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Definition | df-ring 19301* | 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 19331), 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 19302 | Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd} | ||
Theorem | isring 19303* | 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 19304 | A ring is a group. (Contributed by NM, 15-Sep-2011.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | ||
Theorem | ringmgp 19305 | A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐺 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) | ||
Theorem | iscrng 19306 | 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 19307 | A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ 𝐺 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → 𝐺 ∈ CMnd) | ||
Theorem | ringmnd 19308 | A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd) | ||
Theorem | ringmgm 19309 | A ring is a magma. (Contributed by AV, 31-Jan-2020.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mgm) | ||
Theorem | crngring 19310 | A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | ||
Theorem | mgpf 19311 | Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ (mulGrp ↾ Ring):Ring⟶Mnd | ||
Theorem | ringi 19312 | 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 19313 | 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 19314 | 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 19315* | 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 19316 | 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 19317* | 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 19318 | 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 19319 | 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 19320 | 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 19321 | 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 19322 | Lemma for ringlidm 19323 and ringridm 19324. (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 19323 | 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 19324 | 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 19325* | 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 19326* | 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 19327* | 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 19328 | 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 19329 | 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 19330 | 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 19331 | Commutativity of the additive group of a ring. (See also lmodcom 19682.) (Contributed by Gérard Lang, 4-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | ringabl 19332 | A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | ||
Theorem | ringcmn 19333 | A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd) | ||
Theorem | ringpropd 19334* | 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 19335* | 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 19336 | 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 19337* | 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 19338* | 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 19339 | 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 19340 | 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 19341 | Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | ||
Theorem | ring1eq0 19342 | 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 19343 | 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 19344* | 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 19345* | 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 19346 | Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 35221 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 19347 | Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 35222 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 19348 | Negation of a product in a ring. (mulneg1 11078 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 19349 | Negation of a product in a ring. (mulneg2 11079 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 19350 | Double negation of a product in a ring. (mul2neg 11081 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 19351 | Ring multiplication distributes over subtraction. (subdi 11075 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 19352 | Ring multiplication distributes over subtraction. (subdir 11076 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 19353 | 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 19354 | 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) | ||
Theorem | ringn0 19355 | Rings exist. (Contributed by AV, 29-Apr-2019.) |
⊢ Ring ≠ ∅ | ||
Theorem | ringlghm 19356* | Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) | ||
Theorem | ringrghm 19357* | Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 GrpHom 𝑅)) | ||
Theorem | gsummulc1 19358* | A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (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 | gsummulc2 19359* | A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (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 | gsummgp0 19360* | If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.) |
⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) & ⊢ ((𝜑 ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) & ⊢ (𝜑 → ∃𝑖 ∈ 𝑁 𝐵 = 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) | ||
