P. 205 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isirred2 20401*	Expand out the class difference from isirred 20399. (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 20402	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 20403	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 20404	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 20405	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 20406	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 20407	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 20408	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 20409	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 20410	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 20411	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))
10.3.8  Ring primes
Syntax	crpm 20412	Syntax for the ring primes function.
class RPrime
Definition	df-rprm 20413*	Define the function associating with a ring its set of prime elements. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 16683. Prime elements are closely related to irreducible elements (see df-irred 20339). (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ RPrime = (𝑤 ∈ V ↦ ⦋(Base‘𝑤) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g‘𝑤)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑤) / 𝑑](𝑝𝑑(𝑥(.r‘𝑤)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))})
10.3.9  Homomorphisms of non-unital rings
Syntax	crnghm 20414	non-unital ring homomorphisms.
class RngHom
Syntax	crngim 20415	non-unital ring isomorphisms.
class RngIso
Definition	df-rnghm 20416*	Define the set of non-unital ring homomorphisms from 𝑟 to 𝑠. (Contributed by AV, 20-Feb-2020.)
⊢ RngHom = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑m 𝑣) ∣ ∀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-rngim 20417*	Define the set of non-unital ring isomorphisms from 𝑟 to 𝑠. (Contributed by AV, 20-Feb-2020.)
⊢ RngIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RngHom 𝑟)})
Theorem	rnghmrcl 20418	Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020.)
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
Theorem	rnghmfn 20419	The mapping of two non-unital rings to the non-unital ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
⊢ RngHom Fn (Rng × Rng)
Theorem	rnghmval 20420*	The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∗ = (.r‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ✚ (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) ∗ (𝑓‘𝑦)))})
Theorem	isrnghm 20421*	A function is a non-unital ring homomorphism iff it is a group homomorphism and preserves multiplication. (Contributed by AV, 22-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∗ = (.r‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))))
Theorem	isrnghmmul 20422	A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))
Theorem	rnghmmgmhm 20423	A non-unital ring homomorphism is a homomorphism of multiplicative magmas. (Contributed by AV, 27-Feb-2020.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑁 = (mulGrp‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑀 MgmHom 𝑁))
Theorem	rnghmval2 20424	The non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 1-Mar-2020.)
⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆))))
Theorem	isrngim 20425	An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅))))
Theorem	rngimrcl 20426	Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)
⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Theorem	rnghmghm 20427	A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.)
⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
Theorem	rnghmf 20428	A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵⟶𝐶)
Theorem	rnghmmul 20429	A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵)))
Theorem	isrnghm2d 20430*	Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑆 ∈ Rng)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦)))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑅 RngHom 𝑆))
Theorem	isrnghmd 20431*	Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ × = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Rng)    &   ⊢ (𝜑 → 𝑆 ∈ Rng)    &   ⊢ ((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 RngHom 𝑆))
Theorem	rnghmf1o 20432	A non-unital ring homomorphism is bijective iff its converse is also a non-unital ring homomorphism. (Contributed by AV, 27-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))
Theorem	isrngim2 20433	An isomorphism of non-unital rings is a bijective homomorphism. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)))
Theorem	rngimf1o 20434	An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶)
Theorem	rngimrnghm 20435	An isomorphism of non-unital rings is a homomorphism. (Contributed by AV, 23-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹 ∈ (𝑅 RngHom 𝑆))
Theorem	rngimcnv 20436	The converse of an isomorphism of non-unital rings is an isomorphism of non-unital rings. (Contributed by AV, 27-Feb-2025.)
⊢ (𝐹 ∈ (𝑆 RngIso 𝑇) → ◡𝐹 ∈ (𝑇 RngIso 𝑆))
Theorem	rnghmco 20437	The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
⊢ ((𝐹 ∈ (𝑇 RngHom 𝑈) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RngHom 𝑈))
Theorem	idrnghm 20438	The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → ( I ↾ 𝐵) ∈ (𝑅 RngHom 𝑅))
Theorem	c0mgm 20439*	The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))
Theorem	c0mhm 20440*	The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))
Theorem	c0ghm 20441*	The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))
Theorem	c0snmgmhm 20442*	The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (♯‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))
Theorem	c0snmhm 20443*	The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    &   ⊢ 𝑍 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆))
Theorem	c0snghm 20444*	The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    &   ⊢ 𝑍 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
Theorem	rngisomfv1 20445	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is an element of the base set of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹‘ 1 ) ∈ 𝐵)
Theorem	rngisom1 20446*	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is a ring unity of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ · = (.r‘𝑆)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥 ∈ 𝐵 (((𝐹‘ 1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹‘ 1 )) = 𝑥))
Theorem	rngisomring 20447	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then both rings are unital. (Contributed by AV, 27-Feb-2025.)
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑆 ∈ Ring)
Theorem	rngisomring1 20448	If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the ring unity of the second ring is the function value of the ring unity of the first ring for the isomorphism. (Contributed by AV, 16-Mar-2025.)
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (1r‘𝑆) = (𝐹‘(1r‘𝑅)))
10.3.10  Ring homomorphisms
Syntax	crh 20449	Extend class notation with the ring homomorphisms.
class RingHom
Syntax	crs 20450	Extend class notation with the ring isomorphisms.
class RingIso
Syntax	cric 20451	Extend class notation with the ring isomorphism relation.
class ≃𝑟
Definition	df-rhm 20452*	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 20453*	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 20454*	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 20455	Define the ring isomorphism relation, analogous to df-gic 19235: 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 20456	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
Theorem	rhmrcl2 20457	Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
Theorem	isrhm 20458	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 20459	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	rhmisrnghm 20460	Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 RngHom 𝑆))
Theorem	rimrcl 20461	Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
Theorem	isrim0 20462	A ring isomorphism is a homomorphism whose converse is also a homomorphism. Compare isgim2 19240. (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 20463	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 20464	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 20465	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 20466*	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 20467*	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 20468	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 20469	The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅))
Theorem	rhmf1o 20470	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 20471	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 20472	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 20473	A ring isomorphism is a homomorphism. Compare gimghm 19239. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))
Theorem	rimgim 20474	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	rimisrngim 20475	Each unital ring isomorphism is a non-unital ring isomorphism. (Contributed by AV, 30-Mar-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RngIso 𝑆))
Theorem	rhmfn 20476	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 20477	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 20478	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 20479*	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 20480*	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	brric 20481	The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
Theorem	brrici 20482	Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ≃𝑟 𝑆)
Theorem	brric2 20483*	The relation "is isomorphic to" for (unital) rings. This theorem corresponds to Definition df-risc 38304 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 20484	If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑅 ≃𝑔 𝑆)
Theorem	rhmdvdsr 20485	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 20486	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 20487	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 20488	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)))
10.3.11  Nonzero rings and zero rings
Syntax	cnzr 20489	The class of nonzero rings.
class NzRing
Definition	df-nzr 20490	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 20491	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 20492	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 20493	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)
Theorem	nzrringOLD 20494	Obsolete version of nzrring 20493 as of 23-Feb-2025. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Theorem	isnzr2 20495	Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵))
Theorem	isnzr2hash 20496	Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20495. (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)))
Theorem	nzrpropd 20497*	If two structures have the same components (properties), one is a nonzero ring iff the other one is. (Contributed by SN, 21-Jun-2025.)
⊢ (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 ∈ NzRing ↔ 𝐿 ∈ NzRing))
Theorem	opprnzrb 20498	The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 20499. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing)
Theorem	opprnzr 20499	The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
Theorem	ringelnzr 20500	A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing)