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Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremofldlt1 31901 In an ordered field, the ring unity is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0gβ€˜πΉ)    &    1 = (1rβ€˜πΉ)    &    < = (ltβ€˜πΉ)    β‡’   (𝐹 ∈ oField β†’ 0 < 1 )
 
Theoremofldchr 31902 The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.) (Proof shortened by AV, 6-Oct-2020.)
(𝐹 ∈ oField β†’ (chrβ€˜πΉ) = 0)
 
Theoremsuborng 31903 Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝑅 ∈ oRing ∧ (𝑅 β†Ύs 𝐴) ∈ Ring) β†’ (𝑅 β†Ύs 𝐴) ∈ oRing)
 
Theoremsubofld 31904 Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝐹 ∈ oField ∧ (𝐹 β†Ύs 𝐴) ∈ Field) β†’ (𝐹 β†Ύs 𝐴) ∈ oField)
 
Theoremisarchiofld 31905* Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐡 = (Baseβ€˜π‘Š)    &   π» = (β„€RHomβ€˜π‘Š)    &    < = (ltβ€˜π‘Š)    β‡’   (π‘Š ∈ oField β†’ (π‘Š ∈ Archi ↔ βˆ€π‘₯ ∈ 𝐡 βˆƒπ‘› ∈ β„• π‘₯ < (π»β€˜π‘›)))
 
21.3.9.18  Ring homomorphisms - misc additions
 
Theoremrhmdvd 31906 A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
π‘ˆ = (Unitβ€˜π‘†)    &   π‘‹ = (Baseβ€˜π‘…)    &    / = (/rβ€˜π‘†)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) ∧ ((πΉβ€˜π΅) ∈ π‘ˆ ∧ (πΉβ€˜πΆ) ∈ π‘ˆ)) β†’ ((πΉβ€˜π΄) / (πΉβ€˜π΅)) = ((πΉβ€˜(𝐴 Β· 𝐢)) / (πΉβ€˜(𝐡 Β· 𝐢))))
 
Theoremkerunit 31907 If a unit element lies in the kernel of a ring homomorphism, then 0 = 1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.)
π‘ˆ = (Unitβ€˜π‘…)    &    0 = (0gβ€˜π‘†)    &    1 = (1rβ€˜π‘†)    β‡’   ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (π‘ˆ ∩ (◑𝐹 β€œ { 0 })) β‰  βˆ…) β†’ 1 = 0 )
 
21.3.9.19  Scalar restriction operation
 
Syntaxcresv 31908 Extend class notation with the scalar restriction operation.
class β†Ύv
 
Definitiondf-resv 31909* Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.)
β†Ύv = (𝑀 ∈ V, π‘₯ ∈ V ↦ if((Baseβ€˜(Scalarβ€˜π‘€)) βŠ† π‘₯, 𝑀, (𝑀 sSet ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘€) β†Ύs π‘₯)⟩)))
 
Theoremreldmresv 31910 The scalar restriction is a proper operator, so it can be used with ovprc1 7388. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom β†Ύv
 
Theoremresvval 31911 Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (π‘Š β†Ύv 𝐴)    &   πΉ = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = if(𝐡 βŠ† 𝐴, π‘Š, (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩)))
 
Theoremresvid2 31912 General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (π‘Š β†Ύv 𝐴)    &   πΉ = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    β‡’   ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = π‘Š)
 
Theoremresvval2 31913 Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (π‘Š β†Ύv 𝐴)    &   πΉ = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    β‡’   ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ 𝑋 ∧ 𝐴 ∈ π‘Œ) β†’ 𝑅 = (π‘Š sSet ⟨(Scalarβ€˜ndx), (𝐹 β†Ύs 𝐴)⟩))
 
Theoremresvsca 31914 Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (π‘Š β†Ύv 𝐴)    &   πΉ = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐹 β†Ύs 𝐴) = (Scalarβ€˜π‘…))
 
Theoremresvlem 31915 Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝑅 = (π‘Š β†Ύv 𝐴)    &   πΆ = (πΈβ€˜π‘Š)    &   πΈ = Slot (πΈβ€˜ndx)    &   (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)    β‡’   (𝐴 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘…))
 
TheoremresvlemOLD 31916 Obsolete version of resvlem 31915 as of 31-Oct-2024. Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑅 = (π‘Š β†Ύv 𝐴)    &   πΆ = (πΈβ€˜π‘Š)    &   πΈ = Slot 𝑁    &   π‘ ∈ β„•    &   π‘ β‰  5    β‡’   (𝐴 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘…))
 
Theoremresvbas 31917 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 β†Ύv 𝐴)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜π»))
 
TheoremresvbasOLD 31918 Obsolete proof of resvbas 31917 as of 31-Oct-2024. Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝐺 β†Ύv 𝐴)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜π»))
 
Theoremresvplusg 31919 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    + = (+gβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ + = (+gβ€˜π»))
 
TheoremresvplusgOLD 31920 Obsolete proof of resvplusg 31919 as of 31-Oct-2024. +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    + = (+gβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ + = (+gβ€˜π»))
 
Theoremresvvsca 31921 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Proof shortened by AV, 31-Oct-2024.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    Β· = ( ·𝑠 β€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ Β· = ( ·𝑠 β€˜π»))
 
TheoremresvvscaOLD 31922 Obsolete proof of resvvsca 31921 as of 31-Oct-2024. ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    Β· = ( ·𝑠 β€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ Β· = ( ·𝑠 β€˜π»))
 
Theoremresvmulr 31923 .r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    Β· = (.rβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ Β· = (.rβ€˜π»))
 
TheoremresvmulrOLD 31924 Obsolete proof of resvmulr 31923 as of 31-Oct-2024. ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    Β· = (.rβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ Β· = (.rβ€˜π»))
 
Theoremresv0g 31925 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    0 = (0gβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ 0 = (0gβ€˜π»))
 
Theoremresv1r 31926 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺 β†Ύv 𝐴)    &    1 = (1rβ€˜πΊ)    β‡’   (𝐴 ∈ 𝑉 β†’ 1 = (1rβ€˜π»))
 
Theoremresvcmn 31927 Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺 β†Ύv 𝐴)    β‡’   (𝐴 ∈ 𝑉 β†’ (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
 
21.3.9.20  The commutative ring of gaussian integers
 
Theoremgzcrng 31928 The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
(β„‚fld β†Ύs β„€[i]) ∈ CRing
 
21.3.9.21  The archimedean ordered field of real numbers
 
Theoremreofld 31929 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
ℝfld ∈ oField
 
Theoremnn0omnd 31930 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(β„‚fld β†Ύs β„•0) ∈ oMnd
 
Theoremrearchi 31931 The field of the real numbers is Archimedean. See also arch 12343. (Contributed by Thierry Arnoux, 9-Apr-2018.)
ℝfld ∈ Archi
 
Theoremnn0archi 31932 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(β„‚fld β†Ύs β„•0) ∈ Archi
 
Theoremxrge0slmod 31933 The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐺 = (ℝ*𝑠 β†Ύs (0[,]+∞))    &   π‘Š = (𝐺 β†Ύv (0[,)+∞))    β‡’   π‘Š ∈ SLMod
 
21.3.9.22  The quotient map and quotient modules
 
Theoremqusker 31934* The kernel of a quotient map. (Contributed by Thierry Arnoux, 20-May-2023.)
𝑉 = (Baseβ€˜π‘€)    &   πΉ = (π‘₯ ∈ 𝑉 ↦ [π‘₯](𝑀 ~QG 𝐺))    &   π‘ = (𝑀 /s (𝑀 ~QG 𝐺))    &    0 = (0gβ€˜π‘)    β‡’   (𝐺 ∈ (NrmSGrpβ€˜π‘€) β†’ (◑𝐹 β€œ { 0 }) = 𝐺)
 
Theoremeqgvscpbl 31935 The left coset equivalence relation is compatible with the scalar multiplication operation. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐡 = (Baseβ€˜π‘€)    &    ∼ = (𝑀 ~QG 𝐺)    &   π‘† = (Baseβ€˜(Scalarβ€˜π‘€))    &    Β· = ( ·𝑠 β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ LMod)    &   (πœ‘ β†’ 𝐺 ∈ (LSubSpβ€˜π‘€))    &   (πœ‘ β†’ 𝐾 ∈ 𝑆)    β‡’   (πœ‘ β†’ (𝑋 ∼ π‘Œ β†’ (𝐾 Β· 𝑋) ∼ (𝐾 Β· π‘Œ)))
 
Theoremqusvscpbl 31936* The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐡 = (Baseβ€˜π‘€)    &    ∼ = (𝑀 ~QG 𝐺)    &   π‘† = (Baseβ€˜(Scalarβ€˜π‘€))    &    Β· = ( ·𝑠 β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ LMod)    &   (πœ‘ β†’ 𝐺 ∈ (LSubSpβ€˜π‘€))    &   (πœ‘ β†’ 𝐾 ∈ 𝑆)    &   π‘ = (𝑀 /s (𝑀 ~QG 𝐺))    &    βˆ™ = ( ·𝑠 β€˜π‘)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ [π‘₯](𝑀 ~QG 𝐺))    &   (πœ‘ β†’ π‘ˆ ∈ 𝐡)    &   (πœ‘ β†’ 𝑉 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((πΉβ€˜π‘ˆ) = (πΉβ€˜π‘‰) β†’ (πΉβ€˜(𝐾 Β· π‘ˆ)) = (πΉβ€˜(𝐾 Β· 𝑉))))
 
Theoremqusscaval 31937 Value of the scalar multiplication operation on the quotient structure. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐡 = (Baseβ€˜π‘€)    &    ∼ = (𝑀 ~QG 𝐺)    &   π‘† = (Baseβ€˜(Scalarβ€˜π‘€))    &    Β· = ( ·𝑠 β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ LMod)    &   (πœ‘ β†’ 𝐺 ∈ (LSubSpβ€˜π‘€))    &   (πœ‘ β†’ 𝐾 ∈ 𝑆)    &   π‘ = (𝑀 /s (𝑀 ~QG 𝐺))    &    βˆ™ = ( ·𝑠 β€˜π‘)    β‡’   ((πœ‘ ∧ 𝐾 ∈ 𝑆 ∧ 𝑋 ∈ 𝐡) β†’ (𝐾 βˆ™ [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 Β· 𝑋)](𝑀 ~QG 𝐺))
 
Theoremimaslmod 31938* The image structure of a left module is a left module. (Contributed by Thierry Arnoux, 15-May-2023.)
(πœ‘ β†’ 𝑁 = (𝐹 β€œs 𝑀))    &   π‘‰ = (Baseβ€˜π‘€)    &   π‘† = (Baseβ€˜(Scalarβ€˜π‘€))    &    + = (+gβ€˜π‘€)    &    Β· = ( ·𝑠 β€˜π‘€)    &    0 = (0gβ€˜π‘€)    &   (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)    &   ((πœ‘ ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ π‘ž ∈ 𝑉)) β†’ (((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ∧ (πΉβ€˜π‘) = (πΉβ€˜π‘ž)) β†’ (πΉβ€˜(π‘Ž + 𝑏)) = (πΉβ€˜(𝑝 + π‘ž))))    &   ((πœ‘ ∧ (π‘˜ ∈ 𝑆 ∧ π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) β†’ ((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) β†’ (πΉβ€˜(π‘˜ Β· π‘Ž)) = (πΉβ€˜(π‘˜ Β· 𝑏))))    &   (πœ‘ β†’ 𝑀 ∈ LMod)    β‡’   (πœ‘ β†’ 𝑁 ∈ LMod)
 
Theoremquslmod 31939 If 𝐺 is a submodule in 𝑀, then 𝑁 = 𝑀 / 𝐺 is a left module, called the quotient module of 𝑀 by 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.)
𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &   π‘‰ = (Baseβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ LMod)    &   (πœ‘ β†’ 𝐺 ∈ (LSubSpβ€˜π‘€))    β‡’   (πœ‘ β†’ 𝑁 ∈ LMod)
 
Theoremquslmhm 31940* If 𝐺 is a submodule of 𝑀, then the "natural map" from elements to their cosets is a left module homomorphism from 𝑀 to 𝑀 / 𝐺. (Contributed by Thierry Arnoux, 18-May-2023.)
𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &   π‘‰ = (Baseβ€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ LMod)    &   (πœ‘ β†’ 𝐺 ∈ (LSubSpβ€˜π‘€))    &   πΉ = (π‘₯ ∈ 𝑉 ↦ [π‘₯](𝑀 ~QG 𝐺))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑀 LMHom 𝑁))
 
Theoremecxpid 31941 The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
(𝑋 ∈ 𝐴 β†’ [𝑋](𝐴 Γ— 𝐴) = 𝐴)
 
Theoremeqg0el 31942 Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
∼ = (𝐺 ~QG 𝐻)    β‡’   ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrpβ€˜πΊ)) β†’ ([𝑋] ∼ = 𝐻 ↔ 𝑋 ∈ 𝐻))
 
Theoremqsxpid 31943 The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)
(𝐴 β‰  βˆ… β†’ (𝐴 / (𝐴 Γ— 𝐴)) = {𝐴})
 
Theoremqusxpid 31944 The Group quotient equivalence relation for the whole group is the cartesian product, i.e. all elements are in the same equivalence class. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝐡 = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ Grp β†’ (𝐺 ~QG 𝐡) = (𝐡 Γ— 𝐡))
 
