HomeTheorem List (p. 336 of 505)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 336 of 505) | < Previous Next > |
| Bad symbols? Try the
GIF version. |
||
|
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
||
| Color key: | Metamath Proof Explorer
(1-31179) |
Hilbert Space Explorer
(31180-32702) |
Users' Mathboxes
(32703-50434) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | rlocbas 33501 | The base set of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ 𝑊 = (𝐵 × 𝑆) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿)) | ||
| Theorem | rlocaddval 33502 | Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐻 ∈ 𝑆) & ⊢ ⊕ = (+g‘𝐿) ⇒ ⊢ (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] ∼ ) | ||
| Theorem | rlocmulval 33503 | Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝑆) & ⊢ (𝜑 → 𝐻 ∈ 𝑆) & ⊢ ⊗ = (.r‘𝐿) ⇒ ⊢ (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊗ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨(𝐸 · 𝐹), (𝐺 · 𝐻)⟩] ∼ ) | ||
| Theorem | rloccring 33504 | The ring localization 𝐿 of a commutative ring 𝑅 by a multiplicatively closed set 𝑆 is itself a commutative ring. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) ⇒ ⊢ (𝜑 → 𝐿 ∈ CRing) | ||
| Theorem | rloc0g 33505 | The zero of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ 𝑂 = [⟨ 0 , 1 ⟩] ∼ ⇒ ⊢ (𝜑 → 𝑂 = (0g‘𝐿)) | ||
| Theorem | rloc1r 33506 | The multiplicative identity of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ 𝐼 = [⟨ 1 , 1 ⟩] ∼ ⇒ ⊢ (𝜑 → 𝐼 = (1r‘𝐿)) | ||
| Theorem | rlocf1 33507* | The embedding 𝐹 of a ring 𝑅 into its localization 𝐿. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝑆 ⊆ (RLReg‘𝑅)) ⇒ ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿))) | ||
| Theorem | rlocinvunit 33508 | In the localization of a ring 𝑅 at 𝑆, inverses of elements of 𝑆 are units. (Contributed by Thierry Arnoux, 6-Jun-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ 𝑊 = (Unit‘𝐿) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ (𝜑 → 𝑄 ∈ 𝑆) ⇒ ⊢ (𝜑 → [⟨ 1 , 𝑄⟩] ∼ ∈ 𝑊) | ||
| Theorem | rlocisunit 33509* | Characterize the units of the localization 𝐿 of a ring 𝑅 at 𝑆 as the elements with a "numerator" 𝑃 in the saturation 𝑇 of 𝑆. (Contributed by Thierry Arnoux, 6-Jun-2026.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐿 = (𝑅 RLocal 𝑆) & ⊢ 𝑊 = (Unit‘𝐿) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅))) & ⊢ ∼ = (𝑅 ~RL 𝑆) & ⊢ (𝜑 → 𝑃 ∈ 𝐵) & ⊢ (𝜑 → 𝑄 ∈ 𝑆) & ⊢ 𝑇 = {𝑟 ∈ 𝐵 ∣ ∃𝑠 ∈ 𝐵 (𝑟 · 𝑠) ∈ 𝑆} ⇒ ⊢ (𝜑 → ([⟨𝑃, 𝑄⟩] ∼ ∈ 𝑊 ↔ 𝑃 ∈ 𝑇)) | ||
| Theorem | domnmuln0rd 33510 | In a domain, factors of a nonzero product are nonzero. (Contributed by Thierry Arnoux, 8-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑋 · 𝑌) ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) | ||
| Theorem | domnprodn0 33511 | In a domain, a finite product of nonzero terms is nonzero. (Contributed by Thierry Arnoux, 6-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝐹 ∈ Word (𝐵 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑀 Σg 𝐹) ≠ 0 ) | ||
| Theorem | domnprodeq0 33512 | A product over a domain is zero exactly when one of the factors is zero. Generalization of domneq0 20784 for any number of factors. See also domnprodn0 33511. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → ((𝑀 Σg 𝐹) = 0 ↔ 0 ∈ ran 𝐹)) | ||
| Theorem | domnpropd 33513* | If two structures have the same components (properties), one is a domain iff the other one is. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ (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 ∈ Domn ↔ 𝐿 ∈ Domn)) | ||
| Theorem | idompropd 33514* | If two structures have the same components (properties), one is a integral domain iff the other one is. See also domnpropd 33513. (Contributed by Thierry Arnoux, 13-Oct-2025.) |
| ⊢ (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 ∈ IDomn ↔ 𝐿 ∈ IDomn)) | ||
| Theorem | idomrcan 33515 | Right-cancellation law for integral domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 })) & ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍)) ⇒ ⊢ (𝜑 → 𝑋 = 𝑌) | ||
| Theorem | 1rrg 33516 | The multiplicative identity is a left-regular element. (Contributed by Thierry Arnoux, 6-May-2025.) |
| ⊢ 1 = (1r‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 1 ∈ 𝐸) | ||
| Theorem | rrgsubm 33517 | The left regular elements of a ring form a submonoid of the multiplicative group. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝐸 ∈ (SubMnd‘𝑀)) | ||
| Theorem | subrdom 33518 | A subring of a domain is a domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ Domn) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ Domn) | ||
| Theorem | subridom 33519 | A subring of an integral domain is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn) | ||
| Theorem | subrfld 33520 | A subring of a field is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn) | ||
| Theorem | ricnzr1 33521 | A ring isomorphism maps a nonzero ring to a nonzero ring. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ NzRing) → 𝑆 ∈ NzRing) | ||
| Theorem | ricdomn1 33522 | A ring isomorphism maps a domain to a domain. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ Domn) → 𝑆 ∈ Domn) | ||
| Theorem | ricdomn 33523 | A ring is a domain if and only if an isomorphic ring is a domain. (Contributed by Thierry Arnoux, 4-May-2026.) |
| ⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ Domn ↔ 𝑆 ∈ Domn)) | ||
| Syntax | ceuf 33524 | Declare the syntax for the Euclidean function index extractor. |
| class EuclF | ||
| Definition | df-euf 33525 | Define the Euclidean function. (Contributed by Thierry Arnoux, 22-Mar-2025.) Use its index-independent form eufid 33527 instead. (New usage is discouraged.) |
| ⊢ EuclF = Slot ;21 | ||
| Theorem | eufndx 33526 | Index value of the Euclidean function slot. Use ndxarg 17246. (Contributed by Thierry Arnoux, 22-Mar-2025.) (New usage is discouraged.) |
| ⊢ (EuclF‘ndx) = ;21 | ||
| Theorem | eufid 33527 | Utility theorem: index-independent form of df-euf 33525. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ EuclF = Slot (EuclF‘ndx) | ||
| Syntax | cedom 33528 | Declare the syntax for the Euclidean Domain. |
| class EDomn | ||
| Definition | df-edom 33529* | Define Euclidean Domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) |
| ⊢ EDomn = {𝑑 ∈ IDomn ∣ [(EuclF‘𝑑) / 𝑒][(Base‘𝑑) / 𝑣](Fun 𝑒 ∧ (𝑒 “ (𝑣 ∖ {(0g‘𝑑)})) ⊆ (0[,)+∞) ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ (𝑣 ∖ {(0g‘𝑑)})∃𝑞 ∈ 𝑣 ∃𝑟 ∈ 𝑣 (𝑎 = ((𝑏(.r‘𝑑)𝑞)(+g‘𝑑)𝑟) ∧ (𝑟 = (0g‘𝑑) ∨ (𝑒‘𝑟) < (𝑒‘𝑏))))} | ||
| Theorem | ringinveu 33530 | If a ring unit element 𝑋 admits both a left inverse 𝑌 and a right inverse 𝑍, they are equal. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → (𝑌 · 𝑋) = 1 ) & ⊢ (𝜑 → (𝑋 · 𝑍) = 1 ) ⇒ ⊢ (𝜑 → 𝑍 = 𝑌) | ||
| Theorem | isdrng4 33531* | A division ring is a ring in which 1 ≠ 0 and every nonzero element has a left and right inverse. (Contributed by Thierry Arnoux, 2-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑅 ∈ DivRing ↔ ( 1 ≠ 0 ∧ ∀𝑥 ∈ (𝐵 ∖ { 0 })∃𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 1 ∧ (𝑦 · 𝑥) = 1 )))) | ||
| Theorem | rndrhmcl 33532 | The image of a division ring by a ring homomorphism is a division ring. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 𝑅 = (𝑁 ↾s ran 𝐹) & ⊢ 0 = (0g‘𝑁) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁)) & ⊢ (𝜑 → ran 𝐹 ≠ { 0 }) & ⊢ (𝜑 → 𝑀 ∈ DivRing) ⇒ ⊢ (𝜑 → 𝑅 ∈ DivRing) | ||
| Theorem | qfld 33533 | The field of rational numbers is a field. (Contributed by Thierry Arnoux, 19-Oct-2025.) |
| ⊢ 𝑄 = (ℂfld ↾s ℚ) ⇒ ⊢ 𝑄 ∈ Field | ||
| Theorem | subsdrg 33534 | A subring of a sub-division-ring is a sub-division-ring. See also subsubrg 20674. (Contributed by Thierry Arnoux, 26-Oct-2025.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅)) ⇒ ⊢ (𝜑 → (𝐵 ∈ (SubDRing‘𝑆) ↔ (𝐵 ∈ (SubDRing‘𝑅) ∧ 𝐵 ⊆ 𝐴))) | ||
| Theorem | sdrgdvcl 33535 | A sub-division-ring is closed under the ring division operation. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ / = (/r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐴 ∈ (SubDRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 / 𝑌) ∈ 𝐴) | ||
| Theorem | sdrginvcl 33536 | A sub-division-ring is closed under the ring inverse operation. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐼 = (invr‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝐴 ∈ (SubDRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐴) | ||
| Theorem | primefldchr 33537 | The characteristic of a prime field is the same as the characteristic of the main field. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
| ⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅)) ⇒ ⊢ (𝑅 ∈ DivRing → (chr‘𝑃) = (chr‘𝑅)) | ||
| Syntax | cfrac 33538 | Syntax for the field of fractions of a given integral domain. |
| class Frac | ||
| Definition | df-frac 33539 | Define the field of fractions of a given integral domain. (Contributed by Thierry Arnoux, 26-Apr-2025.) |
| ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟))) | ||
| Theorem | fracval 33540 | Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.) |
| ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)) | ||
| Theorem | fracbas 33541 | The base of the field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 𝐹 = ( Frac ‘𝑅) & ⊢ ∼ = (𝑅 ~RL 𝐸) ⇒ ⊢ ((𝐵 × 𝐸) / ∼ ) = (Base‘𝐹) | ||
| Theorem | fracerl 33542 | Rewrite the ring localization equivalence relation in the case of a field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ∼ = (𝑅 ~RL (RLReg‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (RLReg‘𝑅)) & ⊢ (𝜑 → 𝐻 ∈ (RLReg‘𝑅)) ⇒ ⊢ (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹))) | ||
| Theorem | fracf1 33543* | The embedding of a commutative ring 𝑅 into its field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐸 = (RLReg‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ ∼ = (𝑅 ~RL 𝐸) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ ) ⇒ ⊢ (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝐸) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom ( Frac ‘𝑅)))) | ||
| Theorem | fracfld 33544 | The field of fractions of an integral domain is a field. (Contributed by Thierry Arnoux, 4-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → ( Frac ‘𝑅) ∈ Field) | ||
| Theorem | idomsubr 33545* | Every integral domain is isomorphic with a subring of some field. (Proposed by Gerard Lang, 10-May-2025.) (Contributed by Thierry Arnoux, 10-May-2025.) |
| ⊢ (𝜑 → 𝑅 ∈ IDomn) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠)) | ||
| Syntax | cfldgen 33546 | Syntax for a function generating sub-fields. |
| class fldGen | ||
| Definition | df-fldgen 33547* | Define a function generating the smallest sub-division-ring of a given ring containing a given set. If the base structure is a division ring, then this is also a division ring (see fldgensdrg 33550). If the base structure is a field, this is a subfield (see fldgenfld 33556 and fldsdrgfld 20870). In general this will be used in the context of fields, hence the name fldGen. (Contributed by Saveliy Skresanov and Thierry Arnoux, 9-Jan-2025.) |
| ⊢ fldGen = (𝑓 ∈ V, 𝑠 ∈ V ↦ ∩ {𝑎 ∈ (SubDRing‘𝑓) ∣ 𝑠 ⊆ 𝑎}) | ||
| Theorem | fldgenval 33548* | Value of the field generating function: (𝐹 fldGen 𝑆) is the smallest sub-division-ring of 𝐹 containing 𝑆. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) = ∩ {𝑎 ∈ (SubDRing‘𝐹) ∣ 𝑆 ⊆ 𝑎}) | ||
| Theorem | fldgenssid 33549 | The field generated by a set of elements contains those elements. See lspssid 21075. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 ⊆ (𝐹 fldGen 𝑆)) | ||
