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Type | Label | Description |
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Statement | ||
Theorem | 2zrngnring 48101* | R is not a unital ring. (Contributed by AV, 6-Jan-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ 𝑅 ∉ Ring | ||
Theorem | cznrnglem 48102 | Lemma for cznrng 48104: 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 48103 | 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 48104* | 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 48105* | 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) | ||
As an alternative to df-rngc 20633, 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 20644. | ||
Syntax | crngcALTV 48106 | Extend class notation to include the category Rng. (New usage is discouraged.) |
class RngCatALTV | ||
Definition | df-rngcALTV 48107* | 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 48108* | 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 48109 | 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 48110* | 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 48111 | 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 48112 | 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 48113* | 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 48114 | 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 48115* | Lemma for rngccatALTV 48116. (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 48116 | 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 48117 | 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 48118 | 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 48119 | 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 48120 | 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 48121* | 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 48122 | 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 48123 | 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 48124 | Lemma 1 for rhmsubcALTV 48128. (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 48125 | Lemma 2 for rhmsubcALTV 48128. (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 48126* | Lemma 3 for rhmsubcALTV 48128. (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 48127* | Lemma 4 for rhmsubcALTV 48128. (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 48128 | According to df-subc 17859, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17890 and subcss2 17893). 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 48129 | 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) | ||
As an alternative to df-ringc 20662, 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 20673. | ||
Syntax | cringcALTV 48130 | Extend class notation to include the category Ring. (New usage is discouraged.) |
class RingCatALTV | ||
Definition | df-ringcALTV 48131* | 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 48132* | 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 48133* | Lemma 1 for funcringcsetcALTV2 48142. (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 48134* | Lemma 2 for funcringcsetcALTV2 48142. (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 48135* | Lemma 3 for funcringcsetcALTV2 48142. (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 48136* | Lemma 4 for funcringcsetcALTV2 48142. (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 48137* | Lemma 5 for funcringcsetcALTV2 48142. (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 48138* | Lemma 6 for funcringcsetcALTV2 48142. (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 48139* | Lemma 7 for funcringcsetcALTV2 48142. (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 48140* | Lemma 8 for funcringcsetcALTV2 48142. (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 48141* | Lemma 9 for funcringcsetcALTV2 48142. (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 48142* | 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 48143 | 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 48144* | 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 48145 | 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 48146 | 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 48147* | 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 48148 | 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 48149* | Lemma for ringccatALTV 48150. (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 48150 | 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 48151 | 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 48152 | 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 48153 | 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 48154 | 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 48155 | 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 48156* | Lemma 1 for funcringcsetcALTV 48165. (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 48157* | Lemma 2 for funcringcsetcALTV 48165. (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 48158* | Lemma 3 for funcringcsetcALTV 48165. (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 48159* | Lemma 4 for funcringcsetcALTV 48165. (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 48160* | Lemma 5 for funcringcsetcALTV 48165. (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 48161* | Lemma 6 for funcringcsetcALTV 48165. (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 48162* | Lemma 7 for funcringcsetcALTV 48165. (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 48163* | Lemma 8 for funcringcsetcALTV 48165. (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 48164* | Lemma 9 for funcringcsetcALTV 48165. (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 48165* | 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 48166* | Lemma 1 for srhmsubcALTV 48168. (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 48167* | Lemma 2 for srhmsubcALTV 48168. (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 48168* | According to df-subc 17859, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17890 and subcss2 17893). 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 48169* | 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 48170* | According to df-subc 17859, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17890 and subcss2 17893). 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 48171* | 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 48172* | According to df-subc 17859, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17890 and subcss2 17893). 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 48173* | 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 48174* | 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 48175* | 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 48176* | According to df-subc 17859, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17890 and subcss2 17893). 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))) | ||
Theorem | opeliun2xp 48177 | Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5755. (Contributed by AV, 30-Mar-2019.) |
⊢ (〈𝐶, 𝑦〉 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴)) | ||
Theorem | eliunxp2 48178* | Membership in a union of Cartesian products over its second component, analogous to eliunxp 5850. (Contributed by AV, 30-Mar-2019.) |
⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | ||
Theorem | mpomptx2 48179* | 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 7545. (Contributed by AV, 30-Mar-2019.) |
⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) ⇒ ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | cbvmpox2 48180* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7526 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7525. (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 48181* | 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 8089. (Contributed by AV, 30-Mar-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) | ||
Theorem | mpoexxg2 48182* | 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 8098. (Contributed by AV, 30-Mar-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | ovmpordxf 48183* | Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7582. (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 48184* | Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7582. (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 48185* | The value of an operation class abstraction. Variant of ovmpoga 7586 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 48186* | 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 48187 | 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 48188 | 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 48189 | 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 48190 | 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 48191 | 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 48192 | 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.) |
⊢ ((2 · 6) − (3 · 4)) = 0 | ||
Theorem | ssnn0ssfz 48193* | For any finite subset of ℕ0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 32795. (Contributed by AV, 30-Sep-2019.) |
⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛)) | ||
Theorem | nn0sumltlt 48194 | 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)) | ||
Theorem | bcpascm1 48195 | 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 48196* | 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 48195) instead of the binomial theorem (binom 15862), see altgsumbcALT 48197. (Contributed by AV, 13-Sep-2019.) |
⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) | ||
Theorem | altgsumbcALT 48197* | Alternate proof of altgsumbc 48196, using Pascal's rule (bcpascm1 48195) instead of the binomial theorem (binom 15862). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0) | ||
Theorem | zlmodzxzlmod 48198 | 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 48199 | 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 48200 | 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) |
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