HomeHome Metamath Proof Explorer
Theorem List (p. 315 of 465)
< 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  Metamath Proof Explorer
(1-29288)
  Hilbert Space Explorer  Hilbert Space Explorer
(29289-30811)
  Users' Mathboxes  Users' Mathboxes
(30812-46499)
 

Theorem List for Metamath Proof Explorer - 31401-31500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcycpmco2lem1 31401 Lemma for cycpmco2 31408. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑊)‘𝐽))
 
Theoremcycpmco2lem2 31402 Lemma for cycpmco2 31408. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → (𝑈𝐸) = 𝐼)
 
Theoremcycpmco2lem3 31403 Lemma for cycpmco2 31408. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
 
Theoremcycpmco2lem4 31404 Lemma for cycpmco2 31408. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑈)‘𝐼))
 
Theoremcycpmco2lem5 31405 Lemma for cycpmco2 31408. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑 → (𝑈𝐾) = ((♯‘𝑈) − 1))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2lem6 31406 Lemma for cycpmco2 31408. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐼)    &   (𝜑 → (𝑈𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2lem7 31407 Lemma for cycpmco2 31408. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐽)    &   (𝜑 → (𝑈𝐾) ∈ (0..^𝐸))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
Theoremcycpmco2 31408 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀𝑈))
 
Theoremcyc2fvx 31409 Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
 
Theoremcycpm3cl 31410 Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
 
Theoremcycpm3cl2 31411 Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (𝐶 “ (♯ “ {3})))
 
Theoremcyc3fv1 31412 Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
 
Theoremcyc3fv2 31413 Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
 
Theoremcyc3fv3 31414 Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
 
Theoremcyc3co2 31415 Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)    &    · = (+g𝑆)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))
 
Theoremcycpmconjvlem 31416 Lemma for cycpmconjv 31417. (Contributed by Thierry Arnoux, 9-Oct-2023.)
(𝜑𝐹:𝐷1-1-onto𝐷)    &   (𝜑𝐵𝐷)       (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
 
Theoremcycpmconjv 31417 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)    &    + = (+g𝑆)    &    = (-g𝑆)    &   𝐵 = (Base‘𝑆)       ((𝐷𝑉𝐺𝐵𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀𝑊)) 𝐺) = (𝑀‘(𝐺𝑊)))
 
Theoremcycpmrn 31418 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)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → 1 < (♯‘𝑊))       (𝜑 → ran 𝑊 = dom ((𝑀𝑊) ∖ I ))
 
Theoremtocyccntz 31419* 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)    &   (𝜑𝐷𝑉)    &   (𝜑Disj 𝑥𝐴 ran 𝑥)    &   (𝜑𝐴 ⊆ dom 𝑀)       (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
 
20.3.9.9  The Alternating Group
 
Theoremevpmval 31420 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEven‘𝐷)       (𝐷𝑉𝐴 = ((pmSgn‘𝐷) “ {1}))
 
Theoremcnmsgn0g 31421 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 = (0g𝑈)
 
Theoremevpmsubg 31422 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))
 
Theoremevpmid 31423 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷))
 
Theoremaltgnsg 31424 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))
 
Theoremcyc3evpm 31425 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = ((toCyc‘𝐷) “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐶𝐴)
 
Theoremcyc3genpmlem 31426* Lemma for cyc3genpm 31427. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    · = (+g𝑆)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐿𝐷)    &   (𝜑𝐸 = (𝑀‘⟨“𝐼𝐽”⟩))    &   (𝜑𝐹 = (𝑀‘⟨“𝐾𝐿”⟩))    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐽)    &   (𝜑𝐾𝐿)       (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐))
 
Theoremcyc3genpm 31427* 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 → (𝑄𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
 
Theoremcycpmgcl 31428 Cyclic permutations are permutations, similar to cycpmcl 31391, 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)       ((𝐷𝑉𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
 
Theoremcycpmconjslem1 31429 Lemma for cycpmconjs 31431. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → (♯‘𝑊) = 𝑃)       (𝜑 → ((𝑊 ∘ (𝑀𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1))
 
Theoremcycpmconjslem2 31430* Lemma for cycpmconjs 31431. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)       (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
 
Theoremcycpmconjs 31431* 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)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
Theoremcyc3conja 31432* 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)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑 → 5 ≤ 𝑁)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐴 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
20.3.9.10  Signum in an ordered monoid
 
Syntaxcsgns 31433 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 31434* Signum function for a structure. See also df-sgn 14808 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
 
Theoremsgnsv 31435* The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
 
Theoremsgnsval 31436 The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       ((𝑅𝑉𝑋𝐵) → (𝑆𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
 
Theoremsgnsf 31437 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆:𝐵⟶{-1, 0, 1})
 
20.3.9.11  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 31438 Class notation for the infinitesimal relation.
class
 
Syntaxcarchi 31439 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 31440* 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)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
 
Definitiondf-archi 31441 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
 
Theoreminftmrel 31442 The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)       (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
 
Theoremisinftm 31443* Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    · = (.g𝑊)    &    < = (lt‘𝑊)       ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
 
Theoremisarchi 31444* Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (⋘‘𝑊)       (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))
 
Theorempnfinf 31445 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)+∞)
 
Theoremxrnarchi 31446 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
¬ ℝ*𝑠 ∈ Archi
 
Theoremisarchi2 31447* Alternative way to express the predicate "𝑊 is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    · = (.g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)       ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 (𝑛 · 𝑥))))
 
Theoremsubmarchi 31448 A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)
 
Theoremisarchi3 31449* This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)       (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))
 
Theoremarchirng 31450* Property of Archimedean ordered groups, framing positive 𝑌 between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    = (le‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑0 < 𝑌)       (𝜑 → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
 
Theoremarchirngz 31451* Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    = (le‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑 → (oppg𝑊) ∈ oGrp)       (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))
 
Theoremarchiexdiv 31452* In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)       (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋𝐵𝑌𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋))
 
Theoremarchiabllem1a 31453* Lemma for archiabl 31460: 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 . (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)    &   (𝜑𝑋𝐵)    &   (𝜑0 < 𝑋)       (𝜑 → ∃𝑛 ∈ ℕ 𝑋 = (𝑛 · 𝑈))
 
Theoremarchiabllem1b 31454* Lemma for archiabl 31460. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)       ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
 
Theoremarchiabllem1 31455* Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)       (𝜑𝑊 ∈ Abel)
 
Theoremarchiabllem2a 31456* Lemma for archiabl 31460, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))    &   (𝜑𝑋𝐵)    &   (𝜑0 < 𝑋)       (𝜑 → ∃𝑐𝐵 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) 𝑋))
 
Theoremarchiabllem2c 31457* Lemma for archiabl 31460. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ¬ (𝑋 + 𝑌) < (𝑌 + 𝑋))
 
Theoremarchiabllem2b 31458* Lemma for archiabl 31460. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theoremarchiabllem2 31459* Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))       (𝜑𝑊 ∈ Abel)
 
Theoremarchiabl 31460 Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
 
20.3.9.12  Semiring left modules
 
Syntaxcslmd 31461 Extend class notation with class of all semimodules.
class SLMod
 
Definitiondf-slmd 31462* 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][(+g𝑔) / 𝑎][( ·𝑠𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞𝑘𝑟𝑘𝑥𝑣𝑤𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r𝑓)𝑠𝑤) = 𝑤 ∧ ((0g𝑓)𝑠𝑤) = (0g𝑔))))}
 
Theoremisslmd 31463* 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)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   𝑂 = (0g𝐹)       (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((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 ))))
 
Theoremslmdlema 31464 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)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   𝑂 = (0g𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑌𝑉)) → (((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 )))
 
Theoremlmodslmd 31465 Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ LMod → 𝑊 ∈ SLMod)
 
Theoremslmdcmn 31466 A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
 
Theoremslmdmnd 31467 A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
 
Theoremslmdsrg 31468 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)
 
Theoremslmdbn0 31469 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 ≠ ∅)
 
Theoremslmdacl 31470 Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
 
Theoremslmdmcl 31471 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)    &    · = (.r𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
 
Theoremslmdsn0 31472 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 ≠ ∅)
 
Theoremslmdvacl 31473 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)    &    + = (+g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
 
Theoremslmdass 31474 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)    &    + = (+g𝑊)       ((𝑊 ∈ SLMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
Theoremslmdvscl 31475 Closure of scalar product for a semiring left module. (hvmulcl 29383 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)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ SLMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
 
Theoremslmdvsdi 31476 Distributive law for scalar product. (ax-hvdistr1 29378 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
 
Theoremslmdvsdir 31477 Distributive law for scalar product. (ax-hvdistr1 29378 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
 
Theoremslmdvsass 31478 Associative law for scalar product. (ax-hvmulass 29377 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)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
 
Theoremslmd0cl 31479 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 = (0g𝐹)       (𝑊 ∈ SLMod → 0𝐾)
 
Theoremslmd1cl 31480 The ring unit 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 = (1r𝐹)       (𝑊 ∈ SLMod → 1𝐾)
 
Theoremslmdvs1 31481 Scalar product with ring unit. (ax-hvmulid 29376 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)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 1 · 𝑋) = 𝑋)
 
Theoremslmd0vcl 31482 The zero vector is a vector. (ax-hv0cl 29373 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 = (0g𝑊)       (𝑊 ∈ SLMod → 0𝑉)
 
Theoremslmd0vlid 31483 Left identity law for the zero vector. (hvaddid2 29393 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)
 
Theoremslmd0vrid 31484 Right identity law for the zero vector. (ax-hvaddid 29374 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
 
Theoremslmd0vs 31485 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 29380 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)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )
 
Theoremslmdvs0 31486 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 29394 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )
 
Theoremgsumvsca1 31487* 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑃𝐾)    &   ((𝜑𝑘𝐴) → 𝑄𝐵)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘𝐴𝑄))))
 
Theoremgsumvsca2 31488* 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 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑄𝐵)    &   ((𝜑𝑘𝐴) → 𝑃𝐾)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘𝐴𝑃)) · 𝑄))
 
20.3.9.13  Simple groups
 
Theoremprmsimpcyc 31489 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))
 
20.3.9.14  Rings - misc additions
 
Theoremrngurd 31490* Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)       (𝜑1 = (1r𝑅))
 
Theoremdvdschrmulg 31491 In a ring, any multiple of the characteristics annihilates all elements. (Contributed by Thierry Arnoux, 6-Sep-2016.)
𝐶 = (chr‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.g𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐶𝑁𝐴𝐵) → (𝑁 · 𝐴) = 0 )
 
Theoremfreshmansdream 31492 For a prime number 𝑃, if 𝑋 and 𝑌 are members of a commutative ring 𝑅 of characteristic 𝑃, then ((𝑋 + 𝑌)↑𝑃) = ((𝑋𝑃) + (𝑌𝑃)). This theorem is sometimes referred to as "the freshman's dream" . (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.g‘(mulGrp‘𝑅))    &   𝑃 = (chr‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑃 (𝑋 + 𝑌)) = ((𝑃 𝑋) + (𝑃 𝑌)))
 
Theoremfrobrhm 31493* In a commutative ring with prime characteristic, the Frobenius function 𝐹 is a ring endomorphism, thus named the Frobenius endomorphism. (Contributed by Thierry Arnoux, 31-May-2024.)
𝐵 = (Base‘𝑅)    &   𝑃 = (chr‘𝑅)    &    = (.g‘(mulGrp‘𝑅))    &   𝐹 = (𝑥𝐵 ↦ (𝑃 𝑥))    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑃 ∈ ℙ)       (𝜑𝐹 ∈ (𝑅 RingHom 𝑅))
 
Theoremress1r 31494 1r is unaffected by restriction. This is a bit more generic than subrg1 20044. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 1𝐴𝐴𝐵) → 1 = (1r𝑆))
 
Theoremdvrdir 31495 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝑈)) → ((𝑋 + 𝑌) / 𝑍) = ((𝑋 / 𝑍) + (𝑌 / 𝑍)))
 
Theoremrdivmuldivd 31496 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝑈)       (𝜑 → ((𝑋 / 𝑌) · (𝑍 / 𝑊)) = ((𝑋 · 𝑍) / (𝑌 · 𝑊)))
 
Theoremringinvval 31497* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
𝐵 = (Base‘𝑅)    &    = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑈 = (Unit‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) = (𝑦𝑈 (𝑦 𝑋) = 1 ))
 
Theoremdvrcan5 31498 Cancellation law for common factor in ratio. (divcan5 11687 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝑈𝑍𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌))
 
Theoremsubrgchr 31499 If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
(𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅s 𝐴)) = (chr‘𝑅))
 
Theoremrmfsupp2 31500* 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)    &   (𝜑𝑀 ∈ Ring)    &   (𝜑𝑉𝑋)    &   ((𝜑𝑣𝑉) → 𝐶𝑅)    &   (𝜑𝐴:𝑉𝑅)    &   (𝜑𝐴 finSupp (0g𝑀))       (𝜑 → (𝑣𝑉 ↦ ((𝐴𝑣)(.r𝑀)𝐶)) finSupp (0g𝑀))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46499
  Copyright terms: Public domain < Previous  Next >