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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | subrgsubm 20601 | 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 20602 | 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 20603 | 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 20604 | 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 20605 | 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 20606 | 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 20607 | 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 20608* | 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))) | ||
Theorem | opprsubrg 20609 | Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubRing‘𝑅) = (SubRing‘𝑂) | ||
Theorem | subrgnzr 20610 | A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing) | ||
Theorem | subrgint 20611 | The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅)) | ||
Theorem | subrgin 20612 | The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.) |
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRing‘𝑅)) | ||
Theorem | subrgmre 20613 | The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (Moore‘𝐵)) | ||
Theorem | subsubrg 20614 | A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴))) | ||
Theorem | subsubrg2 20615 | The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) ⇒ ⊢ (𝐴 ∈ (SubRing‘𝑅) → (SubRing‘𝑆) = ((SubRing‘𝑅) ∩ 𝒫 𝐴)) | ||
Theorem | issubrg3 20616 | A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀)))) | ||
Theorem | resrhm 20617 | Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.) |
⊢ 𝑈 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 RingHom 𝑇)) | ||
Theorem | resrhm2b 20618 | Restriction of the codomain of a (ring) homomorphism. resghm2b 19264 analog. (Contributed by SN, 7-Feb-2025.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈))) | ||
Theorem | rhmeql 20619 | The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubRing‘𝑆)) | ||
Theorem | rhmima 20620 | The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRing‘𝑁)) | ||
Theorem | rnrhmsubrg 20621 | The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.) |
⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁)) | ||
Theorem | cntzsubr 20622 | Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (Cntz‘𝑀) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRing‘𝑅)) | ||
Theorem | pwsdiagrhm 20623* | Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 RingHom 𝑌)) | ||
Theorem | subrgpropd 20624* | If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-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 → (SubRing‘𝐾) = (SubRing‘𝐿)) | ||
Theorem | rhmpropd 20625* | Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐽)𝑦) = (𝑥(.r‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝑀)𝑦)) ⇒ ⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀)) | ||
Syntax | crgspn 20626 | Extend class notation with span of a set of elements over a ring. |
class RingSpan | ||
Definition | df-rgspn 20627* | 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 | rgspnval 20628* | Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) ⇒ ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡}) | ||
Theorem | rgspncl 20629 | The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) ⇒ ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑅)) | ||
Theorem | rgspnssid 20630 | The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝑈) | ||
Theorem | rgspnmin 20631 | The ring-span is contained in all subrings which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅)) & ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴)) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) ⇒ ⊢ (𝜑 → 𝑈 ⊆ 𝑆) | ||
The "category of non-unital rings" RngCat is the category of all non-unital rings Rng in a universe and non-unital ring homomorphisms RngHom between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to non-unital rings and the morphisms to the non-unital ring homomorphisms, while the composition of morphisms is preserved, see df-rngc 20633. Alternately, the category of non-unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see dfrngc2 20644. Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are a subset of the non-unital rings (relativized to a subset or "universe" 𝑢) (𝑢 ∩ Rng), see rngcbas 20637, and the morphisms/arrows are the non-unital ring homomorphisms restricted to this subset of the non-unital rings ( RngHom ↾ (𝐵 × 𝐵)), see rngchomfval 20638, whereas the composition is the ordinary composition of functions, see rngccofval 20642 and rngcco 20643. By showing that the non-unital ring homomorphisms between non-unital rings are a subcategory subset (⊆cat) of the mappings between base sets of extensible structures, see rnghmsscmap 20646, it can be shown that the non-unital ring homomorphisms between non-unital rings are a subcategory (Subcat) of the category of extensible structures, see rnghmsubcsetc 20649. It follows that the category of non-unital rings RngCat is actually a category, see rngccat 20650 with the identity function as identity arrow, see rngcid 20651. | ||
Syntax | crngc 20632 | Extend class notation to include the category Rng. |
class RngCat | ||
Definition | df-rngc 20633 | 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. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) | ||
Theorem | rngcval 20634 | Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) & ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) | ||
Theorem | rnghmresfn 20635 | The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020.) |
⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) & ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) | ||
Theorem | rnghmresel 20636 | An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.) |
⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHom 𝑌)) | ||
Theorem | rngcbas 20637 | Set of objects of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | ||
Theorem | rngchomfval 20638 | Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) | ||
Theorem | rngchom 20639 | Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHom 𝑌)) | ||
Theorem | elrngchom 20640 | A morphism of non-unital rings is a function. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐶 = (RngCat‘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 | rngchomfeqhom 20641 | The functionalized Hom-set operation equals the Hom-set operation in the category of non-unital rings (in a universe). (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) | ||
Theorem | rngccofval 20642 | Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) | ||
Theorem | rngcco 20643 | Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)) & ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | dfrngc2 20644 | Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) & ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | rnghmsscmap2 20645* | The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) | ||
Theorem | rnghmsscmap 20646* | The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) | ||
Theorem | rnghmsubcsetclem1 20647 | Lemma 1 for rnghmsubcsetc 20649. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rnghmsubcsetclem2 20648* | Lemma 2 for rnghmsubcsetc 20649. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rnghmsubcsetc 20649 | The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) | ||
Theorem | rngccat 20650 | The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 9-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | rngcid 20651 | The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 10-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑆 = (Base‘𝑋) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) | ||
Theorem | rngcsect 20652 | A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) |
⊢ 𝐶 = (RngCat‘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 | rngcinv 20653 | An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) |
⊢ 𝐶 = (RngCat‘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 | rngciso 20654 | An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIso 𝑌))) | ||
Theorem | rngcifuestrc 20655* | The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.) |
⊢ 𝑅 = (RngCat‘𝑈) & ⊢ 𝐸 = (ExtStrCat‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺) | ||
Theorem | funcrngcsetc 20656* | The "natural forgetful functor" from the category of non-unital rings into the category of sets which sends each non-unital ring to its underlying set (base set) and the morphisms (non-unital ring homomorphisms) to mappings of the corresponding base sets. An alternate proof is provided in funcrngcsetcALT 20657, using cofuval2 17937 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 20655, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18204. (Contributed by AV, 26-Mar-2020.) |
⊢ 𝑅 = (RngCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | funcrngcsetcALT 20657* | Alternate proof of funcrngcsetc 20656, using cofuval2 17937 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 20655, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18204. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 20656. (Contributed by AV, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑅 = (RngCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | zrinitorngc 20658 | The zero ring is an initial object in the category of non-unital rings. (Contributed by AV, 18-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶)) | ||
Theorem | zrtermorngc 20659 | The zero ring is a terminal object in the category of non-unital rings. (Contributed by AV, 17-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶)) | ||
Theorem | zrzeroorngc 20660 | The zero ring is a zero object in the category of non-unital rings. (Contributed by AV, 18-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (ZeroO‘𝐶)) | ||
The "category of unital rings" RingCat is the category of all (unital) rings Ring in a universe and (unital) ring homomorphisms RingHom between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to (unital) rings and the morphisms to the (unital) ring homomorphisms, while the composition of morphisms is preserved, see df-ringc 20662. Alternately, the category of unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see dfringc2 20673. In the following, we omit the predicate "unital", so that "ring" and "ring homomorphism" (without predicate) always mean "unital ring" and "unital ring homomorphism". Since we consider only "small categories" (i.e., categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are a subset of the rings (relativized to a subset or "universe" 𝑢) (𝑢 ∩ Ring), see ringcbas 20666, and the morphisms/arrows are the ring homomorphisms restricted to this subset of the rings ( RingHom ↾ (𝐵 × 𝐵)), see ringchomfval 20667, whereas the composition is the ordinary composition of functions, see ringccofval 20671 and ringcco 20672. By showing that the ring homomorphisms between rings are a subcategory subset (⊆cat) of the mappings between base sets of extensible structures, see rhmsscmap 20675, it can be shown that the ring homomorphisms between rings are a subcategory (Subcat) of the category of extensible structures, see rhmsubcsetc 20678. It follows that the category of rings RingCat is actually a category, see ringccat 20679 with the identity function as identity arrow, see ringcid 20680. Furthermore, it is shown that the ring homomorphisms between rings are a subcategory subset of the non-unital ring homomorphisms between non-unital rings, see rhmsscrnghm 20681, and that the ring homomorphisms between rings are a subcategory of the category of non-unital rings, see rhmsubcrngc 20684. By this, the restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings, see rngcresringcat 20685: ((RngCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))) = (RingCat‘𝑈)). Finally, it is shown that the "natural forgetful functor" from the category of rings into the category of sets is the function which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets, see funcringcsetc 20690. | ||
Syntax | cringc 20661 | Extend class notation to include the category Ring. |
class RingCat | ||
Definition | df-ringc 20662 | Definition of the category Ring, relativized to a subset 𝑢. See also the note in [Lang] p. 91, and the item Rng in [Adamek] p. 478. This is the category of all 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. (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))) | ||
Theorem | ringcval 20663 | Value of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) | ||
Theorem | rhmresfn 20664 | The class of unital ring homomorphisms restricted to subsets of unital rings is a function. (Contributed by AV, 10-Mar-2020.) |
⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) | ||
Theorem | rhmresel 20665 | An element of the unital ring homomorphisms restricted to a subset of unital rings is a unital ring homomorphism. (Contributed by AV, 10-Mar-2020.) |
⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌)) | ||
Theorem | ringcbas 20666 | Set of objects of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | ||
Theorem | ringchomfval 20667 | Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | ||
Theorem | ringchom 20668 | Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) | ||
Theorem | elringchom 20669 | A morphism of unital rings is a function. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘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 | ringchomfeqhom 20670 | The functionalized Hom-set operation equals the Hom-set operation in the category of unital rings (in a universe). (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) | ||
Theorem | ringccofval 20671 | Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) | ||
Theorem | ringcco 20672 | Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)) & ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | dfringc2 20673 | Alternate definition of the category of unital rings (in a universe). (Contributed by AV, 16-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | rhmsscmap2 20674* | The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) | ||
Theorem | rhmsscmap 20675* | The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))) | ||
Theorem | rhmsubcsetclem1 20676 | Lemma 1 for rhmsubcsetc 20678. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rhmsubcsetclem2 20677* | Lemma 2 for rhmsubcsetc 20678. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rhmsubcsetc 20678 | The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) | ||
Theorem | ringccat 20679 | The category of unital rings is a category. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 9-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | ringcid 20680 | The identity arrow in the category of unital rings is the identity function. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 10-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 1 = (Id‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑆 = (Base‘𝑋) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) | ||
Theorem | rhmsscrnghm 20681 | The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the non-unital ring homomorphisms between non-unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝑆 = (Rng ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ (𝑆 × 𝑆))) | ||
Theorem | rhmsubcrngclem1 20682 | Lemma 1 for rhmsubcrngc 20684. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rhmsubcrngclem2 20683* | Lemma 2 for rhmsubcrngc 20684. (Contributed by AV, 12-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rhmsubcrngc 20684 | The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of non-unital rings. (Contributed by AV, 12-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) | ||
Theorem | rngcresringcat 20685 | The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) | ||
Theorem | ringcsect 20686 | A section in the category of unital rings, written out. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘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 | ringcinv 20687 | An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘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 | ringciso 20688 | An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) | ||
Theorem | ringcbasbas 20689 | An element of the base set of the base set of the category of unital rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ WUni) ⇒ ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (Base‘𝑅) ∈ 𝑈) | ||
Theorem | funcringcsetc 20690* | 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, 26-Mar-2020.) |
⊢ 𝑅 = (RingCat‘𝑈) & ⊢ 𝑆 = (SetCat‘𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (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 | zrtermoringc 20691 | The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶)) | ||
Theorem | zrninitoringc 20692* | The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → ∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing) ⇒ ⊢ (𝜑 → 𝑍 ∉ (InitO‘𝐶)) | ||
Theorem | srhmsubclem1 20693* | Lemma 1 for srhmsubc 20696. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) | ||
Theorem | srhmsubclem2 20694* | Lemma 2 for srhmsubc 20696. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈))) | ||
Theorem | srhmsubclem3 20695* | Lemma 3 for srhmsubc 20696. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCat‘𝑈))𝑌)) | ||
Theorem | srhmsubc 20696* | 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.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) | ||
Theorem | sringcat 20697* | The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | crhmsubc 20698* | 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.) |
⊢ 𝐶 = (𝑈 ∩ CRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) | ||
Theorem | cringcat 20699* | 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.) |
⊢ 𝐶 = (𝑈 ∩ CRing) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | rngcrescrhm 20700 | 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet 〈(Hom ‘ndx), 𝐻〉)) |
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