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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | unitrinv 18501 | A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 ) | ||
Theorem | 1rinv 18502 | The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝐼 = (invr‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) = 1 ) | ||
Theorem | 0unit 18503 | The additive identity is a unit if and only if 1 = 0, i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ( 0 ∈ 𝑈 ↔ 1 = 0 )) | ||
Theorem | unitnegcl 18504 | The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) | ||
Syntax | cdvr 18505 | Extend class notation with ring division. |
class /r | ||
Definition | df-dvr 18506* | Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.) |
⊢ /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r‘𝑟)((invr‘𝑟)‘𝑦)))) | ||
Theorem | dvrfval 18507* | Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) | ||
Theorem | dvrval 18508 | Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼‘𝑌))) | ||
Theorem | dvrcl 18509 | Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) ∈ 𝐵) | ||
Theorem | unitdvcl 18510 | The units are closed under division. (Contributed by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) ∈ 𝑈) | ||
Theorem | dvrid 18511 | A cancellation law for division. (divid 10593 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 / 𝑋) = 1 ) | ||
Theorem | dvr1 18512 | A cancellation law for division. (div1 10595 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 / 1 ) = 𝑋) | ||
Theorem | dvrass 18513 | An associative law for division. (divass 10582 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑌) / 𝑍) = (𝑋 · (𝑌 / 𝑍))) | ||
Theorem | dvrcan1 18514 | A cancellation law for division. (divcan1 10573 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) · 𝑌) = 𝑋) | ||
Theorem | dvrcan3 18515 | A cancellation law for division. (divcan3 10590 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 · 𝑌) / 𝑌) = 𝑋) | ||
Theorem | dvreq1 18516 | A cancellation law for division. (diveq1 10597 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑋 / 𝑌) = 1 ↔ 𝑋 = 𝑌)) | ||
Theorem | ringinvdv 18517 | Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = ( 1 / 𝑋)) | ||
Theorem | rngidpropd 18518* | The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) | ||
Theorem | dvdsrpropd 18519* | The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (∥r‘𝐾) = (∥r‘𝐿)) | ||
Theorem | unitpropd 18520* | The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) | ||
Theorem | invrpropd 18521* | The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) | ||
Theorem | isirred 18522* | An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝑁 = (𝐵 ∖ 𝑈) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥 · 𝑦) ≠ 𝑋)) | ||
Theorem | isnirred 18523* | The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝑁 = (𝐵 ∖ 𝑈) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐵 → (¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑈 ∨ ∃𝑥 ∈ 𝑁 ∃𝑦 ∈ 𝑁 (𝑥 · 𝑦) = 𝑋))) | ||
Theorem | isirred2 18524* | Expand out the class difference from isirred 18522. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈)))) | ||
Theorem | opprirred 18525 | Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑆 = (oppr‘𝑅) & ⊢ 𝐼 = (Irred‘𝑅) ⇒ ⊢ 𝐼 = (Irred‘𝑆) | ||
Theorem | irredn0 18526 | The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ≠ 0 ) | ||
Theorem | irredcl 18527 | An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵) | ||
Theorem | irrednu 18528 | An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐼 → ¬ 𝑋 ∈ 𝑈) | ||
Theorem | irredn1 18529 | The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → 𝑋 ≠ 1 ) | ||
Theorem | irredrmul 18530 | The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ 𝐼) | ||
Theorem | irredlmul 18531 | The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝐼) → (𝑋 · 𝑌) ∈ 𝐼) | ||
Theorem | irredmul 18532 | If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈)) | ||
Theorem | irredneg 18533 | The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) → (𝑁‘𝑋) ∈ 𝐼) | ||
Theorem | irrednegb 18534 | An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝐼 = (Irred‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝐼 ↔ (𝑁‘𝑋) ∈ 𝐼)) | ||
Syntax | crh 18535 | Extend class notation with the ring homomorphisms. |
class RingHom | ||
Syntax | crs 18536 | Extend class notation with the ring isomorphisms. |
class RingIso | ||
Syntax | cric 18537 | Extend class notation with the ring isomorphism relation. |
class ≃𝑟 | ||
Definition | df-rnghom 18538* | Define the set of ring homomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Base‘𝑟) / 𝑣⦌⦋(Base‘𝑠) / 𝑤⦌{𝑓 ∈ (𝑤 ↑𝑚 𝑣) ∣ ((𝑓‘(1r‘𝑟)) = (1r‘𝑠) ∧ ∀𝑥 ∈ 𝑣 ∀𝑦 ∈ 𝑣 ((𝑓‘(𝑥(+g‘𝑟)𝑦)) = ((𝑓‘𝑥)(+g‘𝑠)(𝑓‘𝑦)) ∧ (𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥)(.r‘𝑠)(𝑓‘𝑦))))}) | ||
Definition | df-rngiso 18539* | Define the set of ring isomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ ◡𝑓 ∈ (𝑠 RingHom 𝑟)}) | ||
Theorem | dfrhm2 18540* | The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠)))) | ||
Definition | df-ric 18541 | Define the ring isomorphism relation, analogous to df-gic 17525: Two (unital) rings are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic rings share all global ring properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by AV, 24-Dec-2019.) |
⊢ ≃𝑟 = (◡ RingIso “ (V ∖ 1𝑜)) | ||
Theorem | rhmrcl1 18542 | Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring) | ||
Theorem | rhmrcl2 18543 | Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | ||
Theorem | isrhm 18544 | A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁)))) | ||
Theorem | rhmmhm 18545 | A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑁 = (mulGrp‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑀 MndHom 𝑁)) | ||
Theorem | isrim0 18546 | An isomorphism of rings is a homomorphism whose converse is also a homomorphism . (Contributed by AV, 22-Oct-2019.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RingHom 𝑅)))) | ||
Theorem | rimrcl 18547 | Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V)) | ||
Theorem | rhmghm 18548 | A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | ||
Theorem | rhmf 18549 | A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵⟶𝐶) | ||
Theorem | rhmmul 18550 | A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹‘𝐴) × (𝐹‘𝐵))) | ||
Theorem | isrhm2d 18551* | Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (1r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 GrpHom 𝑆)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | isrhmd 18552* | Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (1r‘𝑆) & ⊢ · = (.r‘𝑅) & ⊢ × = (.r‘𝑆) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → (𝐹‘ 1 ) = 𝑁) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) × (𝐹‘𝑦))) & ⊢ 𝐶 = (Base‘𝑆) & ⊢ + = (+g‘𝑅) & ⊢ ⨣ = (+g‘𝑆) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | rhm1 18553 | Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.) |
⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (1r‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹‘ 1 ) = 𝑁) | ||
Theorem | idrhm 18554 | The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅)) | ||
Theorem | rhmf1o 18555 | A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RingHom 𝑅))) | ||
Theorem | isrim 18556 | An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶))) | ||
Theorem | rimf1o 18557 | An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) | ||
Theorem | rimrhm 18558 | An isomorphism of rings is a homomorphism. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆)) | ||
Theorem | rimgim 18559 | An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆)) | ||
Theorem | rhmco 18560 | The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ ((𝐹 ∈ (𝑇 RingHom 𝑈) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 RingHom 𝑈)) | ||
Theorem | pwsco1rhm 18561* | Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐴) & ⊢ 𝑍 = (𝑅 ↑s 𝐵) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝑔 ∈ 𝐶 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 RingHom 𝑌)) | ||
Theorem | pwsco2rhm 18562* | Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐴) & ⊢ 𝑍 = (𝑆 ↑s 𝐴) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) ⇒ ⊢ (𝜑 → (𝑔 ∈ 𝐵 ↦ (𝐹 ∘ 𝑔)) ∈ (𝑌 RingHom 𝑍)) | ||
Theorem | f1rhm0to0 18563 | If a ring homomorphism 𝐹 is injective, it maps the zero of one ring (and only the zero) to the zero of the other ring. (Contributed by AV, 24-Oct-2019.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑆) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | ||
Theorem | f1rhm0to0ALT 18564 | Alternate proof for f1rhm0to0 18563. Using ghmf1 17512 does not make the proof shorter and requires disjoint variable restrictions! (Contributed by AV, 24-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑆) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | ||
Theorem | rim0to0 18565 | A ring isomorphism maps the zero of one ring (and only the zero) to the zero of the other ring. (Contributed by AV, 24-Oct-2019.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑆) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | ||
Theorem | kerf1hrm 18566 | A ring homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁})) | ||
Theorem | brric 18567 | The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝑅 ≃𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅) | ||
Theorem | brric2 18568* | The relation "is isomorphic to" for (unital) rings. This theorem corresponds to the definition df-risc 32952 of the ring isomorphism relation in JM's mathbox. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆))) | ||