Theorem | gsumdixp 19361* | Distribute a binary product of sums to a sum of binary products in a ring. (Contributed by Mario Carneiro, 8-Mar-2015.) (Revised by AV, 10-Jul-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ 𝑌) finSupp 0 ) ⇒ ⊢ (𝜑 → ((𝑅 Σg (𝑥 ∈ 𝐼 ↦ 𝑋)) · (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑌))) = (𝑅 Σg (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)))) | ||
Theorem | prdsmgp 19362 | The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝑀 = (mulGrp‘𝑌) & ⊢ 𝑍 = (𝑆Xs(mulGrp ∘ 𝑅)) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 Fn 𝐼) ⇒ ⊢ (𝜑 → ((Base‘𝑀) = (Base‘𝑍) ∧ (+g‘𝑀) = (+g‘𝑍))) | ||
Theorem | prdsmulrcl 19363 | A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) | ||
Theorem | prdsringd 19364 | A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Ring) ⇒ ⊢ (𝜑 → 𝑌 ∈ Ring) | ||
Theorem | prdscrngd 19365 | A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶CRing) ⇒ ⊢ (𝜑 → 𝑌 ∈ CRing) | ||
Theorem | prds1 19366 | Value of the ring unit in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Ring) ⇒ ⊢ (𝜑 → (1r ∘ 𝑅) = (1r‘𝑌)) | ||
Theorem | pwsring 19367 | A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Ring) | ||
Theorem | pws1 19368 | Value of the ring unit in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) | ||
Theorem | pwscrng 19369 | A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ CRing) | ||
Theorem | pwsmgp 19370 | The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (𝑀 ↑s 𝐼) & ⊢ 𝑁 = (mulGrp‘𝑌) & ⊢ 𝐵 = (Base‘𝑁) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ + = (+g‘𝑁) & ⊢ ✚ = (+g‘𝑍) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐵 = 𝐶 ∧ + = ✚ )) | ||
Theorem | imasring 19371* | The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑈 ∈ Ring ∧ (𝐹‘ 1 ) = (1r‘𝑈))) | ||
Theorem | qusring2 19372* | The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞))) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑈 ∈ Ring ∧ [ 1 ] ∼ = (1r‘𝑈))) | ||
Theorem | crngbinom 19373* | The binomial theorem for commutative rings (special case of csrgbinom 19298): (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). (Contributed by AV, 24-Aug-2019.) |
⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) | ||
Syntax | coppr 19374 | The opposite ring operation. |
class oppr | ||
Definition | df-oppr 19375 | Define an opposite ring, which is the same as the original ring but with multiplication written the other way around. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ oppr = (𝑓 ∈ V ↦ (𝑓 sSet 〈(.r‘ndx), tpos (.r‘𝑓)〉)) | ||
Theorem | opprval 19376 | Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) | ||
Theorem | opprmulfval 19377 | Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∙ = (.r‘𝑂) ⇒ ⊢ ∙ = tpos · | ||
Theorem | opprmul 19378 | Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∙ = (.r‘𝑂) ⇒ ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋) | ||
Theorem | crngoppr 19379 | In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 19449). (Contributed by Mario Carneiro, 14-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∙ = (.r‘𝑂) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑋 ∙ 𝑌)) | ||
Theorem | opprlem 19380 | Lemma for opprbas 19381 and oppradd 19382. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < 3 ⇒ ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) | ||
Theorem | opprbas 19381 | Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
Theorem | oppradd 19382 | Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ + = (+g‘𝑂) | ||
Theorem | opprring 19383 | An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) | ||
Theorem | opprringb 19384 | Bidirectional form of opprring 19383. (Contributed by Mario Carneiro, 6-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring) | ||
Theorem | oppr0 19385 | Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ 0 = (0g‘𝑂) | ||
Theorem | oppr1 19386 | Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ 1 = (1r‘𝑂) | ||
Theorem | opprneg 19387 | The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ 𝑁 = (invg‘𝑂) | ||
Theorem | opprsubg 19388 | Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) | ||
Theorem | mulgass3 19389 | 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)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌))) | ||
Syntax | cdsr 19390 | Ring divisibility relation. |
class ∥r | ||
Syntax | cui 19391 | Ring unit. |
class Unit | ||
Syntax | cir 19392 | Ring irreducibles. |
class Irred | ||
Definition | df-dvdsr 19393* | Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through (∥r‘(oppr‘𝑅)). (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ ∥r = (𝑤 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}) | ||
Definition | df-unit 19394 | Define the set of units in a ring, that is, all elements with a left and right multiplicative inverse. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ Unit = (𝑤 ∈ V ↦ (◡((∥r‘𝑤) ∩ (∥r‘(oppr‘𝑤))) “ {(1r‘𝑤)})) | ||
Definition | df-irred 19395* | Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑤)𝑦) ≠ 𝑧}) | ||
Theorem | reldvdsr 19396 | The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ Rel ∥ | ||
Theorem | dvdsrval 19397* | Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ∥ = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} | ||
Theorem | dvdsr 19398* | Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) | ||
Theorem | dvdsr2 19399* | Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∥ 𝑌 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) | ||
Theorem | dvdsrmul 19400 | A left-multiple of 𝑋 is divisible by 𝑋. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∥ (𝑌 · 𝑋)) |
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