Theoremqustriv 31945 The quotient of a group 𝐺 by itself is trivial. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝐡))    β‡’   (𝐺 ∈ Grp β†’ (Baseβ€˜π‘„) = {𝐡})
 
Theoremqustrivr 31946 Converse of qustriv 31945. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝐻))    β‡’   ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrpβ€˜πΊ) ∧ (Baseβ€˜π‘„) = {𝐻}) β†’ 𝐻 = 𝐡)
 
21.3.9.23  The ring of integers modulo ` N `
 
Theoremfermltlchr 31947 A generalization of Fermat's little theorem in a commutative ring 𝐹 of prime characteristic. See fermltl 16590. (Contributed by Thierry Arnoux, 9-Jan-2024.)
𝑃 = (chrβ€˜πΉ)    &   π΅ = (Baseβ€˜πΉ)    &    ↑ = (.gβ€˜(mulGrpβ€˜πΉ))    &   π΄ = ((β„€RHomβ€˜πΉ)β€˜πΈ)    &   (πœ‘ β†’ 𝑃 ∈ β„™)    &   (πœ‘ β†’ 𝐸 ∈ β„€)    &   (πœ‘ β†’ 𝐹 ∈ CRing)    β‡’   (πœ‘ β†’ (𝑃 ↑ 𝐴) = 𝐴)
 
Theoremznfermltl 31948 Fermat's little theorem in β„€/nβ„€. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑍 = (β„€/nβ„€β€˜π‘ƒ)    &   π΅ = (Baseβ€˜π‘)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘))    β‡’   ((𝑃 ∈ β„™ ∧ 𝐴 ∈ 𝐡) β†’ (𝑃 ↑ 𝐴) = 𝐴)
 
21.3.9.24  Independent sets and families
 
Theoremislinds5 31949* A set is linearly independent if and only if it has no non-trivial representations of zero. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐡 = (Baseβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘‚ = (0gβ€˜π‘Š)    &    0 = (0gβ€˜πΉ)    β‡’   ((π‘Š ∈ LMod ∧ 𝑉 βŠ† 𝐡) β†’ (𝑉 ∈ (LIndSβ€˜π‘Š) ↔ βˆ€π‘Ž ∈ (𝐾 ↑m 𝑉)((π‘Ž finSupp 0 ∧ (π‘Š Ξ£g (𝑣 ∈ 𝑉 ↦ ((π‘Žβ€˜π‘£) Β· 𝑣))) = 𝑂) β†’ π‘Ž = (𝑉 Γ— { 0 }))))
 
Theoremellspds 31950* Variation on ellspd 21131. (Contributed by Thierry Arnoux, 18-May-2023.)
𝑁 = (LSpanβ€˜π‘€)    &   π΅ = (Baseβ€˜π‘€)    &   πΎ = (Baseβ€˜π‘†)    &   π‘† = (Scalarβ€˜π‘€)    &    0 = (0gβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ LMod)    &   (πœ‘ β†’ 𝑉 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (𝑋 ∈ (π‘β€˜π‘‰) ↔ βˆƒπ‘Ž ∈ (𝐾 ↑m 𝑉)(π‘Ž finSupp 0 ∧ 𝑋 = (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ ((π‘Žβ€˜π‘£) Β· 𝑣))))))
 
Theorem0ellsp 31951 Zero is in all spans. (Contributed by Thierry Arnoux, 8-May-2023.)
0 = (0gβ€˜π‘Š)    &   π΅ = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑆 βŠ† 𝐡) β†’ 0 ∈ (π‘β€˜π‘†))
 
Theorem0nellinds 31952 The group identity cannot be an element of an independent set. (Contributed by Thierry Arnoux, 8-May-2023.)
0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ LVec ∧ 𝐹 ∈ (LIndSβ€˜π‘Š)) β†’ Β¬ 0 ∈ 𝐹)
 
Theoremrspsnel 31953* Membership in a principal ideal. Analogous to lspsnel 20387. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (𝐼 ∈ (πΎβ€˜{𝑋}) ↔ βˆƒπ‘₯ ∈ 𝐡 𝐼 = (π‘₯ Β· 𝑋)))
 
Theoremrspsnid 31954 A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐡) β†’ 𝐺 ∈ (πΎβ€˜{𝐺}))
 
Theoremelrsp 31955* Write the elements of a ring span as finite linear combinations. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑁 = (RSpanβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (𝑋 ∈ (π‘β€˜πΌ) ↔ βˆƒπ‘Ž ∈ (𝐡 ↑m 𝐼)(π‘Ž finSupp 0 ∧ 𝑋 = (𝑅 Ξ£g (𝑖 ∈ 𝐼 ↦ ((π‘Žβ€˜π‘–) Β· 𝑖))))))
 
Theoremrspidlid 31956 The ideal span of an ideal is the ideal itself. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝐾 = (RSpanβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ) β†’ (πΎβ€˜πΌ) = 𝐼)
 
Theorempidlnz 31957 A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ (πΎβ€˜{𝑋}) β‰  { 0 })
 
Theoremlbslsp 31958* Any element of a left module 𝑀 can be expressed as a linear combination of the elements of a basis 𝑉 of 𝑀. (Contributed by Thierry Arnoux, 3-Aug-2023.)
𝐡 = (Baseβ€˜π‘€)    &   πΎ = (Baseβ€˜π‘†)    &   π‘† = (Scalarβ€˜π‘€)    &    0 = (0gβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘€)    &   (πœ‘ β†’ 𝑀 ∈ LMod)    &   (πœ‘ β†’ 𝑉 ∈ (LBasisβ€˜π‘€))    β‡’   (πœ‘ β†’ (𝑋 ∈ 𝐡 ↔ βˆƒπ‘Ž ∈ (𝐾 ↑m 𝑉)(π‘Ž finSupp 0 ∧ 𝑋 = (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ ((π‘Žβ€˜π‘£) Β· 𝑣))))))
 
Theoremlindssn 31959 Any singleton of a nonzero element is an independent set. (Contributed by Thierry Arnoux, 5-Aug-2023.)
𝐡 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   ((π‘Š ∈ LVec ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ {𝑋} ∈ (LIndSβ€˜π‘Š))
 
Theoremlindflbs 31960 Conditions for an independent family to be a basis. (Contributed by Thierry Arnoux, 21-Jul-2023.)
𝐡 = (Baseβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘† = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘‚ = (0gβ€˜π‘Š)    &    0 = (0gβ€˜π‘†)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑆 ∈ NzRing)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:𝐼–1-1→𝐡)    β‡’   (πœ‘ β†’ (ran 𝐹 ∈ (LBasisβ€˜π‘Š) ↔ (𝐹 LIndF π‘Š ∧ (π‘β€˜ran 𝐹) = 𝐡)))
 
Theoremlinds2eq 31961 Deduce equality of elements in an independent set. (Contributed by Thierry Arnoux, 18-Jul-2023.)
𝐹 = (Baseβ€˜(Scalarβ€˜π‘Š))    &    Β· = ( ·𝑠 β€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜(Scalarβ€˜π‘Š))    &   (πœ‘ β†’ π‘Š ∈ LVec)    &   (πœ‘ β†’ 𝐡 ∈ (LIndSβ€˜π‘Š))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝐾 ∈ 𝐹)    &   (πœ‘ β†’ 𝐿 ∈ 𝐹)    &   (πœ‘ β†’ 𝐾 β‰  0 )    &   (πœ‘ β†’ (𝐾 Β· 𝑋) = (𝐿 Β· π‘Œ))    β‡’   (πœ‘ β†’ (𝑋 = π‘Œ ∧ 𝐾 = 𝐿))
 
Theoremlindfpropd 31962* Property deduction for linearly independent families. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))    &   (πœ‘ β†’ (Baseβ€˜(Scalarβ€˜πΎ)) = (Baseβ€˜(Scalarβ€˜πΏ)))    &   (πœ‘ β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))    &   ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ (Baseβ€˜πΎ))    &   ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝐿 ∈ π‘Š)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    β‡’   (πœ‘ β†’ (𝑋 LIndF 𝐾 ↔ 𝑋 LIndF 𝐿))
 
Theoremlindspropd 31963* Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))    &   (πœ‘ β†’ (Baseβ€˜(Scalarβ€˜πΎ)) = (Baseβ€˜(Scalarβ€˜πΏ)))    &   (πœ‘ β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))    &   ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ (Baseβ€˜πΎ))    &   ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜πΎ)) ∧ 𝑦 ∈ (Baseβ€˜πΎ))) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝐾 ∈ 𝑉)    &   (πœ‘ β†’ 𝐿 ∈ π‘Š)    β‡’   (πœ‘ β†’ (LIndSβ€˜πΎ) = (LIndSβ€˜πΏ))
 
21.3.9.25  Subgroup sum / Sumset / Minkowski sum

The sumset (also called the Minkowski sum) of two subsets 𝐴 and 𝐡, is defined to be the set of all sums of an element from 𝐴 with an element from 𝐡.

The sumset operation can be used for both group (additive) operations and ring (multiplicative) operations.

 
Theoremelgrplsmsn 31964* Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑍 ∈ (𝐴 βŠ• {𝑋}) ↔ βˆƒπ‘₯ ∈ 𝐴 𝑍 = (π‘₯ + 𝑋)))
 
Theoremlsmsnorb 31965* The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19019. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    ∼ = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ βˆƒπ‘” ∈ 𝐴 (𝑔 + π‘₯) = 𝑦)}    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐴 βŠ• {𝑋}) = [𝑋] ∼ )
 
Theoremlsmsnorb2 31966* The sumset of a single element with a group is the element's orbit by the group action. See gaorb 19019. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐡 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    ∼ = {⟨π‘₯, π‘¦βŸ© ∣ ({π‘₯, 𝑦} βŠ† 𝐡 ∧ βˆƒπ‘” ∈ 𝐴 (π‘₯ + 𝑔) = 𝑦)}    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ ({𝑋} βŠ• 𝐴) = [𝑋] ∼ )
 
Theoremelringlsm 31967* Membership in a product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    Γ— = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝐸 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐹 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (𝑍 ∈ (𝐸 Γ— 𝐹) ↔ βˆƒπ‘₯ ∈ 𝐸 βˆƒπ‘¦ ∈ 𝐹 𝑍 = (π‘₯ Β· 𝑦)))
 
Theoremelringlsmd 31968 Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    Γ— = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝐸 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐹 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑋 ∈ 𝐸)    &   (πœ‘ β†’ π‘Œ ∈ 𝐹)    β‡’   (πœ‘ β†’ (𝑋 Β· π‘Œ) ∈ (𝐸 Γ— 𝐹))
 
Theoremringlsmss 31969 Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    Γ— = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐸 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐹 βŠ† 𝐡)    β‡’   (πœ‘ β†’ (𝐸 Γ— 𝐹) βŠ† 𝐡)
 
Theoremringlsmss1 31970 The product of an ideal 𝐼 of a commutative ring 𝑅 with some set E is a subset of the ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
𝐡 = (Baseβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    Γ— = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝐸 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜π‘…))    β‡’   (πœ‘ β†’ (𝐼 Γ— 𝐸) βŠ† 𝐼)
 
Theoremringlsmss2 31971 The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.)
𝐡 = (Baseβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    Γ— = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐸 βŠ† 𝐡)    &   (πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜π‘…))    β‡’   (πœ‘ β†’ (𝐸 Γ— 𝐼) βŠ† 𝐼)
 
Theoremlsmsnpridl 31972 The product of the ring with a single element is equal to the principal ideal generated by that element. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    Γ— = (LSSumβ€˜πΊ)    &   πΎ = (RSpanβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐡 Γ— {𝑋}) = (πΎβ€˜{𝑋}))
 
Theoremlsmsnidl 31973 The product of the ring with a single element is a principal ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    Γ— = (LSSumβ€˜πΊ)    &   πΎ = (RSpanβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐡 Γ— {𝑋}) ∈ (LPIdealβ€˜π‘…))
 
Theoremlsmidllsp 31974 The sum of two ideals is the ideal generated by their union. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    βŠ• = (LSSumβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜π‘…))    &   (πœ‘ β†’ 𝐽 ∈ (LIdealβ€˜π‘…))    β‡’   (πœ‘ β†’ (𝐼 βŠ• 𝐽) = (πΎβ€˜(𝐼 βˆͺ 𝐽)))
 
Theoremlsmidl 31975 The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    βŠ• = (LSSumβ€˜π‘…)    &   πΎ = (RSpanβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜π‘…))    &   (πœ‘ β†’ 𝐽 ∈ (LIdealβ€˜π‘…))    β‡’   (πœ‘ β†’ (𝐼 βŠ• 𝐽) ∈ (LIdealβ€˜π‘…))
 
Theoremlsmssass 31976 Group sum is associative, subset version (see lsmass 19380). (Contributed by Thierry Arnoux, 1-Jun-2024.)
βŠ• = (LSSumβ€˜πΊ)    &   π΅ = (Baseβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Mnd)    &   (πœ‘ β†’ 𝑅 βŠ† 𝐡)    &   (πœ‘ β†’ 𝑇 βŠ† 𝐡)    &   (πœ‘ β†’ π‘ˆ βŠ† 𝐡)    β‡’   (πœ‘ β†’ ((𝑅 βŠ• 𝑇) βŠ• π‘ˆ) = (𝑅 βŠ• (𝑇 βŠ• π‘ˆ)))
 
Theoremgrplsm0l 31977 Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐡 = (Baseβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ Grp ∧ 𝐴 βŠ† 𝐡 ∧ 𝐴 β‰  βˆ…) β†’ ({ 0 } βŠ• 𝐴) = 𝐴)
 
Theoremgrplsmid 31978 The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
βŠ• = (LSSumβ€˜πΊ)    β‡’   ((𝐴 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ 𝐴) β†’ ({𝑋} βŠ• 𝐴) = 𝐴)
 
21.3.9.26  The quotient map
 
Theoremquslsm 31979 Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐡 = (Baseβ€˜πΊ)    &    βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑆 ∈ (SubGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ [𝑋](𝐺 ~QG 𝑆) = ({𝑋} βŠ• 𝑆))
 
Theoremqusima 31980* The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝑁))    &    βŠ• = (LSSumβ€˜πΊ)    &   πΈ = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))    &   πΉ = (π‘₯ ∈ 𝐡 ↦ [π‘₯](𝐺 ~QG 𝑁))    &   (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝐻 ∈ 𝑆)    &   (πœ‘ β†’ 𝑆 βŠ† (SubGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ (πΈβ€˜π») = (𝐹 β€œ 𝐻))
 
Theoremnsgqus0 31981 A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    β‡’   ((𝑁 ∈ (NrmSGrpβ€˜πΊ) ∧ 𝐹 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑁 ∈ 𝐹)
 
Theoremnsgmgclem 31982* Lemma for nsgmgc 31983. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝑁))    &    βŠ• = (LSSumβ€˜πΊ)    &   (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))    &   (πœ‘ β†’ 𝐹 ∈ (SubGrpβ€˜π‘„))    β‡’   (πœ‘ β†’ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝐹} ∈ (SubGrpβ€˜πΊ))
 
Theoremnsgmgc 31983* There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘† = {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž}    &   π‘‡ = (SubGrpβ€˜π‘„)    &   π½ = (𝑉MGalConnπ‘Š)    &   π‘‰ = (toIncβ€˜π‘†)    &   π‘Š = (toIncβ€˜π‘‡)    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝑁))    &    βŠ• = (LSSumβ€˜πΊ)    &   πΈ = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))    &   πΉ = (𝑓 ∈ 𝑇 ↦ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})    &   (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ 𝐸𝐽𝐹)
 
Theoremnsgqusf1olem1 31984* Lemma for nsgqusf1o 31987. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘† = {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž}    &   π‘‡ = (SubGrpβ€˜π‘„)    &    ≀ = (leβ€˜(toIncβ€˜π‘†))    &    ≲ = (leβ€˜(toIncβ€˜π‘‡))    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝑁))    &    βŠ• = (LSSumβ€˜πΊ)    &   πΈ = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))    &   πΉ = (𝑓 ∈ 𝑇 ↦ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})    &   (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))    β‡’   (((πœ‘ ∧ β„Ž ∈ (SubGrpβ€˜πΊ)) ∧ 𝑁 βŠ† β„Ž) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ 𝑇)
 
Theoremnsgqusf1olem2 31985* Lemma for nsgqusf1o 31987. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘† = {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž}    &   π‘‡ = (SubGrpβ€˜π‘„)    &    ≀ = (leβ€˜(toIncβ€˜π‘†))    &    ≲ = (leβ€˜(toIncβ€˜π‘‡))    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝑁))    &    βŠ• = (LSSumβ€˜πΊ)    &   πΈ = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))    &   πΉ = (𝑓 ∈ 𝑇 ↦ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})    &   (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ ran 𝐸 = 𝑇)
 
Theoremnsgqusf1olem3 31986* Lemma for nsgqusf1o 31987. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘† = {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž}    &   π‘‡ = (SubGrpβ€˜π‘„)    &    ≀ = (leβ€˜(toIncβ€˜π‘†))    &    ≲ = (leβ€˜(toIncβ€˜π‘‡))    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝑁))    &    βŠ• = (LSSumβ€˜πΊ)    &   πΈ = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))    &   πΉ = (𝑓 ∈ 𝑇 ↦ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})    &   (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ ran 𝐹 = 𝑆)
 
Theoremnsgqusf1o 31987* The canonical projection homomorphism 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐡 = (Baseβ€˜πΊ)    &   π‘† = {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž}    &   π‘‡ = (SubGrpβ€˜π‘„)    &    ≀ = (leβ€˜(toIncβ€˜π‘†))    &    ≲ = (leβ€˜(toIncβ€˜π‘‡))    &   π‘„ = (𝐺 /s (𝐺 ~QG 𝑁))    &    βŠ• = (LSSumβ€˜πΊ)    &   πΈ = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))    &   πΉ = (𝑓 ∈ 𝑇 ↦ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})    &   (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))    β‡’   (πœ‘ β†’ 𝐸:𝑆–1-1-onto→𝑇)
 
21.3.9.27  Ideals
 
Theoremintlidl 31988 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
((𝑅 ∈ Ring ∧ 𝐢 β‰  βˆ… ∧ 𝐢 βŠ† (LIdealβ€˜π‘…)) β†’ ∩ 𝐢 ∈ (LIdealβ€˜π‘…))
 
Theoremrhmpreimaidl 31989 The preimage of an ideal by a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝐼 = (LIdealβ€˜π‘…)    β‡’   ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdealβ€˜π‘†)) β†’ (◑𝐹 β€œ 𝐽) ∈ 𝐼)
 
Theoremkerlidl 31990 The kernel of a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐼 = (LIdealβ€˜π‘…)    &    0 = (0gβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingHom 𝑆) β†’ (◑𝐹 β€œ { 0 }) ∈ 𝐼)
 
Theorem0ringidl 31991 The zero ideal is the only ideal of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (β™―β€˜π΅) = 1) β†’ (LIdealβ€˜π‘…) = {{ 0 }})
 
Theoremelrspunidl 31992* Elementhood to the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝑁 = (RSpanβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑆 βŠ† (LIdealβ€˜π‘…))    β‡’   (πœ‘ β†’ (𝑋 ∈ (π‘β€˜βˆͺ 𝑆) ↔ βˆƒπ‘Ž ∈ (𝐡 ↑m 𝑆)(π‘Ž finSupp 0 ∧ 𝑋 = (𝑅 Ξ£g π‘Ž) ∧ βˆ€π‘˜ ∈ 𝑆 (π‘Žβ€˜π‘˜) ∈ π‘˜)))
 
Theoremlidlincl 31993 Ideals are closed under intersection. (Contributed by Thierry Arnoux, 5-Jul-2024.)
π‘ˆ = (LIdealβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ π‘ˆ ∧ 𝐽 ∈ π‘ˆ) β†’ (𝐼 ∩ 𝐽) ∈ π‘ˆ)
 
Theoremidlinsubrg 31994 The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝑆 = (𝑅 β†Ύs 𝐴)    &   π‘ˆ = (LIdealβ€˜π‘…)    &   π‘‰ = (LIdealβ€˜π‘†)    β‡’   ((𝐴 ∈ (SubRingβ€˜π‘…) ∧ 𝐼 ∈ π‘ˆ) β†’ (𝐼 ∩ 𝐴) ∈ 𝑉)
 
Theoremrhmimaidl 31995 The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝐡 = (Baseβ€˜π‘†)    &   π‘‡ = (LIdealβ€˜π‘…)    &   π‘ˆ = (LIdealβ€˜π‘†)    β‡’   ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐡 ∧ 𝐼 ∈ 𝑇) β†’ (𝐹 β€œ 𝐼) ∈ π‘ˆ)
 
21.3.9.28  Prime Ideals
 
Syntaxcprmidl 31996 Extend class notation with the class of prime ideals.
class PrmIdeal
 
Definitiondf-prmidl 31997* Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐡 βŠ† 𝐼 for ideals 𝐴 and 𝐡, either 𝐴 βŠ† 𝐼 or 𝐡 βŠ† 𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 32002 and isprmidlc 32009. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
PrmIdeal = (π‘Ÿ ∈ Ring ↦ {𝑖 ∈ (LIdealβ€˜π‘Ÿ) ∣ (𝑖 β‰  (Baseβ€˜π‘Ÿ) ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘Ÿ)βˆ€π‘ ∈ (LIdealβ€˜π‘Ÿ)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘Ÿ)𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
 
Theoremprmidlval 31998* The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (PrmIdealβ€˜π‘…) = {𝑖 ∈ (LIdealβ€˜π‘…) ∣ (𝑖 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
 
Theoremisprmidl 31999* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (𝑃 ∈ (PrmIdealβ€˜π‘…) ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ 𝑃 β‰  𝐡 ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘…)βˆ€π‘ ∈ (LIdealβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
 
Theoremprmidlnr 32000 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑃 β‰  𝐡)
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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 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46997
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