Theorem | ricgic 18569 | If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.) |
⊢ (𝑅 ≃𝑟 𝑆 → 𝑅 ≃𝑔 𝑆) | ||
Syntax | cdr 18570 | Extend class notation with class of all division rings. |
class DivRing | ||
Syntax | cfield 18571 | Class of fields. |
class Field | ||
Definition | df-drng 18572 | Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.) |
⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})} | ||
Definition | df-field 18573 | A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ Field = (DivRing ∩ CRing) | ||
Theorem | isdrng 18574 | The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 }))) | ||
Theorem | drngunit 18575 | Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))) | ||
Theorem | drngui 18576 | The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝑅 ∈ DivRing ⇒ ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅) | ||
Theorem | drngring 18577 | A division ring is a ring. (Contributed by NM, 8-Sep-2011.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | ||
Theorem | drnggrp 18578 | A division ring is a group. (Contributed by NM, 8-Sep-2011.) |
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp) | ||
Theorem | isfld 18579 | A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.) |
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing)) | ||
Theorem | isdrng2 18580 | A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp)) | ||
Theorem | drngprop 18581 | If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.) |
⊢ (Base‘𝐾) = (Base‘𝐿) & ⊢ (+g‘𝐾) = (+g‘𝐿) & ⊢ (.r‘𝐾) = (.r‘𝐿) ⇒ ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing) | ||
Theorem | drngmgp 18582 | A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing → 𝐺 ∈ Grp) | ||
Theorem | drngmcl 18583 | The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 })) | ||
Theorem | drngid 18584 | A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 })) ⇒ ⊢ (𝑅 ∈ DivRing → 1 = (0g‘𝐺)) | ||
Theorem | drngunz 18585 | A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.) |
⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 ) | ||
Theorem | drngid2 18586 | Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼)) | ||
Theorem | drnginvrcl 18587 | Closure of the multiplicative inverse in a division ring. (reccl 10571 analog.) (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵) | ||
Theorem | drnginvrn0 18588 | The multiplicative inverse in a division ring is nonzero. (recne0 10577 analog.) (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ≠ 0 ) | ||
Theorem | drnginvrl 18589 | Property of the multiplicative inverse in a division ring. (recid2 10579 analog.) (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ((𝐼‘𝑋) · 𝑋) = 1 ) | ||
Theorem | drnginvrr 18590 | Property of the multiplicative inverse in a division ring. (recid 10578 analog.) (Contributed by NM, 19-Apr-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑋 · (𝐼‘𝑋)) = 1 ) | ||
Theorem | drngmul0or 18591 | A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) | ||
Theorem | drngmulne0 18592 | A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) | ||
Theorem | drngmuleq0 18593 | An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 )) | ||
Theorem | opprdrng 18594 | The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.) |
⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing) | ||
Theorem | isdrngd 18595* | Properties that determine a division ring. 𝐼 (reciprocal) is normally dependent on 𝑥 i.e. read it as 𝐼(𝑥)." (Contributed by NM, 2-Aug-2013.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝐼 · 𝑥) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | isdrngrd 18596* | Properties that determine a division ring. 𝐼 (reciprocal) is normally dependent on 𝑥 i.e. read it as 𝐼(𝑥)." This version of isdrngd 18595 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 0 = (0g‘𝑅)) & ⊢ (𝜑 → 1 = (1r‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝑥 · 𝑦) ≠ 0 ) & ⊢ (𝜑 → 1 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ∈ 𝐵) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → 𝐼 ≠ 0 ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 )) → (𝑥 · 𝐼) = 1 ) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
Theorem | drngpropd 18597* | If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-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 ∈ DivRing ↔ 𝐿 ∈ DivRing)) | ||
Theorem | fldpropd 18598* | If two structures have the same group components (properties), one is a field 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 ∈ Field ↔ 𝐿 ∈ Field)) | ||
Syntax | csubrg 18599 | Extend class notation with all subrings of a ring. |
class SubRing | ||
Syntax | crgspn 18600 | Extend class notation with span of a set of elements over a ring. |
class RingSpan |
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