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Theorem List for Metamath Proof Explorer - 31901-32000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresvval2 31901 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 31902 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 31903 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 31904 Obsolete version of resvlem 31903 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 31905 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 31906 Obsolete proof of resvbas 31905 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 31907 +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 31908 Obsolete proof of resvplusg 31907 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 31909 ·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 31910 Obsolete proof of resvvsca 31909 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 31911 .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 31912 Obsolete proof of resvmulr 31911 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 31913 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 31914 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 31915 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 31916 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 31917 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
ℝfld ∈ oField
 
Theoremnn0omnd 31918 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(β„‚fld β†Ύs β„•0) ∈ oMnd
 
Theoremrearchi 31919 The field of the real numbers is Archimedean. See also arch 12344. (Contributed by Thierry Arnoux, 9-Apr-2018.)
ℝfld ∈ Archi
 
Theoremnn0archi 31920 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(β„‚fld β†Ύs β„•0) ∈ Archi
 
Theoremxrge0slmod 31921 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 31922* 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 31923 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 31924* 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 31925 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 31926* 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 31927 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 31928* 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 31929 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 31930 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 31931 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 31932 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 31933 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 31934 Converse of qustriv 31933. (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 31935 A generalization of Fermat's little theorem in a commutative ring 𝐹 of prime characteristic. See fermltl 16591. (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 31936 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 31937* 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 31938* 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 31939 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 31940 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 31941* 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 31942 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 31943* 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 31944 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 31945 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 31946* 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 31947 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 31948 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 31949 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 31950* 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 31951* 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 31952* 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 31953* 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 31954* 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 31955* 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 31956 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 31957 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 31958 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 31959 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 31960 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 31961 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 31962 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 31963 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 31964 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 31965 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 31966 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 31967 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 31968* 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 31969 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 31970* Lemma for nsgmgc 31971. (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 31971* 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 31972* Lemma for nsgqusf1o 31975. (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 31973* Lemma for nsgqusf1o 31975. (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 31974* Lemma for nsgqusf1o 31975. (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 31975* 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 31976 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 31977 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 31978 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 31979 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 31980* 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 31981 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 31982 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 31983 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 31984 Extend class notation with the class of prime ideals.
class PrmIdeal
 
Definitiondf-prmidl 31985* 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 31990 and isprmidlc 31997. (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 31986* 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 31987* 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 31988 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)
 
Theoremprmidl 31989* The main property of a prime 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‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐽 ∈ (LIdealβ€˜π‘…))) ∧ βˆ€π‘₯ ∈ 𝐼 βˆ€π‘¦ ∈ 𝐽 (π‘₯ Β· 𝑦) ∈ 𝑃) β†’ (𝐼 βŠ† 𝑃 ∨ 𝐽 βŠ† 𝑃))
 
Theoremprmidl2 31990* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 36415 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdealβ€˜π‘…)) ∧ (𝑃 β‰  𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘₯ ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) β†’ 𝑃 ∈ (PrmIdealβ€˜π‘…))
 
Theoremidlmulssprm 31991 Let 𝑃 be a prime ideal containing the product (𝐼 Γ— 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 βŠ† 𝑃 or 𝐽 βŠ† 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
Γ— = (LSSumβ€˜(mulGrpβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑃 ∈ (PrmIdealβ€˜π‘…))    &   (πœ‘ β†’ 𝐼 ∈ (LIdealβ€˜π‘…))    &   (πœ‘ β†’ 𝐽 ∈ (LIdealβ€˜π‘…))    &   (πœ‘ β†’ (𝐼 Γ— 𝐽) βŠ† 𝑃)    β‡’   (πœ‘ β†’ (𝐼 βŠ† 𝑃 ∨ 𝐽 βŠ† 𝑃))
 
Theorempridln1 31992 A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝐼 β‰  𝐡) β†’ Β¬ 1 ∈ 𝐼)
 
Theoremprmidlidl 31993 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) β†’ 𝑃 ∈ (LIdealβ€˜π‘…))
 
Theoremprmidlssidl 31994 Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
(𝑅 ∈ Ring β†’ (PrmIdealβ€˜π‘…) βŠ† (LIdealβ€˜π‘…))
 
Theoremlidlnsg 31995 An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdealβ€˜π‘…)) β†’ 𝐼 ∈ (NrmSGrpβ€˜π‘…))
 
Theoremcringm4 31996 Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑍 ∈ 𝐡 ∧ π‘Š ∈ 𝐡)) β†’ ((𝑋 Β· π‘Œ) Β· (𝑍 Β· π‘Š)) = ((𝑋 Β· 𝑍) Β· (π‘Œ Β· π‘Š)))
 
Theoremisprmidlc 31997* The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ CRing β†’ (𝑃 ∈ (PrmIdealβ€˜π‘…) ↔ (𝑃 ∈ (LIdealβ€˜π‘…) ∧ 𝑃 β‰  𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) ∈ 𝑃 β†’ (π‘₯ ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))
 
Theoremprmidlc 31998 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdealβ€˜π‘…)) ∧ (𝐼 ∈ 𝐡 ∧ 𝐽 ∈ 𝐡 ∧ (𝐼 Β· 𝐽) ∈ 𝑃)) β†’ (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃))
 
Theorem0ringprmidl 31999 The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝐡 = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (β™―β€˜π΅) = 1) β†’ (PrmIdealβ€˜π‘…) = βˆ…)
 
Theoremprmidl0 32000 The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.)
0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdealβ€˜π‘…)) ↔ 𝑅 ∈ IDomn)
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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 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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 46901-46966
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