HomeTheorem List (p. 333 of 501)
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 501) | < 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
(1-30993) |
Hilbert Space Explorer
(30994-32516) |
Users' Mathboxes
(32517-50046) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | cycpm2tr 33201 | A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) = (𝑇‘{𝐼, 𝐽})) | ||
| Theorem | cycpm2cl 33202 | Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆)) | ||
| Theorem | cyc2fv1 33203 | Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽) | ||
| Theorem | cyc2fv2 33204 | Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.) |
| ⊢ 𝐶 = (toCyc‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝐷) & ⊢ (𝜑 → 𝐽 ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ≠ 𝐽) & ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼) | ||
| Theorem | trsp2cyc 33205* | Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.) |
| ⊢ 𝑇 = ran (pmTrsp‘𝐷) & ⊢ 𝐶 = (toCyc‘𝐷) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) → ∃𝑖 ∈ 𝐷 ∃𝑗 ∈ 𝐷 (𝑖 ≠ 𝑗 ∧ 𝑃 = (𝐶‘⟨“𝑖𝑗”⟩))) | ||
| Theorem | cycpmco2f1 33206 | The word U used in cycpmco2 33215 is injective, so it can represent a cycle and form a cyclic permutation (𝑀‘𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → 𝑈:dom 𝑈–1-1→𝐷) | ||
| Theorem | cycpmco2rn 33207 | The orbit of the composition of a cyclic permutation and a well-chosen transposition is one element more than the orbit of the original permutation. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼})) | ||
| Theorem | cycpmco2lem1 33208 | Lemma for cycpmco2 33215. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) | ||
| Theorem | cycpmco2lem2 33209 | Lemma for cycpmco2 33215. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼) | ||
| Theorem | cycpmco2lem3 33210 | Lemma for cycpmco2 33215. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊)) | ||
| Theorem | cycpmco2lem4 33211 | Lemma for cycpmco2 33215. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀‘𝑈)‘𝐼)) | ||
| Theorem | cycpmco2lem5 33212 | Lemma for cycpmco2 33215. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → (◡𝑈‘𝐾) = ((♯‘𝑈) − 1)) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
| Theorem | cycpmco2lem6 33213 | Lemma for cycpmco2 33215. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → 𝐾 ≠ 𝐼) & ⊢ (𝜑 → (◡𝑈‘𝐾) ∈ (𝐸..^((♯‘𝑈) − 1))) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
| Theorem | cycpmco2lem7 33214 | Lemma for cycpmco2 33215. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) & ⊢ (𝜑 → 𝐾 ∈ ran 𝑊) & ⊢ (𝜑 → 𝐾 ≠ 𝐽) & ⊢ (𝜑 → (◡𝑈‘𝐾) ∈ (0..^𝐸)) ⇒ ⊢ (𝜑 → ((𝑀‘𝑈)‘𝐾) = ((𝑀‘𝑊)‘𝐾)) | ||
| Theorem | cycpmco2 33215 | The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-Jan-2024.) |
| ⊢ 𝑀 = (toCyc‘𝐷) & ⊢ 𝑆 = (SymGrp‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) & ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) & ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) & ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) & ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) ⇒ ⊢ (𝜑 → ((𝑀‘𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀‘𝑈)) | ||
| Theorem | cyc2fvx 33216 | Function value of a 2-cycle outside of its orbit. (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) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾) | ||
| Theorem | cycpm3cl 33217 | Closure of the 3-cycles in the permutations. (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) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆)) | ||
| Theorem | cycpm3cl2 33218 | Closure of the 3-cycles in the class of 3-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) ⇒ ⊢ (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (𝐶 “ (◡♯ “ {3}))) | ||
| Theorem | cyc3fv1 33219 | Function value of a 3-cycle at the first point. (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) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽) | ||
| Theorem | cyc3fv2 33220 | Function value of a 3-cycle at the second point. (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) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾) | ||
| Theorem | cyc3fv3 33221 | Function value of a 3-cycle at the third point. (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) ⇒ ⊢ (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼) | ||
| Theorem | cyc3co2 33222 | 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 33223 | Lemma for cycpmconjv 33224. (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 33224 | 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 33225 | 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 33226* | 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 33227 | 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 33228 | 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 33229 | 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 33230 | The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.) |
| ⊢ 𝑆 = (SymGrp‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷)) | ||
| Theorem | altgnsg 33231 | 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 33232 | 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 33233* | Lemma for cyc3genpm 33234. (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 33234* | 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 33235 | Cyclic permutations are permutations, similar to cycpmcl 33198, 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 33236 | Lemma for cycpmconjs 33238. (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 33237* | Lemma for cycpmconjs 33238. (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 33238* | 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 33239* | 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 33240 | Extend class notation to include the Signum function. |
| class sgns | ||
| Definition | df-sgns 33241* | Signum function for a structure. See also df-sgn 15010 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 33242* | 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 33243 | 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 33244 | 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 33245 | Extend class notation with the fixed points operation. |
| class FixPts | ||
| Definition | df-fxp 33246* | 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 33247* | 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 33248 | 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 33249* | 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 33250* | 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 33251 | 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 33252* | 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 33253* | 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 33254* | 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 33255* | 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 33256* | 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 33257* | 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 20731), (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 33258 | Class notation for the infinitesimal relation. |
| class ⋘ | ||
| Syntax | carchi 33259 | Class notation for the Archimedean property. |
| class Archi | ||
| Definition | df-inftm 33260* | 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 33261 | A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | ||
| Theorem | inftmrel 33262 | The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
| ⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) | ||
| Theorem | isinftm 33263* | 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 33264* | 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 33265 | 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 33266 | The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.) |
| ⊢ ¬ ℝ*𝑠 ∈ Archi | ||
| Theorem | isarchi2 33267* | 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 33268 | A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
| ⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi) | ||
| Theorem | isarchi3 33269* | 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 33270* | 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 33271* | 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 33272* | 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 33273* | Lemma for archiabl 33280: 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 33274* | Lemma for archiabl 33280. (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 33275* | 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 33276* | Lemma for archiabl 33280, 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 33277* | Lemma for archiabl 33280. (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 33278* | Lemma for archiabl 33280. (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 33279* | 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 33280 | 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 33281* | 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 33282 | Extend class notation with class of all semimodules. |
| class SLMod | ||
| Definition | df-slmd 33283* | 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 33284* | 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 33285 | 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 33286 | Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ (𝑊 ∈ LMod → 𝑊 ∈ SLMod) | ||
| Theorem | slmdcmn 33287 | A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) | ||
| Theorem | slmdmnd 33288 | A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
| ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | ||
| Theorem | slmdsrg 33289 | 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 33290 | 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 33291 | 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 33292 | 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 33293 | 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 33294 | 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 33295 | 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 33296 | Closure of scalar product for a semiring left module. (hvmulcl 31088 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 33297 | Distributive law for scalar product. (ax-hvdistr1 31083 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 33298 | Distributive law for scalar product. (ax-hvdistr1 31083 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 33299 | Associative law for scalar product. (ax-hvmulass 31082 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 33300 | 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) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |