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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dvdsrcl 19401 | Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ (𝑋 ∥ 𝑌 → 𝑋 ∈ 𝐵) | ||
Theorem | dvdsrcl2 19402 | Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∥ 𝑌) → 𝑌 ∈ 𝐵) | ||
Theorem | dvdsrid 19403 | An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ 𝑋) | ||
Theorem | dvdsrtr 19404 | Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∥ 𝑍 ∧ 𝑍 ∥ 𝑋) → 𝑌 ∥ 𝑋) | ||
Theorem | dvdsrmul1 19405 | The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∥ 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍)) | ||
Theorem | dvdsrneg 19406 | An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ (𝑁‘𝑋)) | ||
Theorem | dvdsr01 19407 | In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 20058.) (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ 0 ) | ||
Theorem | dvdsr02 19408 | Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 )) | ||
Theorem | isunit 19409 | Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑆 = (oppr‘𝑅) & ⊢ 𝐸 = (∥r‘𝑆) ⇒ ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 )) | ||
Theorem | 1unit 19410 | The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 1 ∈ 𝑈) | ||
Theorem | unitcl 19411 | A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ 𝐵) | ||
Theorem | unitss 19412 | The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ 𝑈 ⊆ 𝐵 | ||
Theorem | opprunit 19413 | Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑆 = (oppr‘𝑅) ⇒ ⊢ 𝑈 = (Unit‘𝑆) | ||
Theorem | crngunit 19414 | Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∥ 1 )) | ||
Theorem | dvdsunit 19415 | A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ∧ 𝑋 ∈ 𝑈) → 𝑌 ∈ 𝑈) | ||
Theorem | unitmulcl 19416 | The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ 𝑈) | ||
Theorem | unitmulclb 19417 | Reversal of unitmulcl 19416 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) ∈ 𝑈 ↔ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈))) | ||
Theorem | unitgrpbas 19418 | The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ 𝑈 = (Base‘𝐺) | ||
Theorem | unitgrp 19419 | The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) | ||
Theorem | unitabl 19420 | The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel) | ||
Theorem | unitgrpid 19421 | The identity of the multiplicative group is 1r. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 1 = (0g‘𝐺)) | ||
Theorem | unitsubm 19422 | The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀)) | ||
Syntax | cinvr 19423 | Extend class notation with multiplicative inverse. |
class invr | ||
Definition | df-invr 19424 | Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.) |
⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | ||
Theorem | invrfval 19425 | Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ 𝐼 = (invg‘𝐺) | ||
Theorem | unitinvcl 19426 | The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝑈) | ||
Theorem | unitinvinv 19427 | The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘(𝐼‘𝑋)) = 𝑋) | ||
Theorem | ringinvcl 19428 | The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝐵) | ||
Theorem | unitlinv 19429 | 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) → ((𝐼‘𝑋) · 𝑋) = 1 ) | ||
Theorem | unitrinv 19430 | 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 19431 | 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 19432 | 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 19433 | 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 19434 | Extend class notation with ring division. |
class /r | ||
Definition | df-dvr 19435* | 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 19436* | Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ / = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝑈 ↦ (𝑥 · (𝐼‘𝑦))) | ||
Theorem | dvrval 19437 | 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 19438 | 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 19439 | 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 19440 | A cancellation law for division. (divid 11329 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 19441 | A cancellation law for division. (div1 11331 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 19442 | An associative law for division. (divass 11318 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 19443 | A cancellation law for division. (divcan1 11309 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 19444 | A cancellation law for division. (divcan3 11326 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 19445 | A cancellation law for division. (diveq1 11333 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 19446 | 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 19447* | 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 19448* | 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 19449* | 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 19450* | 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 19451* | 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 19452* | 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 19453* | Expand out the class difference from isirred 19451. (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 19454 | 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 19455 | 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 19456 | 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 19457 | 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 19458 | 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 19459 | 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 19460 | 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 19461 | 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 19462 | 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 19463 | 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 | crpm 19464 | Syntax for the ring primes function. |
class RPrime | ||
Definition | df-rprm 19465* | Define the function associating with a ring its set of prime elements. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 16059. Prime elements are closely related to irreducible elements (see df-irred 19395). (Contributed by Mario Carneiro, 17-Feb-2015.) |
⊢ RPrime = (𝑤 ∈ V ↦ ⦋(Base‘𝑤) / 𝑏⦌{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g‘𝑤)})) ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 [(∥r‘𝑤) / 𝑑](𝑝𝑑(𝑥(.r‘𝑤)𝑦) → (𝑝𝑑𝑥 ∨ 𝑝𝑑𝑦))}) | ||
Syntax | crh 19466 | Extend class notation with the ring homomorphisms. |
class RingHom | ||
Syntax | crs 19467 | Extend class notation with the ring isomorphisms. |
class RingIso | ||
Syntax | cric 19468 | Extend class notation with the ring isomorphism relation. |
class ≃𝑟 | ||
Definition | df-rnghom 19469* | 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 19470* | 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 19471* | 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 19472 | Define the ring isomorphism relation, analogous to df-gic 18402: 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 19473 | Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring) | ||
Theorem | rhmrcl2 19474 | Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring) | ||
Theorem | isrhm 19475 | 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 19476 | 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 19477 | 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 19478 | Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.) |
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V)) | ||
Theorem | rhmghm 19479 | 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 19480 | 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 19481 | 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 19482* | 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 19483* | 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 19484 | 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 19485 | The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅)) | ||
Theorem | rhmf1o 19486 | 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 19487 | 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 19488 | 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 19489 | 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 19490 | 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 19491 | 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 19492* | 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 19493* | 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 | f1ghm0to0 19494 | If a group homomorphism 𝐹 is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑆) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | ||
Theorem | f1rhm0to0OLD 19495 | Obsolete version of f1ghm0to0 19494 as of 13-May-2023. (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 | f1rhm0to0ALT 19496 | Alternate proof for f1ghm0to0 19494. Using ghmf1 18389 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 | gim0to0 19497 | A group isomorphism maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 23-May-2023.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑆) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | ||
Theorem | rim0to0OLD 19498 | Obsolete version of gim0to0 19497 as of 13-May-2023. (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 RingIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | ||
Theorem | kerf1ghm 19499 | A group 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.) (Revised by Thierry Arnoux, 13-May-2023.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁})) | ||
Theorem | kerf1hrmOLD 19500 | Obsolete version of kerf1ghm 19499 as of 13-May-2023. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐴 = (Base‘𝑅) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝑁 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴–1-1→𝐵 ↔ (◡𝐹 “ { 0 }) = {𝑁})) |
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