl 18760	The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 }))
Theorem	drngid 18761	A division ring's unit 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 18762	A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 )
Theorem	drngid2 18763	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 18764	Closure of the multiplicative inverse in a division ring. (reccl 10692 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 18765	The multiplicative inverse in a division ring is nonzero. (recne0 10698 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	drnginvrl 18766	Property of the multiplicative inverse in a division ring. (recid2 10700 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 18767	Property of the multiplicative inverse in a division ring. (recid 10699 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	drngmul0or 18768	A product is zero iff one of its factors 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 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 )))
Theorem	drngmulne0 18769	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 18770	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 18771	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 18772*	Properties that determine a division ring. 𝐼 (reciprocal) is normally dependent on 𝑥 i.e. read it as 𝐼(𝑥)." (Contributed by NM, 2-Aug-2013.)
⊢ (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	isdrngrd 18773*	Properties that determine a division ring. 𝐼 (reciprocal) is normally dependent on 𝑥 i.e. read it as 𝐼(𝑥)." This version of isdrngd 18772 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)
⊢ (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 18774*	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 18775*	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))
10.5.2  Subrings of a ring
Syntax	csubrg 18776	Extend class notation with all subrings of a ring.
class SubRing
Syntax	crgspn 18777	Extend class notation with span of a set of elements over a ring.
class RingSpan
Definition	df-subrg 18778*	Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity. The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset (ℤ × {0}) of (ℤ × ℤ) (where multiplication is componentwise) contains the false identity ⟨1, 0⟩ which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})
Definition	df-rgspn 18779*	The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
Theorem	issubrg 18780	The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
Theorem	subrgss 18781	A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵)
Theorem	subrgid 18782	Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
Theorem	subrgring 18783	A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
Theorem	subrgcrng 18784	A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ CRing)
Theorem	subrgrcl 18785	Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Theorem	subrgsubg 18786	A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Theorem	subrg0 18787	A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆))
Theorem	subrg1cl 18788	A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴)
Theorem	subrgbas 18789	Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
Theorem	subrg1 18790	A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆))
Theorem	subrgacl 18791	A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
Theorem	subrgmcl 18792	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
Theorem	subrgsubm 18793	A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀))
Theorem	subrgdvds 18794	If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 𝐸 = (∥r‘𝑆)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )
Theorem	subrguss 18795	A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑉 = (Unit‘𝑆)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
Theorem	subrginv 18796	A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑆)    &   ⊢ 𝐽 = (invr‘𝑆)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))
Theorem	subrgdv 18797	A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑆)    &   ⊢ 𝐸 = (/r‘𝑆)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌))
Theorem	subrgunit 18798	An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑉 = (Unit‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)))
Theorem	subrgugrp 18799	The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑉 = (Unit‘𝑆)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))
Theorem	issubrg2 18800*	Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))