| Theorem | fldgensdrg 33550 | A generated subfield is a sub-division-ring. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) ∈ (SubDRing‘𝐹)) | ||
| Theorem | fldgenssv 33551 | A generated subfield is a subset of the field's base. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) ⊆ 𝐵) | ||
| Theorem | fldgenss 33552 | Generated subfields preserve subset ordering. ( see lspss 21074 and spanss 31609) (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ (𝐹 fldGen 𝑆)) | ||
| Theorem | fldgenidfld 33553 | The subfield generated by a subfield is the subfield itself. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹)) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑆) = 𝑆) | ||
| Theorem | fldgenssp 33554 | The field generated by a set of elements in a division ring is contained in any sub-division-ring which contains those elements. (Contributed by Thierry Arnoux, 25-Feb-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝐹)) & ⊢ (𝜑 → 𝑇 ⊆ 𝑆) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝑇) ⊆ 𝑆) | ||
| Theorem | fldgenid 33555 | The subfield of a field 𝐹 generated by the whole base set of 𝐹 is 𝐹 itself. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → (𝐹 fldGen 𝐵) = 𝐵) | ||
| Theorem | fldgenfld 33556 | A generated subfield is a field. (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 ∈ Field) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ↾s (𝐹 fldGen 𝑆)) ∈ Field) | ||
| Theorem | primefldgen1 33557 | The prime field of a division ring is the subfield generated by the multiplicative identity element. In general, we should write "prime division ring", but since most later usages are in the case where the ambient ring is commutative, we keep the term "prime field". (Contributed by Thierry Arnoux, 11-Jan-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) ⇒ ⊢ (𝜑 → ∩ (SubDRing‘𝑅) = (𝑅 fldGen { 1 })) | ||
| Theorem | 1fldgenq 33558 | The field of rational numbers ℚ is generated by 1 in ℂfld, that is, ℚ is the prime field of ℂfld. (Contributed by Thierry Arnoux, 15-Jan-2025.) |
| ⊢ (ℂfld fldGen {1}) = ℚ | ||
| Theorem | rhmdvd 33559 | A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
| ⊢ 𝑈 = (Unit‘𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ / = (/r‘𝑆) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → ((𝐹‘𝐴) / (𝐹‘𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶)))) | ||
| Theorem | kerunit 33560 | If a unit element lies in the kernel of a ring homomorphism, then 0 = 1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑆) & ⊢ 1 = (1r‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑈 ∩ (◡𝐹 “ { 0 })) ≠ ∅) → 1 = 0 ) | ||
| Syntax | cresv 33561 | Extend class notation with the scalar restriction operation. |
| class ↾v | ||
| Definition | df-resv 33562* | Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.) |
| ⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩))) | ||
| Theorem | reldmresv 33563 | The scalar restriction is a proper operator, so it can be used with ovprc1 7439. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ Rel dom ↾v | ||
| Theorem | resvval 33564 | Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩))) | ||
| Theorem | resvid2 33565 | General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) | ||
| Theorem | resvval2 33566 | Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹 ↾s 𝐴)⟩)) | ||
| Theorem | resvsca 33567 | Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐹 ↾s 𝐴) = (Scalar‘𝑅)) | ||
| Theorem | resvlem 33568 | Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐶 = (𝐸‘𝑊) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (Scalar‘ndx) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) | ||
| Theorem | resvbas 33569 | Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐵 = (Base‘𝐻)) | ||
| Theorem | resvplusg 33570 | +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) | ||
| Theorem | resvvsca 33571 | ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Proof shortened by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻)) | ||
| Theorem | resvmulr 33572 | .r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ · = (.r‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝐻)) | ||
| Theorem | resv0g 33573 | 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 0 = (0g‘𝐻)) | ||
| Theorem | resv1r 33574 | 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 1 = (1r‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻)) | ||
| Theorem | resvcmn 33575 | Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐻 = (𝐺 ↾v 𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd)) | ||
| Theorem | gzcrng 33576 | The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.) |
| ⊢ (ℂfld ↾s ℤ[i]) ∈ CRing | ||
| Theorem | cnfldfld 33577 | The complex numbers form a field. (Contributed by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ ℂfld ∈ Field | ||
| Theorem | reofld 33578 | The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
| ⊢ ℝfld ∈ oField | ||
| Theorem | nn0omnd 33579 | The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
| ⊢ (ℂfld ↾s ℕ0) ∈ oMnd | ||
| Theorem | gsumind 33580 | The group sum of an indicator function of the set 𝐴 gives the size of 𝐴. (Contributed by Thierry Arnoux, 18-Jan-2026.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ 𝑂) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → (ℂfld Σg ((𝟭‘𝑂)‘𝐴)) = (♯‘𝐴)) | ||
| Theorem | rearchi 33581 | The field of the real numbers is Archimedean. See also arch 12492. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
| ⊢ ℝfld ∈ Archi | ||
| Theorem | nn0archi 33582 | The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
| ⊢ (ℂfld ↾s ℕ0) ∈ Archi | ||
| Theorem | xrge0slmod 33583 | The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) & ⊢ 𝑊 = (𝐺 ↾v (0[,)+∞)) ⇒ ⊢ 𝑊 ∈ SLMod | ||
| Theorem | qusker 33584* | The kernel of a quotient map. (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑉 = (Base‘𝑀) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) & ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ 0 = (0g‘𝑁) ⇒ ⊢ (𝐺 ∈ (NrmSGrp‘𝑀) → (◡𝐹 “ { 0 }) = 𝐺) | ||
| Theorem | eqgvscpbl 33585 | The left coset equivalence relation is compatible with the scalar multiplication operation. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ∼ = (𝑀 ~QG 𝐺) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 ∼ 𝑌 → (𝐾 · 𝑋) ∼ (𝐾 · 𝑌))) | ||
| Theorem | qusvscpbl 33586* | The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ∼ = (𝑀 ~QG 𝐺) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ 𝑆) & ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ ∙ = ( ·𝑠 ‘𝑁) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝑀 ~QG 𝐺)) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) & ⊢ (𝜑 → 𝑉 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐹‘𝑈) = (𝐹‘𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉)))) | ||
| Theorem | qusvsval 33587 | Value of the scalar multiplication operation on the quotient structure. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ ∼ = (𝑀 ~QG 𝐺) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ (𝜑 → 𝐾 ∈ 𝑆) & ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ ∙ = ( ·𝑠 ‘𝑁) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐾 ∙ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺)) | ||
| Theorem | imaslmod 33588* | The image structure of a left module is a left module. (Contributed by Thierry Arnoux, 15-May-2023.) |
| ⊢ (𝜑 → 𝑁 = (𝐹 “s 𝑀)) & ⊢ 𝑉 = (Base‘𝑀) & ⊢ 𝑆 = (Base‘(Scalar‘𝑀)) & ⊢ + = (+g‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ 0 = (0g‘𝑀) & ⊢ (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 ∧ (𝑘 ∈ 𝑆 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘 · 𝑎)) = (𝐹‘(𝑘 · 𝑏)))) & ⊢ (𝜑 → 𝑀 ∈ LMod) ⇒ ⊢ (𝜑 → 𝑁 ∈ LMod) | ||
| Theorem | imasmhm 33589* | Given a function 𝐹 with homomorphic properties, build the image of a monoid. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑊 ∈ Mnd) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Mnd ∧ 𝐹 ∈ (𝑊 MndHom (𝐹 “s 𝑊)))) | ||
| Theorem | imasghm 33590* | Given a function 𝐹 with homomorphic properties, build the image of a group. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ (𝜑 → 𝑊 ∈ Grp) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Grp ∧ 𝐹 ∈ (𝑊 GrpHom (𝐹 “s 𝑊)))) | ||
| Theorem | imasrhm 33591* | Given a function 𝐹 with homomorphic properties, build the image of a ring. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ · = (.r‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑊 ∈ Ring) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Ring ∧ 𝐹 ∈ (𝑊 RingHom (𝐹 “s 𝑊)))) | ||
| Theorem | imaslmhm 33592* | Given a function 𝐹 with homomorphic properties, build the image of a left module. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) & ⊢ + = (+g‘𝑊) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ 𝐷 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐷) & ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐾 ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝐹‘𝑎) = (𝐹‘𝑏) → (𝐹‘(𝑘 × 𝑎)) = (𝐹‘(𝑘 × 𝑏)))) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ × = ( ·𝑠 ‘𝑊) ⇒ ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ LMod ∧ 𝐹 ∈ (𝑊 LMHom (𝐹 “s 𝑊)))) | ||
| Theorem | quslmod 33593 | If 𝐺 is a submodule in 𝑀, then 𝑁 = 𝑀 / 𝐺 is a left module, called the quotient module of 𝑀 by 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ 𝑉 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) ⇒ ⊢ (𝜑 → 𝑁 ∈ LMod) | ||
| Theorem | quslmhm 33594* | If 𝐺 is a submodule of 𝑀, then the "natural map" from elements to their cosets is a left module homomorphism from 𝑀 to 𝑀 / 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑁 = (𝑀 /s (𝑀 ~QG 𝐺)) & ⊢ 𝑉 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝑀)) & ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥](𝑀 ~QG 𝐺)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑀 LMHom 𝑁)) | ||
| Theorem | quslvec 33595 | If 𝑆 is a vector subspace in 𝑊, then 𝑄 = 𝑊 / 𝑆 is a vector space, called the quotient space of 𝑊 by 𝑆. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑄 = (𝑊 /s (𝑊 ~QG 𝑆)) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑆 ∈ (LSubSp‘𝑊)) ⇒ ⊢ (𝜑 → 𝑄 ∈ LVec) | ||
| Theorem | znfermltl 33596 | Fermat's little theorem in ℤ/nℤ. (Contributed by Thierry Arnoux, 24-Jul-2024.) |
| ⊢ 𝑍 = (ℤ/nℤ‘𝑃) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ ↑ = (.g‘(mulGrp‘𝑍)) ⇒ ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ 𝐵) → (𝑃 ↑ 𝐴) = 𝐴) | ||
| Theorem | islinds5 33597* | A set is linearly independent if and only if it has no non-trivial representations of zero. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝑊) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑉 ⊆ 𝐵) → (𝑉 ∈ (LIndS‘𝑊) ↔ ∀𝑎 ∈ (𝐾 ↑m 𝑉)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣))) = 𝑂) → 𝑎 = (𝑉 × { 0 })))) | ||
| Theorem | ellspds 33598* | Variation on ellspd 21912. (Contributed by Thierry Arnoux, 18-May-2023.) |
| ⊢ 𝑁 = (LSpan‘𝑀) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝑉 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘𝑉) ↔ ∃𝑎 ∈ (𝐾 ↑m 𝑉)(𝑎 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝑎‘𝑣) · 𝑣)))))) | ||
| Theorem | 0ellsp 33599 | Zero is in all spans. (Contributed by Thierry Arnoux, 8-May-2023.) |
| ⊢ 0 = (0g‘𝑊) & ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ⊆ 𝐵) → 0 ∈ (𝑁‘𝑆)) | ||
| Theorem | 0nellinds 33600 | The group identity cannot be an element of an independent set. (Contributed by Thierry Arnoux, 8-May-2023.) |
| ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) → ¬ 0 ∈ 𝐹) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |