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Theorem List for Metamath Proof Explorer - 32801-32900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfzto1st1 32801* Special case where the permutation defined in psgnfzto1st 32804 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    β‡’   (𝐼 = 1 β†’ 𝑃 = ( I β†Ύ 𝐷))
 
Theoremfzto1st 32802* The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   πΊ = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐼 ∈ 𝐷 β†’ 𝑃 ∈ 𝐡)
 
Theoremfzto1stinvn 32803* Value of the inverse of our permutation 𝑃 at 𝐼. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   πΊ = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐼 ∈ 𝐷 β†’ (β—‘π‘ƒβ€˜πΌ) = 1)
 
Theorempsgnfzto1st 32804* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   π‘ƒ = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≀ 𝐼, (𝑖 βˆ’ 1), 𝑖)))    &   πΊ = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜πΊ)    &   π‘† = (pmSgnβ€˜π·)    β‡’   (𝐼 ∈ 𝐷 β†’ (π‘†β€˜π‘ƒ) = (-1↑(𝐼 + 1)))
 
21.3.9.8  Permutation cycles
 
Syntaxctocyc 32805 Extend class notation with the permutation cycle builder.
class toCyc
 
Definitiondf-tocyc 32806* Define a convenience permutation cycle builder. Given a list of elements to be cycled, in the form of a word, this function produces the corresponding permutation cycle. See definition in [Lang] p. 30. (Contributed by Thierry Arnoux, 19-Sep-2023.)
toCyc = (𝑑 ∈ V ↦ (𝑀 ∈ {𝑒 ∈ Word 𝑑 ∣ 𝑒:dom 𝑒–1-1→𝑑} ↦ (( I β†Ύ (𝑑 βˆ– ran 𝑀)) βˆͺ ((𝑀 cyclShift 1) ∘ ◑𝑀))))
 
Theoremtocycval 32807* Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐢 = (𝑀 ∈ {𝑒 ∈ Word 𝐷 ∣ 𝑒:dom 𝑒–1-1→𝐷} ↦ (( I β†Ύ (𝐷 βˆ– ran 𝑀)) βˆͺ ((𝑀 cyclShift 1) ∘ ◑𝑀))))
 
Theoremtocycfv 32808 Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    β‡’   (πœ‘ β†’ (πΆβ€˜π‘Š) = (( I β†Ύ (𝐷 βˆ– ran π‘Š)) βˆͺ ((π‘Š cyclShift 1) ∘ β—‘π‘Š)))
 
Theoremtocycfvres1 32809 A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š) β†Ύ ran π‘Š) = ((π‘Š cyclShift 1) ∘ β—‘π‘Š))
 
Theoremtocycfvres2 32810 A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š) β†Ύ (𝐷 βˆ– ran π‘Š)) = ( I β†Ύ (𝐷 βˆ– ran π‘Š)))
 
Theoremcycpmfvlem 32811 Lemma for cycpmfv1 32812 and cycpmfv2 32813. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 𝑁 ∈ (0..^(β™―β€˜π‘Š)))    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜(π‘Šβ€˜π‘)) = (((π‘Š cyclShift 1) ∘ β—‘π‘Š)β€˜(π‘Šβ€˜π‘)))
 
Theoremcycpmfv1 32812 Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 𝑁 ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)))    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜(π‘Šβ€˜π‘)) = (π‘Šβ€˜(𝑁 + 1)))
 
Theoremcycpmfv2 32813 Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 0 < (β™―β€˜π‘Š))    &   (πœ‘ β†’ 𝑁 = ((β™―β€˜π‘Š) βˆ’ 1))    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜(π‘Šβ€˜π‘)) = (π‘Šβ€˜0))
 
Theoremcycpmfv3 32814 Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 𝑋 ∈ 𝐷)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ ran π‘Š)    β‡’   (πœ‘ β†’ ((πΆβ€˜π‘Š)β€˜π‘‹) = 𝑋)
 
Theoremcycpmcl 32815 Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ (πΆβ€˜π‘Š) ∈ (Baseβ€˜π‘†))
 
Theoremtocycf 32816* The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐢:{𝑀 ∈ Word 𝐷 ∣ 𝑀:dom 𝑀–1-1→𝐷}⟢𝐡)
 
Theoremtocyc01 32817 Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.)
𝐢 = (toCycβ€˜π·)    β‡’   ((𝐷 ∈ 𝑉 ∧ π‘Š ∈ (dom 𝐢 ∩ (β—‘β™― β€œ {0, 1}))) β†’ (πΆβ€˜π‘Š) = ( I β†Ύ 𝐷))
 
Theoremcycpm2tr 32818 A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘‡ = (pmTrspβ€˜π·)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©) = (π‘‡β€˜{𝐼, 𝐽}))
 
Theoremcycpm2cl 32819 Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©) ∈ (Baseβ€˜π‘†))
 
Theoremcyc2fv1 32820 Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΌ) = 𝐽)
 
Theoremcyc2fv2 32821 Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   π‘† = (SymGrpβ€˜π·)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜π½) = 𝐼)
 
Theoremtrsp2cyc 32822* Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝑇 = ran (pmTrspβ€˜π·)    &   πΆ = (toCycβ€˜π·)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝑇) β†’ βˆƒπ‘– ∈ 𝐷 βˆƒπ‘— ∈ 𝐷 (𝑖 β‰  𝑗 ∧ 𝑃 = (πΆβ€˜βŸ¨β€œπ‘–π‘—β€βŸ©)))
 
Theoremcycpmco2f1 32823 The word U used in cycpmco2 32832 is injective, so it can represent a cycle and form a cyclic permutation (π‘€β€˜π‘ˆ). (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ π‘ˆ:dom π‘ˆβ€“1-1→𝐷)
 
Theoremcycpmco2rn 32824 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 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ran π‘ˆ = (ran π‘Š βˆͺ {𝐼}))
 
Theoremcycpmco2lem1 32825 Lemma for cycpmco2 32832. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘Š)β€˜((π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΌ)) = ((π‘€β€˜π‘Š)β€˜π½))
 
Theoremcycpmco2lem2 32826 Lemma for cycpmco2 32832. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ (π‘ˆβ€˜πΈ) = 𝐼)
 
Theoremcycpmco2lem3 32827 Lemma for cycpmco2 32832. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((β™―β€˜π‘ˆ) βˆ’ 1) = (β™―β€˜π‘Š))
 
Theoremcycpmco2lem4 32828 Lemma for cycpmco2 32832. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘Š)β€˜((π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΌ)) = ((π‘€β€˜π‘ˆ)β€˜πΌ))
 
Theoremcycpmco2lem5 32829 Lemma for cycpmco2 32832. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    &   (πœ‘ β†’ 𝐾 ∈ ran π‘Š)    &   (πœ‘ β†’ (β—‘π‘ˆβ€˜πΎ) = ((β™―β€˜π‘ˆ) βˆ’ 1))    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘ˆ)β€˜πΎ) = ((π‘€β€˜π‘Š)β€˜πΎ))
 
Theoremcycpmco2lem6 32830 Lemma for cycpmco2 32832. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    &   (πœ‘ β†’ 𝐾 ∈ ran π‘Š)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    &   (πœ‘ β†’ (β—‘π‘ˆβ€˜πΎ) ∈ (𝐸..^((β™―β€˜π‘ˆ) βˆ’ 1)))    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘ˆ)β€˜πΎ) = ((π‘€β€˜π‘Š)β€˜πΎ))
 
Theoremcycpmco2lem7 32831 Lemma for cycpmco2 32832. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    &   (πœ‘ β†’ 𝐾 ∈ ran π‘Š)    &   (πœ‘ β†’ 𝐾 β‰  𝐽)    &   (πœ‘ β†’ (β—‘π‘ˆβ€˜πΎ) ∈ (0..^𝐸))    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘ˆ)β€˜πΎ) = ((π‘€β€˜π‘Š)β€˜πΎ))
 
Theoremcycpmco2 32832 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 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ dom 𝑀)    &   (πœ‘ β†’ 𝐼 ∈ (𝐷 βˆ– ran π‘Š))    &   (πœ‘ β†’ 𝐽 ∈ ran π‘Š)    &   πΈ = ((β—‘π‘Šβ€˜π½) + 1)    &   π‘ˆ = (π‘Š splice ⟨𝐸, 𝐸, βŸ¨β€œπΌβ€βŸ©βŸ©)    β‡’   (πœ‘ β†’ ((π‘€β€˜π‘Š) ∘ (π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©)) = (π‘€β€˜π‘ˆ))
 
Theoremcyc2fvx 32833 Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)β€˜πΎ) = 𝐾)
 
Theoremcycpm3cl 32834 Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©) ∈ (Baseβ€˜π‘†))
 
Theoremcycpm3cl2 32835 Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©) ∈ (𝐢 β€œ (β—‘β™― β€œ {3})))
 
Theoremcyc3fv1 32836 Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©)β€˜πΌ) = 𝐽)
 
Theoremcyc3fv2 32837 Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©)β€˜π½) = 𝐾)
 
Theoremcyc3fv3 32838 Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    β‡’   (πœ‘ β†’ ((πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©)β€˜πΎ) = 𝐼)
 
Theoremcyc3co2 32839 Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐢 = (toCycβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐾 β‰  𝐼)    &    Β· = (+gβ€˜π‘†)    β‡’   (πœ‘ β†’ (πΆβ€˜βŸ¨β€œπΌπ½πΎβ€βŸ©) = ((πΆβ€˜βŸ¨β€œπΌπΎβ€βŸ©) Β· (πΆβ€˜βŸ¨β€œπΌπ½β€βŸ©)))
 
Theoremcycpmconjvlem 32840 Lemma for cycpmconjv 32841. (Contributed by Thierry Arnoux, 9-Oct-2023.)
(πœ‘ β†’ 𝐹:𝐷–1-1-onto→𝐷)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐷)    β‡’   (πœ‘ β†’ ((𝐹 β†Ύ (𝐷 βˆ– 𝐡)) ∘ ◑𝐹) = ( I β†Ύ (𝐷 βˆ– ran (𝐹 β†Ύ 𝐡))))
 
Theoremcycpmconjv 32841 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 = (SymGrpβ€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝐺 ∈ 𝐡 ∧ π‘Š ∈ dom 𝑀) β†’ ((𝐺 + (π‘€β€˜π‘Š)) βˆ’ 𝐺) = (π‘€β€˜(𝐺 ∘ π‘Š)))
 
Theoremcycpmrn 32842 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 = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ 1 < (β™―β€˜π‘Š))    β‡’   (πœ‘ β†’ ran π‘Š = dom ((π‘€β€˜π‘Š) βˆ– I ))
 
Theoremtocyccntz 32843* All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π‘ = (Cntzβ€˜π‘†)    &   π‘€ = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ Disj π‘₯ ∈ 𝐴 ran π‘₯)    &   (πœ‘ β†’ 𝐴 βŠ† dom 𝑀)    β‡’   (πœ‘ β†’ (𝑀 β€œ 𝐴) βŠ† (π‘β€˜(𝑀 β€œ 𝐴)))
 
21.3.9.9  The Alternating Group
 
Theoremevpmval 32844 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ 𝑉 β†’ 𝐴 = (β—‘(pmSgnβ€˜π·) β€œ {1}))
 
Theoremcnmsgn0g 32845 The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.)
π‘ˆ = ((mulGrpβ€˜β„‚fld) β†Ύs {1, -1})    β‡’   1 = (0gβ€˜π‘ˆ)
 
Theoremevpmsubg 32846 The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑆 = (SymGrpβ€˜π·)    &   π΄ = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ 𝐴 ∈ (SubGrpβ€˜π‘†))
 
Theoremevpmid 32847 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrpβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ ( I β†Ύ 𝐷) ∈ (pmEvenβ€˜π·))
 
Theoremaltgnsg 32848 The alternating group (pmEvenβ€˜π·) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrpβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ (pmEvenβ€˜π·) ∈ (NrmSGrpβ€˜π‘†))
 
Theoremcyc3evpm 32849 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = ((toCycβ€˜π·) β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ 𝐢 βŠ† 𝐴)
 
Theoremcyc3genpmlem 32850* Lemma for cyc3genpm 32851. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    Β· = (+gβ€˜π‘†)    &   (πœ‘ β†’ 𝐼 ∈ 𝐷)    &   (πœ‘ β†’ 𝐽 ∈ 𝐷)    &   (πœ‘ β†’ 𝐾 ∈ 𝐷)    &   (πœ‘ β†’ 𝐿 ∈ 𝐷)    &   (πœ‘ β†’ 𝐸 = (π‘€β€˜βŸ¨β€œπΌπ½β€βŸ©))    &   (πœ‘ β†’ 𝐹 = (π‘€β€˜βŸ¨β€œπΎπΏβ€βŸ©))    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 β‰  𝐽)    &   (πœ‘ β†’ 𝐾 β‰  𝐿)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ Word 𝐢(𝐸 Β· 𝐹) = (𝑆 Ξ£g 𝑐))
 
Theoremcyc3genpm 32851* The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    β‡’   (𝐷 ∈ Fin β†’ (𝑄 ∈ 𝐴 ↔ βˆƒπ‘€ ∈ Word 𝐢𝑄 = (𝑆 Ξ£g 𝑀)))
 
Theoremcycpmgcl 32852 Cyclic permutations are permutations, similar to cycpmcl 32815, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ (0...𝑁)) β†’ 𝐢 βŠ† 𝐡)
 
Theoremcycpmconjslem1 32853 Lemma for cycpmconjs 32855. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   (πœ‘ β†’ 𝐷 ∈ 𝑉)    &   (πœ‘ β†’ π‘Š ∈ Word 𝐷)    &   (πœ‘ β†’ π‘Š:dom π‘Šβ€“1-1→𝐷)    &   (πœ‘ β†’ (β™―β€˜π‘Š) = 𝑃)    β‡’   (πœ‘ β†’ ((β—‘π‘Š ∘ (π‘€β€˜π‘Š)) ∘ π‘Š) = (( I β†Ύ (0..^𝑃)) cyclShift 1))
 
Theoremcycpmconjslem2 32854* Lemma for cycpmconjs 32855. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 𝑃 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ž(π‘ž:(0..^𝑁)–1-1-onto→𝐷 ∧ ((β—‘π‘ž ∘ 𝑄) ∘ π‘ž) = ((( I β†Ύ (0..^𝑃)) cyclShift 1) βˆͺ ( I β†Ύ (𝑃..^𝑁)))))
 
Theoremcycpmconjs 32855* All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {𝑃}))    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &   π΅ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 𝑃 ∈ (0...𝑁))    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    &   (πœ‘ β†’ 𝑇 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐡 𝑄 = ((𝑝 + 𝑇) βˆ’ 𝑝))
 
Theoremcyc3conja 32856* 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.)
𝐢 = (𝑀 β€œ (β—‘β™― β€œ {3}))    &   π΄ = (pmEvenβ€˜π·)    &   π‘† = (SymGrpβ€˜π·)    &   π‘ = (β™―β€˜π·)    &   π‘€ = (toCycβ€˜π·)    &    + = (+gβ€˜π‘†)    &    βˆ’ = (-gβ€˜π‘†)    &   (πœ‘ β†’ 5 ≀ 𝑁)    &   (πœ‘ β†’ 𝐷 ∈ Fin)    &   (πœ‘ β†’ 𝑄 ∈ 𝐢)    &   (πœ‘ β†’ 𝑇 ∈ 𝐢)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐴 𝑄 = ((𝑝 + 𝑇) βˆ’ 𝑝))
 
21.3.9.10  Signum in an ordered monoid
 
Syntaxcsgns 32857 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 32858* Signum function for a structure. See also df-sgn 15058 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
sgns = (π‘Ÿ ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘Ÿ) ↦ if(π‘₯ = (0gβ€˜π‘Ÿ), 0, if((0gβ€˜π‘Ÿ)(ltβ€˜π‘Ÿ)π‘₯, 1, -1))))
 
Theoremsgnsv 32859* The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑆 = (π‘₯ ∈ 𝐡 ↦ if(π‘₯ = 0 , 0, if( 0 < π‘₯, 1, -1))))
 
Theoremsgnsval 32860 The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡) β†’ (π‘†β€˜π‘‹) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
 
Theoremsgnsf 32861 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    < = (ltβ€˜π‘…)    &   π‘† = (sgnsβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑆:𝐡⟢{-1, 0, 1})
 
21.3.9.11  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 32862 Class notation for the infinitesimal relation.
class β‹˜
 
Syntaxcarchi 32863 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 32864* Define the relation "π‘₯ is infinitesimal with respect to 𝑦 " for a structure 𝑀. (Contributed by Thierry Arnoux, 30-Jan-2018.)
β‹˜ = (𝑀 ∈ V ↦ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (Baseβ€˜π‘€) ∧ 𝑦 ∈ (Baseβ€˜π‘€)) ∧ ((0gβ€˜π‘€)(ltβ€˜π‘€)π‘₯ ∧ βˆ€π‘› ∈ β„• (𝑛(.gβ€˜π‘€)π‘₯)(ltβ€˜π‘€)𝑦))})
 
Definitiondf-archi 32865 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi = {𝑀 ∣ (β‹˜β€˜π‘€) = βˆ…}
 
Theoreminftmrel 32866 The infinitesimal relation for a structure π‘Š. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ (β‹˜β€˜π‘Š) βŠ† (𝐡 Γ— 𝐡))
 
Theoremisinftm 32867* Express π‘₯ is infinitesimal with respect to 𝑦 for a structure π‘Š. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    β‡’   ((π‘Š ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋(β‹˜β€˜π‘Š)π‘Œ ↔ ( 0 < 𝑋 ∧ βˆ€π‘› ∈ β„• (𝑛 Β· 𝑋) < π‘Œ)))
 
Theoremisarchi 32868* Express the predicate "π‘Š is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (β‹˜β€˜π‘Š)    β‡’   (π‘Š ∈ 𝑉 β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 Β¬ π‘₯ < 𝑦))
 
Theorempnfinf 32869 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+ β†’ 𝐴(β‹˜β€˜β„*𝑠)+∞)
 
Theoremxrnarchi 32870 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
Β¬ ℝ*𝑠 ∈ Archi
 
Theoremisarchi2 32871* Alternative way to express the predicate "π‘Š is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    β‡’   ((π‘Š ∈ Toset ∧ π‘Š ∈ Mnd) β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ( 0 < π‘₯ β†’ βˆƒπ‘› ∈ β„• 𝑦 ≀ (𝑛 Β· π‘₯))))
 
Theoremsubmarchi 32872 A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(((π‘Š ∈ Toset ∧ π‘Š ∈ Archi) ∧ 𝐴 ∈ (SubMndβ€˜π‘Š)) β†’ (π‘Š β†Ύs 𝐴) ∈ Archi)
 
Theoremisarchi3 32873* This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    β‡’   (π‘Š ∈ oGrp β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ( 0 < π‘₯ β†’ βˆƒπ‘› ∈ β„• 𝑦 < (𝑛 Β· π‘₯))))
 
Theoremarchirng 32874* Property of Archimedean ordered groups, framing positive π‘Œ between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    &   (πœ‘ β†’ 0 < π‘Œ)    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ β„•0 ((𝑛 Β· 𝑋) < π‘Œ ∧ π‘Œ ≀ ((𝑛 + 1) Β· 𝑋)))
 
Theoremarchirngz 32875* Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ β„€ ((𝑛 Β· 𝑋) < π‘Œ ∧ π‘Œ ≀ ((𝑛 + 1) Β· 𝑋)))
 
Theoremarchiexdiv 32876* In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    β‡’   (((π‘Š ∈ oGrp ∧ π‘Š ∈ Archi) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 0 < 𝑋) β†’ βˆƒπ‘› ∈ β„• π‘Œ < (𝑛 Β· 𝑋))
 
Theoremarchiabllem1a 32877* Lemma for archiabl 32884: In case an archimedean group π‘Š admits a smallest positive element π‘ˆ, then any positive element 𝑋 of π‘Š can be written as (𝑛 Β· π‘ˆ) with 𝑛 ∈ β„•. Since the reciprocal holds for negative elements, π‘Š is then isomorphic to β„€. (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ 0 < π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 0 < π‘₯) β†’ π‘ˆ ≀ π‘₯)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    β‡’   (πœ‘ β†’ βˆƒπ‘› ∈ β„• 𝑋 = (𝑛 Β· π‘ˆ))
 
Theoremarchiabllem1b 32878* Lemma for archiabl 32884. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ 0 < π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 0 < π‘₯) β†’ π‘ˆ ≀ π‘₯)    β‡’   ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ βˆƒπ‘› ∈ β„€ 𝑦 = (𝑛 Β· π‘ˆ))
 
Theoremarchiabllem1 32879* Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ 0 < π‘ˆ)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 0 < π‘₯) β†’ π‘ˆ ≀ π‘₯)    β‡’   (πœ‘ β†’ π‘Š ∈ Abel)
 
Theoremarchiabllem2a 32880* Lemma for archiabl 32884, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 0 < 𝑋)    β‡’   (πœ‘ β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) ≀ 𝑋))
 
Theoremarchiabllem2c 32881* Lemma for archiabl 32884. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ Β¬ (𝑋 + π‘Œ) < (π‘Œ + 𝑋))
 
Theoremarchiabllem2b 32882* Lemma for archiabl 32884. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
 
Theoremarchiabllem2 32883* Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &    ≀ = (leβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    &    Β· = (.gβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ oGrp)    &   (πœ‘ β†’ π‘Š ∈ Archi)    &    + = (+gβ€˜π‘Š)    &   (πœ‘ β†’ (oppgβ€˜π‘Š) ∈ oGrp)    &   ((πœ‘ ∧ π‘Ž ∈ 𝐡 ∧ 0 < π‘Ž) β†’ βˆƒπ‘ ∈ 𝐡 ( 0 < 𝑏 ∧ 𝑏 < π‘Ž))    β‡’   (πœ‘ β†’ π‘Š ∈ Abel)
 
Theoremarchiabl 32884 Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
((π‘Š ∈ oGrp ∧ (oppgβ€˜π‘Š) ∈ oGrp ∧ π‘Š ∈ Archi) β†’ π‘Š ∈ Abel)
 
21.3.9.12  Semiring left modules
 
Syntaxcslmd 32885 Extend class notation with class of all semimodules.
class SLMod
 
Definitiondf-slmd 32886* 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 ∣ [(Baseβ€˜π‘”) / 𝑣][(+gβ€˜π‘”) / π‘Ž][( ·𝑠 β€˜π‘”) / 𝑠][(Scalarβ€˜π‘”) / 𝑓][(Baseβ€˜π‘“) / π‘˜][(+gβ€˜π‘“) / 𝑝][(.rβ€˜π‘“) / 𝑑](𝑓 ∈ SRing ∧ βˆ€π‘ž ∈ π‘˜ βˆ€π‘Ÿ ∈ π‘˜ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘€ ∈ 𝑣 (((π‘Ÿπ‘ π‘€) ∈ 𝑣 ∧ (π‘Ÿπ‘ (π‘€π‘Žπ‘₯)) = ((π‘Ÿπ‘ π‘€)π‘Ž(π‘Ÿπ‘ π‘₯)) ∧ ((π‘žπ‘π‘Ÿ)𝑠𝑀) = ((π‘žπ‘ π‘€)π‘Ž(π‘Ÿπ‘ π‘€))) ∧ (((π‘žπ‘‘π‘Ÿ)𝑠𝑀) = (π‘žπ‘ (π‘Ÿπ‘ π‘€)) ∧ ((1rβ€˜π‘“)𝑠𝑀) = 𝑀 ∧ ((0gβ€˜π‘“)𝑠𝑀) = (0gβ€˜π‘”))))}
 
Theoremisslmd 32887* 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   π‘‚ = (0gβ€˜πΉ)    β‡’   (π‘Š ∈ SLMod ↔ (π‘Š ∈ CMnd ∧ 𝐹 ∈ SRing ∧ βˆ€π‘ž ∈ 𝐾 βˆ€π‘Ÿ ∈ 𝐾 βˆ€π‘₯ ∈ 𝑉 βˆ€π‘€ ∈ 𝑉 (((π‘Ÿ Β· 𝑀) ∈ 𝑉 ∧ (π‘Ÿ Β· (𝑀 + π‘₯)) = ((π‘Ÿ Β· 𝑀) + (π‘Ÿ Β· π‘₯)) ∧ ((π‘ž ⨣ π‘Ÿ) Β· 𝑀) = ((π‘ž Β· 𝑀) + (π‘Ÿ Β· 𝑀))) ∧ (((π‘ž Γ— π‘Ÿ) Β· 𝑀) = (π‘ž Β· (π‘Ÿ Β· 𝑀)) ∧ ( 1 Β· 𝑀) = 𝑀 ∧ (𝑂 Β· 𝑀) = 0 ))))
 
Theoremslmdlema 32888 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    ⨣ = (+gβ€˜πΉ)    &    Γ— = (.rβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &   π‘‚ = (0gβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((𝑅 Β· π‘Œ) ∈ 𝑉 ∧ (𝑅 Β· (π‘Œ + 𝑋)) = ((𝑅 Β· π‘Œ) + (𝑅 Β· 𝑋)) ∧ ((𝑄 ⨣ 𝑅) Β· π‘Œ) = ((𝑄 Β· π‘Œ) + (𝑅 Β· π‘Œ))) ∧ (((𝑄 Γ— 𝑅) Β· π‘Œ) = (𝑄 Β· (𝑅 Β· π‘Œ)) ∧ ( 1 Β· π‘Œ) = π‘Œ ∧ (𝑂 Β· π‘Œ) = 0 )))
 
Theoremlmodslmd 32889 Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(π‘Š ∈ LMod β†’ π‘Š ∈ SLMod)
 
Theoremslmdcmn 32890 A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(π‘Š ∈ SLMod β†’ π‘Š ∈ CMnd)
 
Theoremslmdmnd 32891 A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(π‘Š ∈ SLMod β†’ π‘Š ∈ Mnd)
 
Theoremslmdsrg 32892 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 = (Scalarβ€˜π‘Š)    β‡’   (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
 
Theoremslmdbn0 32893 The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.)
𝐡 = (Baseβ€˜π‘Š)    β‡’   (π‘Š ∈ SLMod β†’ 𝐡 β‰  βˆ…)
 
Theoremslmdacl 32894 Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    + = (+gβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐾) β†’ (𝑋 + π‘Œ) ∈ 𝐾)
 
Theoremslmdmcl 32895 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 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    Β· = (.rβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐾) β†’ (𝑋 Β· π‘Œ) ∈ 𝐾)
 
Theoremslmdsn0 32896 The set of scalars in a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) (Proof shortened by AV, 10-Jan-2023.)
𝐹 = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    β‡’   (π‘Š ∈ SLMod β†’ 𝐡 β‰  βˆ…)
 
Theoremslmdvacl 32897 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) β†’ (𝑋 + π‘Œ) ∈ 𝑉)
 
Theoremslmdass 32898 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 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) β†’ ((𝑋 + π‘Œ) + 𝑍) = (𝑋 + (π‘Œ + 𝑍)))
 
Theoremslmdvscl 32899 Closure of scalar product for a semiring left module. (hvmulcl 30810 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑅 Β· 𝑋) ∈ 𝑉)
 
Theoremslmdvsdi 32900 Distributive law for scalar product. (ax-hvdistr1 30805 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑅 Β· (𝑋 + π‘Œ)) = ((𝑅 Β· 𝑋) + (𝑅 Β· π‘Œ)))
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