HomeTheorem List (p. 333 of 503)
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. 333 of 503) | < 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
(1-30989) |
Hilbert Space Explorer
(30990-32512) |
Users' Mathboxes
(32513-50280) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cyc3co2 33201 | Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) & ⊢ · = (+g‘𝑆) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩))) | ||
| Theorem | cycpmconjvlem 33202 | Lemma for cycpmconjv 33203. (Contributed by Thierry Arnoux, 9-Oct-2023.) |
| ⊢ (𝜑 → 𝐹:𝐷–1-1-onto→𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐷) ⇒ ⊢ (𝜑 → ((𝐹 ↾ (𝐷 ∖ 𝐵)) ∘ ◡𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹 ↾ 𝐵)))) | ||
| Theorem | cycpmconjv 33203 | A formula for computing conjugacy classes of cyclic permutations. Formula in property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 9-Oct-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐵 ∧ 𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀‘𝑊)) − 𝐺) = (𝑀‘(𝐺 ∘ 𝑊))) | ||
| Theorem | cycpmrn 33204 | The range of the word used to build a cycle is the cycle's orbit, i.e., the set of points it moves. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → 1 < (♯‘𝑊)) ⇒ ⊢ (𝜑 → ran 𝑊 = dom ((𝑀‘𝑊) ∖ I )) | ||
| Theorem | tocyccntz 33205* | All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑍 = (Cntz‘𝑆) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ran 𝑥) & ⊢ (𝜑 → 𝐴 ⊆ dom 𝑀) ⇒ ⊢ (𝜑 → (𝑀 “ 𝐴) ⊆ (𝑍‘(𝑀 “ 𝐴))) | ||
| Theorem | evpmval 33206 | Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
| ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ 𝑉 → 𝐴 = (◡(pmSgn‘𝐷) “ {1})) | ||
| Theorem | cnmsgn0g 33207 | The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
| ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) ⇒ ⊢ 1 = (0g‘𝑈) | ||
| Theorem | evpmsubg 33208 | The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆)) | ||
| Theorem | evpmid 33209 | The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷)) | ||
| Theorem | altgnsg 33210 | The alternating group (pmEven‘𝐷) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆)) | ||
| Theorem | cyc3evpm 33211 | 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = ((toCyc‘𝐷) “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴) | ||
| Theorem | cyc3genpmlem 33212* | Lemma for cyc3genpm 33213. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ · = (+g‘𝑆) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐾 ∈ 𝐷) & ⊢ (𝜑 → 𝐿 ∈ 𝐷) & ⊢ (𝜑 → 𝐸 = (𝑀‘⟨“𝐼𝐽”⟩)) & ⊢ (𝜑 → 𝐹 = (𝑀‘⟨“𝐾𝐿”⟩)) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ (𝜑 → 𝐾 ≠ 𝐿) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐)) | ||
| Theorem | cyc3genpm 33213* | The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))) | ||
| Theorem | cycpmgcl 33214 | Cyclic permutations are permutations, similar to cycpmcl 33177, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ (0...𝑁)) → 𝐶 ⊆ 𝐵) | ||
| Theorem | cycpmconjslem1 33215 | Lemma for cycpmconjs 33217. (Contributed by Thierry Arnoux, 14-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) & ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) & ⊢ (𝜑 → (♯‘𝑊) = 𝑃) ⇒ ⊢ (𝜑 → ((◡𝑊 ∘ (𝑀‘𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1)) | ||
| Theorem | cycpmconjslem2 33216* | Lemma for cycpmconjs 33217. (Contributed by Thierry Arnoux, 14-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 𝑃 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto→𝐷 ∧ ((◡𝑞 ∘ 𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁))))) | ||
| Theorem | cycpmconjs 33217* | All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {𝑃})) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 𝑃 ∈ (0...𝑁)) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐵 𝑄 = ((𝑝 + 𝑇) − 𝑝)) | ||
| Theorem | cyc3conja 33218* | All 3-cycles are conjugate in the alternating group An for n>= 5. Property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 15-Oct-2023.) |
| ⊢ 𝐶 = (𝑀 “ (◡♯ “ {3})) & ⊢ 𝐴 = (pmEven‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ 𝑁 = (♯‘𝐷) & ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ + = (+g‘𝑆) & ⊢ − = (-g‘𝑆) & ⊢ (𝜑 → 5 ≤ 𝑁) & ⊢ (𝜑 → 𝐷 ∈ Fin) & ⊢ (𝜑 → 𝑄 ∈ 𝐶) & ⊢ (𝜑 → 𝑇 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 𝑄 = ((𝑝 + 𝑇) − 𝑝)) | ||
| Syntax | csgns 33219 | Extend class notation to include the Signum function. |
| class sgns | ||
| Definition | df-sgns 33220* | Signum function for a structure. See also df-sgn 15049 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.) |
| ⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1)))) | ||
| Theorem | sgnsv 33221* | The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ 𝑆 = (sgns‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1)))) | ||
| Theorem | sgnsval 33222 | The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ 𝑆 = (sgns‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑆‘𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1))) | ||
| Theorem | sgnsf 33223 | The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ 𝑆 = (sgns‘𝑅) ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1}) | ||
| Syntax | cfxp 33224 | Extend class notation with the fixed points operation. |
| class FixPts | ||
| Definition | df-fxp 33225* | Define the set of fixed points left unchanged by a group action. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ FixPts = (𝑏 ∈ V, 𝑎 ∈ V ↦ {𝑥 ∈ 𝑏 ∣ ∀𝑝 ∈ dom dom 𝑎(𝑝𝑎𝑥) = 𝑥}) | ||
| Theorem | fxpval 33226* | Value of the set of fixed points. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐵FixPts𝐴) = {𝑥 ∈ 𝐵 ∣ ∀𝑝 ∈ dom dom 𝐴(𝑝𝐴𝑥) = 𝑥}) | ||
| Theorem | fxpss 33227 | The set of fixed points is a subset of the set acted upon. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐵FixPts𝐴) ⊆ 𝐵) | ||
| Theorem | fxpgaval 33228* | Value of the set of fixed points for a group action 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝑈 = (Base‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶)) ⇒ ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥}) | ||
| Theorem | isfxp 33229* | Property of being a fixed point. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝑈 = (Base‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶)) & ⊢ (𝜑 → 𝑋 ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝐶FixPts𝐴) ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋)) | ||
| Theorem | fxpgaeq 33230 | A fixed point 𝑋 is invariant under group action 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝑈 = (Base‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶)) & ⊢ (𝜑 → 𝑋 ∈ (𝐶FixPts𝐴)) & ⊢ (𝜑 → 𝑃 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋) | ||
| Theorem | conjga 33231* | Group conjugation induces a group action. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ − = (-g‘𝑀) & ⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥)) ⇒ ⊢ (𝑀 ∈ Grp → ⊕ ∈ (𝑀 GrpAct 𝐵)) | ||
| Theorem | cntrval2 33232* | Express the center 𝑍 of a group 𝑀 as the set of fixed points of the conjugation operation ⊕. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ − = (-g‘𝑀) & ⊢ ⊕ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥 + 𝑦) − 𝑥)) & ⊢ 𝑍 = (Cntr‘𝑀) ⇒ ⊢ (𝑀 ∈ Grp → 𝑍 = (𝐵FixPts ⊕ )) | ||
| Theorem | fxpsubm 33233* | Provided the group action 𝐴 induces monoid automorphisms, the set of fixed points of 𝐴 on a monoid 𝑊 is a submonoid, which could be called the fixed submonoid under 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = (Base‘𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥)) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 MndHom 𝑊)) ⇒ ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubMnd‘𝑊)) | ||
| Theorem | fxpsubg 33234* | The fixed points of a group action 𝐴 on a group 𝑊 is a subgroup. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = (Base‘𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥)) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 GrpHom 𝑊)) ⇒ ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubGrp‘𝑊)) | ||
| Theorem | fxpsubrg 33235* | The fixed points of a group action 𝐴 on a ring 𝑊 is a subgring. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = (Base‘𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥)) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 RingHom 𝑊)) ⇒ ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubRing‘𝑊)) | ||
| Theorem | fxpsdrg 33236* | The fixed points of a group action 𝐴 on a division ring 𝑊 is a sub-division-ring. Since sub-division-rings of fields are subfields (see fldsdrgfld 20775), (𝐶FixPts𝐴) might be called the fixed subfield under 𝐴. (Contributed by Thierry Arnoux, 18-Nov-2025.) |
| ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝐶 = (Base‘𝑊) & ⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ (𝑝𝐴𝑥)) & ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶)) & ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝐹 ∈ (𝑊 RingHom 𝑊)) & ⊢ (𝜑 → 𝑊 ∈ DivRing) ⇒ ⊢ (𝜑 → (𝐶FixPts𝐴) ∈ (SubDRing‘𝑊)) | ||
| Syntax | cinftm 33237 | Class notation for the infinitesimal relation. |
| class ⋘ | ||
| Syntax | carchi 33238 | Class notation for the Archimedean property. |
| class Archi | ||
| Definition | df-inftm 33239* | Define the relation "𝑥 is infinitesimal with respect to 𝑦 " for a structure 𝑤. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ ⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))}) | ||
| Definition | df-archi 33240 | A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | ||
| Theorem | inftmrel 33241 | The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) | ||
| Theorem | isinftm 33242* | Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ < = (lt‘𝑊) ⇒ ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌))) | ||
| Theorem | isarchi 33243* | Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ < = (⋘‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦)) | ||
| Theorem | pnfinf 33244 | Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018.) |
| ⊢ (𝐴 ∈ ℝ+ → 𝐴(⋘‘ℝ*𝑠)+∞) | ||
| Theorem | xrnarchi 33245 | The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.) |
| ⊢ ¬ ℝ*𝑠 ∈ Archi | ||
| Theorem | isarchi2 33246* | Alternative way to express the predicate "𝑊 is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) ⇒ ⊢ ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 ≤ (𝑛 · 𝑥)))) | ||
| Theorem | submarchi 33247 | A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
| ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi) | ||
