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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | c0ghm 41701* | The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇)) | ||
Theorem | c0rhm 41702* | The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Ring ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RingHom 𝑇)) | ||
Theorem | c0rnghm 41703* | The constant mapping to zero is a nonunital ring homomorphism from any nonunital ring to the zero ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHomo 𝑇)) | ||
Theorem | c0snmgmhm 41704* | The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆)) | ||
Theorem | c0snmhm 41705* | The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) & ⊢ 𝑍 = (0g‘𝑇) ⇒ ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆)) | ||
Theorem | c0snghm 41706* | The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) & ⊢ 𝑍 = (0g‘𝑇) ⇒ ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 GrpHom 𝑆)) | ||
Theorem | zrrnghm 41707* | The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.) |
⊢ 𝐵 = (Base‘𝑇) & ⊢ 0 = (0g‘𝑆) & ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 ) ⇒ ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆)) | ||
Theorem | rhmfn 41708 | The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.) |
⊢ RingHom Fn (Ring × Ring) | ||
Theorem | rhmval 41709 | The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.) |
⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆)))) | ||
Theorem | rhmisrnghm 41710 | Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 RngHomo 𝑆)) | ||
Theorem | lidldomn1 41711* | If a (left) ideal (which is not the zero ideal) of a domain has a multiplicative identity element, the identity element is the identity of the domain. (Contributed by AV, 17-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 }) ∧ 𝐼 ∈ 𝑈) → (∀𝑥 ∈ 𝑈 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥) → 𝐼 = 1 )) | ||
Theorem | lidlssbas 41712 | The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) ⇒ ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅)) | ||
Theorem | lidlbas 41713 | A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) ⇒ ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) = 𝑈) | ||
Theorem | lidlabl 41714 | A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Abel) | ||
Theorem | lidlmmgm 41715 | The multiplicative group of a (left) ideal of a ring is a magma. (Contributed by AV, 17-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (mulGrp‘𝐼) ∈ Mgm) | ||
Theorem | lidlmsgrp 41716 | The multiplicative group of a (left) ideal of a ring is a semigroup. (Contributed by AV, 17-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → (mulGrp‘𝐼) ∈ SGrp) | ||
Theorem | lidlrng 41717 | A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng) | ||
Theorem | zlidlring 41718 | The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring) | ||
Theorem | uzlidlring 41719 | Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵))) | ||
Theorem | lidldomnnring 41720 | A (left) ideal of a domain which is neither the zero ideal nor the unit ideal is not a unital ring. (Contributed by AV, 18-Feb-2020.) |
⊢ 𝐿 = (LIdeal‘𝑅) & ⊢ 𝐼 = (𝑅 ↾s 𝑈) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → 𝐼 ∉ Ring) | ||
Theorem | 0even 41721* | 0 is an even integer. (Contributed by AV, 11-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ⇒ ⊢ 0 ∈ 𝐸 | ||
Theorem | 1neven 41722* | 1 is not an even integer. (Contributed by AV, 12-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ⇒ ⊢ 1 ∉ 𝐸 | ||
Theorem | 2even 41723* | 2 is an even integer. (Contributed by AV, 12-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ⇒ ⊢ 2 ∈ 𝐸 | ||
Theorem | 2zlidl 41724* | The even integers are a (left) ideal of the ring of integers. (Contributed by AV, 20-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑈 = (LIdeal‘ℤring) ⇒ ⊢ 𝐸 ∈ 𝑈 | ||
Theorem | 2zrng 41725* | The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Remark: the structure of the complementary subset of the set of integers, the odd integers, is not even a magma, see oddinmgm 41605. (Contributed by AV, 20-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑈 = (LIdeal‘ℤring) & ⊢ 𝑅 = (ℤring ↾s 𝐸) ⇒ ⊢ 𝑅 ∈ Rng | ||
Theorem | 2zrngbas 41726* | The base set of R is the set of all even integers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 𝐸 = (Base‘𝑅) | ||
Theorem | 2zrngadd 41727* | The group addition operation of R is the addition of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ + = (+g‘𝑅) | ||
Theorem | 2zrng0 41728* | The additive identity of R is the complex number 0. (Contributed by AV, 11-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 0 = (0g‘𝑅) | ||
Theorem | 2zrngamgm 41729* | R is an (additive) magma. (Contributed by AV, 6-Jan-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 𝑅 ∈ Mgm | ||
Theorem | 2zrngasgrp 41730* | R is an (additive) semigroup. (Contributed by AV, 4-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 𝑅 ∈ SGrp | ||
Theorem | 2zrngamnd 41731* | R is an (additive) monoid. (Contributed by AV, 11-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 𝑅 ∈ Mnd | ||
Theorem | 2zrngacmnd 41732* | R is a commutative (additive) monoid. (Contributed by AV, 11-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 𝑅 ∈ CMnd | ||
Theorem | 2zrngagrp 41733* | R is an (additive) group. (Contributed by AV, 6-Jan-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 𝑅 ∈ Grp | ||
Theorem | 2zrngaabl 41734* | R is an (additive) abelian group. (Contributed by AV, 11-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ 𝑅 ∈ Abel | ||
Theorem | 2zrngmul 41735* | The ring multiplication operation of R is the multiplication on complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) ⇒ ⊢ · = (.r‘𝑅) | ||
Theorem | 2zrngmmgm 41736* | R is a (multiplicative) magma. (Contributed by AV, 11-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ 𝑀 ∈ Mgm | ||
Theorem | 2zrngmsgrp 41737* | R is a (multiplicative) semigroup. (Contributed by AV, 4-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ 𝑀 ∈ SGrp | ||
Theorem | 2zrngALT 41738* | The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Alternate version of 2zrng 41725, based on a restriction of the field of the complex numbers. The proof is based on the facts that the ring of even integers is an additive abelian group (see 2zrngaabl 41734) and a multiplicative semigroup (see 2zrngmsgrp 41737). (Contributed by AV, 11-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ 𝑅 ∈ Rng | ||
Theorem | 2zrngnmlid 41739* | R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ ∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 | ||
Theorem | 2zrngnmrid 41740* | R has no multiplicative (right) identity. (Contributed by AV, 12-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑎 · 𝑏) ≠ 𝑎 | ||
Theorem | 2zrngnmlid2 41741* | R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.) |
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} & ⊢ 𝑅 = (ℂfld ↾s 𝐸) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 | ||
Theorem | 2zrngnring 41742* | 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 | plusgndxnmulrndx 41743 | The slot for the group (addition) operation is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
⊢ (+g‘ndx) ≠ (.r‘ndx) | ||
Theorem | basendxnmulrndx 41744 | The slot for the base set is not the slot for the ring (multiplication) operation in an extensible structure. (Contributed by AV, 16-Feb-2020.) |
⊢ (Base‘ndx) ≠ (.r‘ndx) | ||
Theorem | cznrnglem 41745 | Lemma for cznrng 41747: 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 41746 | 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 41747* | 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 41748* | 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) | ||
The "category of non-unital rings" RngCat is the category of all non-unital rings Rng in a universe and non-unital ring homomorphisms RngHomo 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 41751. Alternatively, 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 df-rngcALTV 41752 or dfrngc2 41764. 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 41757, and the morphisms/arrows are the non-unital ring homomorphisms restricted to this subset of the non-unital rings ( RngHomo ↾ (𝐵 × 𝐵)), see rngchomfval 41758, whereas the composition is the ordinary composition of functions, see rngccofval 41762 and rngcco 41763. 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 41766, 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 41769. It follows that the category of non-unital rings RngCat is actually a category, see rngccat 41770 with the identity function as identity arrow, see rngcid 41771. | ||
Syntax | crngc 41749 | Extend class notation to include the category Rng. |
class RngCat | ||
Syntax | crngcALTV 41750 | Extend class notation to include the category Rng. (New usage is discouraged.) |
class RngCatALTV | ||
Definition | df-rngc 41751 | 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 ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))) | ||
Definition | df-rngcALTV 41752* | 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 RngHomo 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) | ||
Theorem | rngcvalALTV 41753* | 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 RngHomo 𝑦))) & ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | rngcval 41754 | 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 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) | ||
Theorem | rnghmresfn 41755 | 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 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) | ||
Theorem | rnghmresel 41756 | 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 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌)) | ||
Theorem | rngcbas 41757 | 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 41758 | 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 = ( RngHomo ↾ (𝐵 × 𝐵))) | ||
Theorem | rngchom 41759 | 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 RngHomo 𝑌)) | ||
Theorem | elrngchom 41760 | 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 41761 | 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 41762 | 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 41763 | 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 41764 | 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 = ( RngHomo ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | rnghmsscmap2 41765* | 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 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | ||
Theorem | rnghmsscmap 41766* | 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 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | ||
Theorem | rnghmsubcsetclem1 41767 | Lemma 1 for rnghmsubcsetc 41769. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rnghmsubcsetclem2 41768* | Lemma 2 for rnghmsubcsetc 41769. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rnghmsubcsetc 41769 | 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 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) | ||
Theorem | rngccat 41770 | 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 41771 | 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 41772 | 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 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) | ||
Theorem | rngcinv 41773 | 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 RngIsom 𝑌) ∧ 𝐺 = ◡𝐹))) | ||
Theorem | rngciso 41774 | 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 RngIsom 𝑌))) | ||
Theorem | rngcbasALTV 41775 | 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 41776* | 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 RngHomo 𝑦))) | ||
Theorem | rngchomALTV 41777 | 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 RngHomo 𝑌)) | ||
Theorem | elrngchomALTV 41778 | 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 41779* | 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) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) | ||
Theorem | rngccoALTV 41780 | 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 RngHomo 𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌 RngHomo 𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | rngccatidALTV 41781* | Lemma for rngccatALTV 41782. (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 41782 | 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 41783 | 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 41784 | 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 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) | ||
Theorem | rngcinvALTV 41785 | 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 RngIsom 𝑌) ∧ 𝐺 = ◡𝐹))) | ||
Theorem | rngcisoALTV 41786 | 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 RngIsom 𝑌))) | ||
Theorem | rngchomffvalALTV 41787* | 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 RngHomo 𝑦))) | ||
Theorem | rngchomrnghmresALTV 41788 | 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 = ( RngHomo ↾ (𝐵 × 𝐵))) | ||
Theorem | rngcifuestrc 41789* | 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 RngHomo 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺) | ||
Theorem | funcrngcsetc 41790* | 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 41791, using cofuval2 16370 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 41789, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16612. (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 RngHomo 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | funcrngcsetcALT 41791* | Alternate proof of funcrngcsetc 41790, using cofuval2 16370 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 41789, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16612. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 41790. (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 RngHomo 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | zrinitorngc 41792 | The zero ring is an initial object in the category of nonunital 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 41793 | The zero ring is a terminal object in the category of nonunital 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 41794 | 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 41797. Alternatively, 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 41810. 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 41803, and the morphisms/arrows are the ring homomorphisms restricted to this subset of the rings ( RingHom ↾ (𝐵 × 𝐵)), see ringchomfval 41804, whereas the composition is the ordinary composition of functions, see ringccofval 41808 and ringcco 41809. By showing that the ring homomorphisms between rings are a subcategory subset (⊆cat) of the mappings between base sets of extensible structures, see rhmsscmap 41812, it can be shown that the ring homomorphisms between rings are a subcategory (Subcat) of the category of extensible structures, see rhmsubcsetc 41815. It follows that the category of rings RingCat is actually a category, see ringccat 41816 with the identity function as identity arrow, see ringcid 41817. 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 41818, and that the ring homomorphisms between rings are a subcategory of the category of non-unital rings, see rhmsubcrngc 41821. 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 41822: ((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 41827. | ||
Syntax | cringc 41795 | Extend class notation to include the category Ring. |
class RingCat | ||
Syntax | cringcALTV 41796 | Extend class notation to include the category Ring. (New usage is discouraged.) |
class RingCatALTV | ||
Definition | df-ringc 41797 | 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))))) | ||
Definition | df-ringcALTV 41798* | 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 41799* | 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 | ringcval 41800 | 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)) |
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