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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ringidval 18901 | The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ 1 = (0g‘𝐺) | ||
Theorem | dfur2 18902* | The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ 1 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑒) = 𝑥))) | ||
Syntax | csrg 18903 | Extend class notation with the class of all semirings. |
class SRing | ||
Definition | df-srg 18904* | Define class of all semirings. A semiring is a set equipped with two everywhere-defined internal operations, whose first one is an additive commutative monoid structure and the second one is a multiplicative monoid structure, and where multiplication is (left- and right-) distributive over addition. Compared to the definition of a ring, this definition also adds that the additive identity is an absorbing element of the multiplicative law, as this cannot be deduced from distributivity alone. Definition of [Golan] p. 1. Note that our semirings are unital. Such semirings are sometimes called "rigs", being "rings without negatives". (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ SRing = {𝑓 ∈ CMnd ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡][(0g‘𝑓) / 𝑛]∀𝑥 ∈ 𝑟 (∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))) ∧ ((𝑛𝑡𝑥) = 𝑛 ∧ (𝑥𝑡𝑛) = 𝑛)))} | ||
Theorem | issrg 18905* | The predicate "is a semiring." (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) ∧ (( 0 · 𝑥) = 0 ∧ (𝑥 · 0 ) = 0 )))) | ||
Theorem | srgcmn 18906 | A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) | ||
Theorem | srgmnd 18907 | A semiring is a monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | ||
Theorem | srgmgp 18908 | A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
⊢ 𝐺 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) | ||
Theorem | srgi 18909 | Properties of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))) | ||
Theorem | srgcl 18910 | Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) | ||
Theorem | srgass 18911 | Associative law for the multiplication operation of a semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍))) | ||
Theorem | srgideu 18912* | The unit element of a semiring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ SRing → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥)) | ||
Theorem | srgfcl 18913 | Functionality of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ · Fn (𝐵 × 𝐵)) → · :(𝐵 × 𝐵)⟶𝐵) | ||
Theorem | srgdi 18914 | Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) | ||
Theorem | srgdir 18915 | Distributive law for the multiplication operation of a semiring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) | ||
Theorem | srgidcl 18916 | The unit element of a semiring belongs to the base set of the semiring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ SRing → 1 ∈ 𝐵) | ||
Theorem | srg0cl 18917 | The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ SRing → 0 ∈ 𝐵) | ||
Theorem | srgidmlem 18918 | Lemma for srglidm 18919 and srgridm 18920. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) | ||
Theorem | srglidm 18919 | The unit element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋) | ||
Theorem | srgridm 18920 | The unit element of a semiring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) | ||
Theorem | issrgid 18921* | Properties showing that an element 𝐼 is the unity element of a semiring. (Contributed by NM, 7-Aug-2013.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ SRing → ((𝐼 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼)) | ||
Theorem | srgacl 18922 | Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | ||
Theorem | srgcom 18923 | Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | srgrz 18924 | The zero of a semiring is a right-absorbing element. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) | ||
Theorem | srglz 18925 | The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 ) | ||
Theorem | srgisid 18926* | In a semiring, the only left-absorbing element is the additive identity. Remark in [Golan] p. 1. (Contributed by Thierry Arnoux, 1-May-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍 · 𝑥) = 𝑍) ⇒ ⊢ (𝜑 → 𝑍 = 0 ) | ||
Theorem | srg1zr 18927 | The only semiring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) ⇒ ⊢ (((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | srgen1zr 18928 | The only semiring with one element is the zero ring (at least if its operations are internal binary operations). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ ∗ = (.r‘𝑅) & ⊢ 𝑍 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {〈〈𝑍, 𝑍〉, 𝑍〉} ∧ ∗ = {〈〈𝑍, 𝑍〉, 𝑍〉}))) | ||
Theorem | srgmulgass 18929 | An associative property between group multiple and ring multiplication for semirings. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ × = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))) | ||
Theorem | srgpcomp 18930 | If two elements of a semiring commute, they also commute if one of the elements is raised to a higher power. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) ⇒ ⊢ (𝜑 → ((𝐾 ↑ 𝐵) × 𝐴) = (𝐴 × (𝐾 ↑ 𝐵))) | ||
Theorem | srgpcompp 18931 | If two elements of a semiring commute, they also commute if the elements are raised to a higher power. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵)) × 𝐴) = (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵))) | ||
Theorem | srgpcomppsc 18932 | If two elements of a semiring commute, they also commute if the elements are raised to a higher power and a scalar multiplication is involved. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ · = (.g‘𝑅) & ⊢ (𝜑 → 𝐶 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐶 · ((𝑁 ↑ 𝐴) × (𝐾 ↑ 𝐵))) × 𝐴) = (𝐶 · (((𝑁 + 1) ↑ 𝐴) × (𝐾 ↑ 𝐵)))) | ||
Theorem | srglmhm 18933* | Left-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringlghm 19002. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 MndHom 𝑅)) | ||
Theorem | srgrmhm 18934* | Right-multiplication in a semiring by a fixed element of the ring is a monoid homomorphism, analogous to ringrghm 19003. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 MndHom 𝑅)) | ||
Theorem | srgsummulcr 18935* | A finite semiring sum multiplied by a constant, analogous to gsummulc1 19004. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌)) | ||
Theorem | sgsummulcl 18936* | A finite semiring sum multiplied by a constant, analogous to gsummulc2 19005. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | ||
Theorem | srg1expzeq1 18937 | The exponentiation (by a nonnegative integer) of the unity element of a (semi)ring, analogous to mulgnn0z 17964. (Contributed by AV, 25-Nov-2019.) |
⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ · = (.g‘𝐺) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) → (𝑁 · 1 ) = 1 ) | ||
In this section, we prove the binomial theorem for semirings, srgbinom 18943, which is a generalization of the binomial theorem for complex numbers, binom 14975: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). Notice that the binomial theorem would also hold in the non-unital case (that is, in a "rg") and actually, the additive unit is not needed in its proof either. Therefore, it could be proven for even more general cases. An example would be the integrable nonnegative (resp. positive) bounded functions on ℝ. Special cases of the binomial theorem are csrgbinom 18944 (binomial theorem for commutative semirings) and crngbinom 19019 (binomial theorem for commutative rings). | ||
Theorem | srgbinomlem1 18938 | Lemma 1 for srgbinomlem 18942. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ ((𝜑 ∧ (𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵)) ∈ 𝑆) | ||
Theorem | srgbinomlem2 18939 | Lemma 2 for srgbinomlem 18942. (Contributed by AV, 23-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0 ∧ 𝐸 ∈ ℕ0)) → (𝐶 · ((𝐷 ↑ 𝐴) × (𝐸 ↑ 𝐵))) ∈ 𝑆) | ||
Theorem | srgbinomlem3 18940* | Lemma 3 for srgbinomlem 18942. (Contributed by AV, 23-Aug-2019.) (Proof shortened by AV, 27-Oct-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜓 → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 ↑ (𝐴 + 𝐵)) × 𝐴) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C𝑘) · ((((𝑁 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) | ||
Theorem | srgbinomlem4 18941* | Lemma 4 for srgbinomlem 18942. (Contributed by AV, 24-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜓 → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 ↑ (𝐴 + 𝐵)) × 𝐵) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ ((𝑁C(𝑘 − 1)) · ((((𝑁 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) | ||
Theorem | srgbinomlem 18942* | Lemma for srgbinom 18943. Inductive step, analogous to binomlem 14974. (Contributed by AV, 24-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) & ⊢ (𝜑 → 𝑅 ∈ SRing) & ⊢ (𝜑 → 𝐴 ∈ 𝑆) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) & ⊢ (𝜑 → (𝐴 × 𝐵) = (𝐵 × 𝐴)) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜓 → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 + 1) ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...(𝑁 + 1)) ↦ (((𝑁 + 1)C𝑘) · ((((𝑁 + 1) − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) | ||
Theorem | srgbinom 18943* | The binomial theorem for commuting elements of a semiring: (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)) (generalization of binom 14975). (Contributed by AV, 24-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) ⇒ ⊢ (((𝑅 ∈ SRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐵 × 𝐴))) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) | ||
Theorem | csrgbinom 18944* | The binomial theorem for commutative semirings. (Contributed by AV, 24-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) ⇒ ⊢ (((𝑅 ∈ SRing ∧ 𝐺 ∈ CMnd ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) | ||
Syntax | crg 18945 | Extend class notation with class of all (unital) rings. |
class Ring | ||
Syntax | ccrg 18946 | Extend class notation with class of all (unital) commutative rings. |
class CRing | ||
Definition | df-ring 18947* | Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 18977), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡]∀𝑥 ∈ 𝑟 ∀𝑦 ∈ 𝑟 ∀𝑧 ∈ 𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))} | ||
Definition | df-cring 18948 | Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd} | ||
Theorem | isring 18949* | The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))))) | ||
Theorem | ringgrp 18950 | A ring is a group. (Contributed by NM, 15-Sep-2011.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | ||
Theorem | ringmgp 18951 | A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐺 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) | ||
Theorem | iscrng 18952 | A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝐺 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd)) | ||
Theorem | crngmgp 18953 | A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝐺 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) | ||
Theorem | ringmnd 18954 | A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | ||
Theorem | ringmgm 18955 | A ring is a magma. (Contributed by AV, 31-Jan-2020.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mgm) | ||
Theorem | crngring 18956 | A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | ||
Theorem | mgpf 18957 | Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ (mulGrp ↾ Ring):Ring⟶Mnd | ||
Theorem | ringi 18958 | Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))) | ||
Theorem | ringcl 18959 | Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) | ||
Theorem | crngcom 18960 | A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑌 · 𝑋)) | ||
Theorem | iscrng2 18961* | A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 · 𝑦) = (𝑦 · 𝑥))) | ||
Theorem | ringass 18962 | Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍))) | ||
Theorem | ringideu 18963* | The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥)) | ||
Theorem | ringdi 18964 | Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍))) | ||
Theorem | ringdir 18965 | Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))) | ||
Theorem | ringidcl 18966 | The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐵) | ||
Theorem | ring0cl 18967 | The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) | ||
Theorem | ringidmlem 18968 | Lemma for ringlidm 18969 and ringridm 18970. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋)) | ||
Theorem | ringlidm 18969 | The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 1 · 𝑋) = 𝑋) | ||
Theorem | ringridm 18970 | The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) | ||
Theorem | isringid 18971* | Properties showing that an element 𝐼 is the unity element of a ring. (Contributed by NM, 7-Aug-2013.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ((𝐼 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼)) | ||
Theorem | ringid 18972* | The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑢 ∈ 𝐵 ((𝑢 · 𝑋) = 𝑋 ∧ (𝑋 · 𝑢) = 𝑋)) | ||
Theorem | ringadd2 18973* | A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋)) | ||
Theorem | rngo2times 18974 | A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unit with itself. (Contributed by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴)) | ||
Theorem | ringidss 18975 | A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝑀 = ((mulGrp‘𝑅) ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴) → 1 = (0g‘𝑀)) | ||
Theorem | ringacl 18976 | Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | ||
Theorem | ringcom 18977 | Commutativity of the additive group of a ring. (See also lmodcom 19312.) (Contributed by Gérard Lang, 4-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | ringabl 18978 | A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | ||
Theorem | ringcmn 18979 | A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | ||
Theorem | ringpropd 18980* | If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-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 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) | ||
Theorem | crngpropd 18981* | If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-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 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing)) | ||
Theorem | ringprop 18982 | If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) |
⊢ (Base‘𝐾) = (Base‘𝐿) & ⊢ (+g‘𝐾) = (+g‘𝐿) & ⊢ (.r‘𝐾) = (.r‘𝐿) ⇒ ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring) | ||
Theorem | isringd 18983* | Properties that determine a ring. (Contributed by NM, 2-Aug-2013.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) & ⊢ (𝜑 → 1 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥) ⇒ ⊢ (𝜑 → 𝑅 ∈ Ring) | ||
Theorem | iscrngd 18984* | Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → + = (+g‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Grp) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) & ⊢ (𝜑 → 1 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) ⇒ ⊢ (𝜑 → 𝑅 ∈ CRing) | ||
Theorem | ringlz 18985 | The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 · 𝑋) = 0 ) | ||
Theorem | ringrz 18986 | The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) | ||
Theorem | ringsrg 18987 | Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | ||
Theorem | ring1eq0 18988 | If one and zero are equal, then any two elements of a ring are equal. Alternately, every ring has one distinct from zero except the zero ring containing the single element {0}. (Contributed by Mario Carneiro, 10-Sep-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( 1 = 0 → 𝑋 = 𝑌)) | ||
Theorem | ring1ne0 18989 | If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 1 ≠ 0 ) | ||
Theorem | ringinvnz1ne0 18990* | In a unitary ring, a left invertible element is different from zero iff 1 ≠ 0. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 ) ⇒ ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) | ||
Theorem | ringinvnzdiv 18991* | In a unitary ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 ) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 )) | ||
Theorem | ringnegl 18992 | Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 34373 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝑋) = (𝑁‘𝑋)) | ||
Theorem | rngnegr 18993 | Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 34374 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · (𝑁‘ 1 )) = (𝑁‘𝑋)) | ||
Theorem | ringmneg1 18994 | Negation of a product in a ring. (mulneg1 10814 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌))) | ||
Theorem | ringmneg2 18995 | Negation of a product in a ring. (mulneg2 10815 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) | ||
Theorem | ringm2neg 18996 | Double negation of a product in a ring. (mul2neg 10817 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌)) | ||
Theorem | ringsubdi 18997 | Ring multiplication distributes over subtraction. (subdi 10811 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍))) | ||
Theorem | rngsubdir 18998 | Ring multiplication distributes over subtraction. (subdir 10812 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍))) | ||
Theorem | mulgass2 18999 | An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ × = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))) | ||
Theorem | ring1 19000 | The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.) |
⊢ 𝑀 = {〈(Base‘ndx), {𝑍}〉, 〈(+g‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉, 〈(.r‘ndx), {〈〈𝑍, 𝑍〉, 𝑍〉}〉} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ Ring) |
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