| Theorem | isarchi3 33248* | This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) ⇒ ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)))) | ||
| Theorem | archirng 33249* | Property of Archimedean ordered groups, framing positive 𝑌 between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑋) & ⊢ (𝜑 → 0 < 𝑌) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋))) | ||
| Theorem | archirngz 33250* | Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑋) & ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋))) | ||
| Theorem | archiexdiv 33251* | In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) ⇒ ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)) | ||
| Theorem | archiabllem1a 33252* | Lemma for archiabl 33259: In case an archimedean group 𝑊 admits a smallest positive element 𝑈, then any positive element 𝑋 of 𝑊 can be written as (𝑛 · 𝑈) with 𝑛 ∈ ℕ. Since the reciprocal holds for negative elements, 𝑊 is then isomorphic to ℤ. (Contributed by Thierry Arnoux, 12-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑈) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑋) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝑋 = (𝑛 · 𝑈)) | ||
| Theorem | archiabllem1b 33253* | Lemma for archiabl 33259. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑈) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥) ⇒ ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈)) | ||
| Theorem | archiabllem1 33254* | Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑈) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥) ⇒ ⊢ (𝜑 → 𝑊 ∈ Abel) | ||
| Theorem | archiabllem2a 33255* | Lemma for archiabl 33259, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑋) ⇒ ⊢ (𝜑 → ∃𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) ≤ 𝑋)) | ||
| Theorem | archiabllem2c 33256* | Lemma for archiabl 33259. (Contributed by Thierry Arnoux, 1-May-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ¬ (𝑋 + 𝑌) < (𝑌 + 𝑋)) | ||
| Theorem | archiabllem2b 33257* | Lemma for archiabl 33259. (Contributed by Thierry Arnoux, 1-May-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
| Theorem | archiabllem2 33258* | Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ≤ = (le‘𝑊) & ⊢ < = (lt‘𝑊) & ⊢ · = (.g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ oGrp) & ⊢ (𝜑 → 𝑊 ∈ Archi) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) ⇒ ⊢ (𝜑 → 𝑊 ∈ Abel) | ||
| Theorem | archiabl 33259 | Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.) |
| ⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel) | ||
| Theorem | isarchiofld 33260* | Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐻 = (ℤRHom‘𝑊) & ⊢ < = (lt‘𝑊) ⇒ ⊢ (𝑊 ∈ oField → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∃𝑛 ∈ ℕ 𝑥 < (𝐻‘𝑛))) | ||
| Syntax | cslmd 33261 | Extend class notation with class of all semimodules. |
| class SLMod | ||
| Definition | df-slmd 33262* | Define the class of all (left) modules over semirings, i.e. semimodules, which are generalizations of left modules. A semimodule is a commutative monoid (=vectors) together with a semiring (=scalars) and a left scalar product connecting them. (0[,]+∞) for example is not a full fledged left module, but is a semimodule. Definition of [Golan] p. 149. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
| ⊢ SLMod = {𝑔 ∈ CMnd ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][( ·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))))} | ||
| Theorem | isslmd 33263* | The predicate "is a semimodule". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) & ⊢ × = (.r‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ 𝑂 = (0g‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤 ∧ (𝑂 · 𝑤) = 0 )))) | ||
| Theorem | slmdlema 33264 | Lemma for properties of a semimodule. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) & ⊢ × = (.r‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ 𝑂 = (0g‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑅 · 𝑌) ∈ 𝑉 ∧ (𝑅 · (𝑌 + 𝑋)) = ((𝑅 · 𝑌) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑌) = ((𝑄 · 𝑌) + (𝑅 · 𝑌))) ∧ (((𝑄 × 𝑅) · 𝑌) = (𝑄 · (𝑅 · 𝑌)) ∧ ( 1 · 𝑌) = 𝑌 ∧ (𝑂 · 𝑌) = 0 ))) | ||
| Theorem | lmodslmd 33265 | Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ (𝑊 ∈ LMod → 𝑊 ∈ SLMod) | ||
| Theorem | slmdcmn 33266 | A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) | ||
| Theorem | slmdmnd 33267 | A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | ||
| Theorem | slmdsrg 33268 | The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) | ||
| Theorem | slmdbn0 33269 | The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.) |
| ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) | ||
| Theorem | slmdacl 33270 | Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ + = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) | ||
| Theorem | slmdmcl 33271 | Closure of ring multiplication for a semimodule. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾) | ||
| Theorem | slmdsn0 33272 | The set of scalars in a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.) |
| ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) | ||
| Theorem | slmdvacl 33273 | Closure of vector addition for a semiring left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉) | ||
| Theorem | slmdass 33274 | Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | ||
| Theorem | slmdvscl 33275 | Closure of scalar product for a semiring left module. (hvmulcl 31084 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉) | ||
| Theorem | slmdvsdi 33276 | Distributive law for scalar product. (ax-hvdistr1 31079 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌))) | ||
| Theorem | slmdvsdir 33277 | Distributive law for scalar product. (ax-hvdistr1 31079 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋))) | ||
| Theorem | slmdvsass 33278 | Associative law for scalar product. (ax-hvmulass 31078 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ × = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) | ||
| Theorem | slmd0cl 33279 | The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾) | ||
| Theorem | slmd1cl 33280 | The ring unity in a semiring left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 1 ∈ 𝐾) | ||
| Theorem | slmdvs1 33281 | Scalar product with ring unity. (ax-hvmulid 31077 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋) | ||
| Theorem | slmd0vcl 33282 | The zero vector is a vector. (ax-hv0cl 31074 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝑉) | ||
| Theorem | slmd0vlid 33283 | Left identity law for the zero vector. (hvaddlid 31094 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) | ||
| Theorem | slmd0vrid 33284 | Right identity law for the zero vector. (ax-hvaddid 31075 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋) | ||
| Theorem | slmd0vs 33285 | Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 31081 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) | ||
| Theorem | slmdvs0 33286 | Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 31095 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) | ||
| Theorem | gsumvsca1 33287* | Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ SLMod) & ⊢ (𝜑 → 𝑃 ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))) | ||
| Theorem | gsumvsca2 33288* | Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) (Proof shortened by AV, 12-Dec-2019.) |
| ⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ SLMod) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄)) | ||
| Theorem | prmsimpcyc 33289 | A group of prime order is cyclic if and only if it is simple. This is the first family of finite simple groups. (Contributed by Thierry Arnoux, 21-Sep-2023.) |
| ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ ((♯‘𝐵) ∈ ℙ → (𝐺 ∈ SimpGrp ↔ 𝐺 ∈ CycGrp)) | ||
| Theorem | ringrngd 33290 | A unital ring is a non-unital ring, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑅 ∈ Rng) | ||
| Theorem | ringdi22 33291 | Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑇 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇)))) | ||
| Theorem | urpropd 33292* | Sufficient condition for ring unities to be equal. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 = (Base‘𝑇)) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘𝑇)𝑦)) ⇒ ⊢ (𝜑 → (1r‘𝑆) = (1r‘𝑇)) | ||
| Theorem | subrgmcld 33293 | A subring is closed under multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025.) |
| ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝐴 ∈ (SubRing‘𝑅)) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐴) | ||
| Theorem | ress1r 33294 | 1r is unaffected by restriction. This is a bit more generic than subrg1 20559. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
| ⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 = (1r‘𝑆)) | ||
| Theorem | ringm1expp1 33295 | Ring exponentiation of minus one: Adding one to the exponent is the same as taking the additive inverse. (Contributed by Thierry Arnoux, 15-Feb-2026.) |
| ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ ↑ = (.g‘(mulGrp‘𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐾 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐾 + 1) ↑ (𝑁‘ 1 )) = (𝑁‘(𝐾 ↑ (𝑁‘ 1 )))) | ||
| Theorem | ringinvval 33296* | The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∗ = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 )) | ||
| Theorem | dvrcan5 33297 | Cancellation law for common factor in ratio. (divcan5 11857 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌)) | ||
| Theorem | subrgchr 33298 | If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
| ⊢ (𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅 ↾s 𝐴)) = (chr‘𝑅)) | ||
| Theorem | rmfsupp2 33299* | A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by Thierry Arnoux, 3-Jun-2023.) |
| ⊢ 𝑅 = (Base‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Ring) & ⊢ (𝜑 → 𝑉 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐶 ∈ 𝑅) & ⊢ (𝜑 → 𝐴:𝑉⟶𝑅) & ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑀)) ⇒ ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)(.r‘𝑀)𝐶)) finSupp (0g‘𝑀)) | ||
| Theorem | unitnz 33300 | In a nonzero ring, a unit cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2025.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ NzRing) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑋 ≠ 0 ) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |