P. 486 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48501-48600   *Has distinct variable group(s)

Type	Label	Description
Statement
21.48.18.4  A constructed not unital ring
Theorem	cznrnglem 48501	Lemma for cznrng 48503: The base set of the ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is the base set of the ℤ/nℤ structure. (Contributed by AV, 16-Feb-2020.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    ⇒   ⊢ 𝐵 = (Base‘𝑋)
Theorem	cznabel 48502	The ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is an abelian group. (Contributed by AV, 16-Feb-2020.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑋 ∈ Abel)
Theorem	cznrng 48503*	The ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is a non-unital ring. (Contributed by AV, 17-Feb-2020.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    &   ⊢ 0 = (0g‘𝑌)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑋 ∈ Rng)
Theorem	cznnring 48504*	The ring constructed from a ℤ/nℤ structure with 1 < 𝑛 by replacing the (multiplicative) ring operation by a constant operation is not a unital ring. (Contributed by AV, 17-Feb-2020.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    &   ⊢ 0 = (0g‘𝑌)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝑋 ∉ Ring)
21.48.18.5  The category of non-unital rings (alternate definition) As an alternative to df-rngc 20550, the "category of non-unital rings" can be defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, according to dfrngc2 20561.
Syntax	crngcALTV 48505	Extend class notation to include the category Rng. (New usage is discouraged.)
class RngCatALTV
Definition	df-rngcALTV 48506*	Definition of the category Rng, relativized to a subset 𝑢. This is the category of all non-unital rings in 𝑢 and homomorphisms between these rings. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ RngCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Rng) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
Theorem	rngcvalALTV 48507*	Value of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	rngcbasALTV 48508	Set of objects of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
Theorem	rngchomfvalALTV 48509*	Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
Theorem	rngchomALTV 48510	Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHom 𝑌))
Theorem	elrngchomALTV 48511	A morphism of non-unital rings is a function. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
Theorem	rngccofvalALTV 48512*	Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
Theorem	rngccoALTV 48513	Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋 RngHom 𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌 RngHom 𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Theorem	rngccatidALTV 48514*	Lemma for rngccatALTV 48515. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Theorem	rngccatALTV 48515	The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	rngcidALTV 48516	The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑆 = (Base‘𝑋)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆))
Theorem	rngcsectALTV 48517	A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐸 = (Base‘𝑋)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
Theorem	rngcinvALTV 48518	An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑁 = (Inv‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = ◡𝐹)))
Theorem	rngcisoALTV 48519	An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIso 𝑌)))
Theorem	rngchomffvalALTV 48520*	The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) in maps-to notation for an operation. (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐹 = (Homf ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
Theorem	rngchomrnghmresALTV 48521	The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) as restriction of the non-unital ring homomorphisms. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Rng ∩ 𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐹 = (Homf ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐹 = ( RngHom ↾ (𝐵 × 𝐵)))
Theorem	rngcrescrhmALTV 48522	The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Theorem	rhmsubcALTVlem1 48523	Lemma 1 for rhmsubcALTV 48527. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅))
Theorem	rhmsubcALTVlem2 48524	Lemma 2 for rhmsubcALTV 48527. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
Theorem	rhmsubcALTVlem3 48525*	Lemma 3 for rhmsubcALTV 48527. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
Theorem	rhmsubcALTVlem4 48526*	Lemma 4 for rhmsubcALTV 48527. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
Theorem	rhmsubcALTV 48527	According to df-subc 17736, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17764 and subcss2 17767). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈)))
Theorem	rhmsubcALTVcat 48528	The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.) (New usage is discouraged.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → ((RngCatALTV‘𝑈) ↾cat 𝐻) ∈ Cat)
21.48.18.6  The category of (unital) rings (alternate definition) As an alternative to df-ringc 20579, the "category of unital rings" can be defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, according to dfringc2 20590.
Syntax	cringcALTV 48529	Extend class notation to include the category Ring. (New usage is discouraged.)
class RingCatALTV
Definition	df-ringcALTV 48530*	Definition of the category Ring, relativized to a subset 𝑢. This is the category of all rings in 𝑢 and homomorphisms between these rings. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
⊢ RingCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Ring) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
Theorem	ringcvalALTV 48531*	Value of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	funcringcsetcALTV2lem1 48532*	Lemma 1 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
Theorem	funcringcsetcALTV2lem2 48533*	Lemma 2 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
Theorem	funcringcsetcALTV2lem3 48534*	Lemma 3 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
Theorem	funcringcsetcALTV2lem4 48535*	Lemma 4 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
Theorem	funcringcsetcALTV2lem5 48536*	Lemma 5 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
Theorem	funcringcsetcALTV2lem6 48537*	Lemma 6 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Theorem	funcringcsetcALTV2lem7 48538*	Lemma 7 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋)))
Theorem	funcringcsetcALTV2lem8 48539*	Lemma 8 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))
Theorem	funcringcsetcALTV2lem9 48540*	Lemma 9 for funcringcsetcALTV2 48541. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
Theorem	funcringcsetcALTV2 48541*	The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
Theorem	ringcbasALTV 48542	Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
Theorem	ringchomfvalALTV 48543*	Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦)))
Theorem	ringchomALTV 48544	Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
Theorem	elringchomALTV 48545	A morphism of rings is a function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
Theorem	ringccofvalALTV 48546*	Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
Theorem	ringccoALTV 48547	Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑋 RingHom 𝑌))    &   ⊢ (𝜑 → 𝐺 ∈ (𝑌 RingHom 𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Theorem	ringccatidALTV 48548*	Lemma for ringccatALTV 48549. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥)))))
Theorem	ringccatALTV 48549	The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	ringcidALTV 48550	The identity arrow in the category of rings is the identity function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑆 = (Base‘𝑋)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆))
Theorem	ringcsectALTV 48551	A section in the category of rings, written out. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐸 = (Base‘𝑋)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
Theorem	ringcinvALTV 48552	An inverse in the category of rings is the converse operation. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑁 = (Inv‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)))
Theorem	ringcisoALTV 48553	An isomorphism in the category of rings is a bijection. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))
Theorem	ringcbasbasALTV 48554	An element of the base set of the base set of the category of rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (RingCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    ⇒   ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (Base‘𝑅) ∈ 𝑈)
Theorem	funcringcsetclem1ALTV 48555*	Lemma 1 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
Theorem	funcringcsetclem2ALTV 48556*	Lemma 2 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
Theorem	funcringcsetclem3ALTV 48557*	Lemma 3 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    ⇒   ⊢ (𝜑 → 𝐹:𝐵⟶𝐶)
Theorem	funcringcsetclem4ALTV 48558*	Lemma 4 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))
Theorem	funcringcsetclem5ALTV 48559*	Lemma 5 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
Theorem	funcringcsetclem6ALTV 48560*	Lemma 6 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
Theorem	funcringcsetclem7ALTV 48561*	Lemma 7 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋)))
Theorem	funcringcsetclem8ALTV 48562*	Lemma 8 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))
Theorem	funcringcsetclem9ALTV 48563*	Lemma 9 for funcringcsetcALTV 48564. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹‘𝑋), (𝐹‘𝑌)⟩(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻)))
Theorem	funcringcsetcALTV 48564*	The "natural forgetful functor" from the category of rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
⊢ 𝑅 = (RingCatALTV‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
Theorem	srhmsubcALTVlem1 48565*	Lemma 1 for srhmsubcALTV 48567. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    ⇒   ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈)))
Theorem	srhmsubcALTVlem2 48566*	Lemma 2 for srhmsubcALTV 48567. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌))
Theorem	srhmsubcALTV 48567*	According to df-subc 17736, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17764 and subcss2 17767). Therefore, the set of special ring homomorphisms (i.e., ring homomorphisms from a special ring to another ring of that kind) is a subcategory of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
Theorem	sringcatALTV 48568*	The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)
Theorem	crhmsubcALTV 48569*	According to df-subc 17736, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17764 and subcss2 17767). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (𝑈 ∩ CRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
Theorem	cringcatALTV 48570*	The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (𝑈 ∩ CRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)
Theorem	drhmsubcALTV 48571*	According to df-subc 17736, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17764 and subcss2 17767). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
Theorem	drngcatALTV 48572*	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.) (New usage is discouraged.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)
Theorem	fldcatALTV 48573*	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.) (New usage is discouraged.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    &   ⊢ 𝐷 = (𝑈 ∩ Field)    &   ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐹) ∈ Cat)
Theorem	fldcALTV 48574*	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.) (New usage is discouraged.)
⊢ 𝐶 = (𝑈 ∩ DivRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    &   ⊢ 𝐷 = (𝑈 ∩ Field)    &   ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → (((RingCatALTV‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)
Theorem	fldhmsubcALTV 48575*	According to df-subc 17736, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17764 and subcss2 17767). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
⊢ 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‘((RingCatALTV‘𝑈) ↾cat 𝐽)))
21.48.19  Basic algebraic structures (extension)
21.48.19.1  Auxiliary theorems
Theorem	eliunxp2 48576*	Membership in a union of Cartesian products over its second component, analogous to eliunxp 5786. (Contributed by AV, 30-Mar-2019.)
⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
Theorem	mpomptx2 48577*	Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐴(𝑦) is not assumed to be constant w.r.t 𝑦, analogous to mpomptx 7471. (Contributed by AV, 30-Mar-2019.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)    ⇒   ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Theorem	cbvmpox2 48578*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7452 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7451. (Contributed by AV, 30-Mar-2019.)
⊢ Ⅎ𝑧𝐴    &   ⊢ Ⅎ𝑦𝐷    &   ⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑤𝐶    &   ⊢ Ⅎ𝑥𝐸    &   ⊢ Ⅎ𝑦𝐸    &   ⊢ (𝑦 = 𝑧 → 𝐴 = 𝐷)    &   ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → 𝐶 = 𝐸)    ⇒   ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ 𝐷, 𝑧 ∈ 𝐵 ↦ 𝐸)
Theorem	dmmpossx2 48579*	The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpossx 8010. (Contributed by AV, 30-Mar-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
Theorem	mpoexxg2 48580*	Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpoexxg 8019. (Contributed by AV, 30-Mar-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	ovmpordxf 48581*	Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7508. (Contributed by AV, 30-Mar-2019.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝑆    &   ⊢ Ⅎ𝑦𝑆    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpordx 48582*	Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7508. (Contributed by AV, 30-Mar-2019.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐿)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Theorem	ovmpox2 48583*	The value of an operation class abstraction. Variant of ovmpoga 7512 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)    &   ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐿)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)
Theorem	fdmdifeqresdif 48584*	The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥)))    ⇒   ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
Theorem	ofaddmndmap 48585	The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)
⊢ 𝑅 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉))
Theorem	mapsnop 48586	A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)
⊢ 𝐹 = {⟨𝑋, 𝑌⟩}    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹 ∈ (𝑅 ↑m {𝑋}))
Theorem	fprmappr 48587	A function with a domain of two elements as element of the mapping operator applied to a pair. (Contributed by AV, 20-May-2024.)
⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ (𝑋 ↑m {𝐴, 𝐵}))
Theorem	mapprop 48588	An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.) (Proof shortened by AV, 2-Jun-2024.)
⊢ 𝐹 = {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩}    ⇒   ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → 𝐹 ∈ (𝑅 ↑m {𝑋, 𝑌}))
Theorem	ztprmneprm 48589	A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
⊢ ((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
Theorem	2t6m3t4e0 48590	2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
⊢ ((2 · 6) − (3 · 4)) = 0
Theorem	ssnn0ssfz 48591*	For any finite subset of ℕ0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 32867. (Contributed by AV, 30-Sep-2019.)
⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛))
Theorem	nn0sumltlt 48592	If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑐 → 𝑏 < 𝑐))
21.48.19.2  The binomial coefficient operation (extension)
Theorem	bcpascm1 48593	Pascal's rule for the binomial coefficient, generalized to all integers 𝐾, shifted down by 1. (Contributed by AV, 8-Sep-2019.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (((𝑁 − 1)C𝐾) + ((𝑁 − 1)C(𝐾 − 1))) = (𝑁C𝐾))
Theorem	altgsumbc 48594*	The sum of binomial coefficients for a fixed positive 𝑁 with alternating signs is zero. Notice that this is not valid for 𝑁 = 0 (since ((-1↑0) · (0C0)) = (1 · 1) = 1). For a proof using Pascal's rule (bcpascm1 48593) instead of the binomial theorem (binom 15753), see altgsumbcALT 48595. (Contributed by AV, 13-Sep-2019.)
⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)
Theorem	altgsumbcALT 48595*	Alternate proof of altgsumbc 48594, using Pascal's rule (bcpascm1 48593) instead of the binomial theorem (binom 15753). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)
21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `
Theorem	zlmodzxzlmod 48596	The ℤ-module ℤ × ℤ is a (left) module with the ring of integers as base set. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    ⇒   ⊢ (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍))
Theorem	zlmodzxzel 48597	An element of the (base set of the) ℤ-module ℤ × ℤ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ (Base‘𝑍))
Theorem	zlmodzxz0 48598	The 0 of the ℤ-module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ 0 = {⟨0, 0⟩, ⟨1, 0⟩}    ⇒   ⊢ 0 = (0g‘𝑍)
Theorem	zlmodzxzscm 48599	The scalar multiplication of the ℤ-module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ ∙ = ( ·𝑠 ‘𝑍)    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∙ {⟨0, 𝐵⟩, ⟨1, 𝐶⟩}) = {⟨0, (𝐴 · 𝐵)⟩, ⟨1, (𝐴 · 𝐶)⟩})
Theorem	zlmodzxzadd 48600	The addition of the ℤ-module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ + = (+g‘𝑍)    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} + {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩})