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Theorem List for Metamath Proof Explorer - 30901-31000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelrhmunit 30901 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹𝐴) ∈ (Unit‘𝑆))

Theoremrhmdvd 30902 A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝑈 = (Unit‘𝑆)    &   𝑋 = (Base‘𝑅)    &    / = (/r𝑆)    &    · = (.r𝑅)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋) ∧ ((𝐹𝐵) ∈ 𝑈 ∧ (𝐹𝐶) ∈ 𝑈)) → ((𝐹𝐴) / (𝐹𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶))))

Theoremrhmunitinv 30903 Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr𝑅)‘𝐴)) = ((invr𝑆)‘(𝐹𝐴)))

Theoremkerunit 30904 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.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑆)    &    1 = (1r𝑆)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑈 ∩ (𝐹 “ { 0 })) ≠ ∅) → 1 = 0 )

20.3.9.18  Scalar restriction operation

Syntaxcresv 30905 Extend class notation with the scalar restriction operation.
class v

Definitiondf-resv 30906* Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.)
v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))

Theoremreldmresv 30907 The scalar restriction is a proper operator, so it can be used with ovprc1 7172. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom ↾v

Theoremresvval 30908 Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))

Theoremresvid2 30909 General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)

Theoremresvval2 30910 Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))

Theoremresvsca 30911 Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       (𝐴𝑉 → (𝐹s 𝐴) = (Scalar‘𝑅))

Theoremresvlem 30912 Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 5       (𝐴𝑉𝐶 = (𝐸𝑅))

Theoremresvbas 30913 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))

Theoremresvplusg 30914 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))

Theoremresvvsca 30915 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))

Theoremresvmulr 30916 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = (.r𝐺)       (𝐴𝑉· = (.r𝐻))

Theoremresv0g 30917 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    0 = (0g𝐺)       (𝐴𝑉0 = (0g𝐻))

Theoremresv1r 30918 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    1 = (1r𝐺)       (𝐴𝑉1 = (1r𝐻))

Theoremresvcmn 30919 Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)       (𝐴𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))

20.3.9.19  The commutative ring of gaussian integers

Theoremgzcrng 30920 The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
(ℂflds ℤ[i]) ∈ CRing

20.3.9.20  The archimedean ordered field of real numbers

Theoremreofld 30921 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ oField

Theoremnn0omnd 30922 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(ℂflds0) ∈ oMnd

Theoremrearchi 30923 The field of the real numbers is Archimedean. See also arch 11873. (Contributed by Thierry Arnoux, 9-Apr-2018.)
fld ∈ Archi

Theoremnn0archi 30924 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(ℂflds0) ∈ Archi

Theoremxrge0slmod 30925 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*𝑠s (0[,]+∞))    &   𝑊 = (𝐺v (0[,)+∞))       𝑊 ∈ SLMod

20.3.9.21  The quotient map and quotient modules

Theoremqusker 30926* The kernel of a quotient map. (Contributed by Thierry Arnoux, 20-May-2023.)
𝑉 = (Base‘𝑀)    &   𝐹 = (𝑥𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))    &   𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &    0 = (0g𝑁)       (𝐺 ∈ (NrmSGrp‘𝑀) → (𝐹 “ { 0 }) = 𝐺)

Theoremeqgvscpbl 30927 The left coset equivalence relation is compatible with the scalar multiplication operation. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑀)    &    = (𝑀 ~QG 𝐺)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))    &   (𝜑𝐾𝑆)       (𝜑 → (𝑋 𝑌 → (𝐾 · 𝑋) (𝐾 · 𝑌)))

Theoremqusvscpbl 30928* The quotient map distributes over the scalar multiplication. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑀)    &    = (𝑀 ~QG 𝐺)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))    &   (𝜑𝐾𝑆)    &   𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &    = ( ·𝑠𝑁)    &   𝐹 = (𝑥𝐵 ↦ [𝑥](𝑀 ~QG 𝐺))    &   (𝜑𝑈𝐵)    &   (𝜑𝑉𝐵)       (𝜑 → ((𝐹𝑈) = (𝐹𝑉) → (𝐹‘(𝐾 · 𝑈)) = (𝐹‘(𝐾 · 𝑉))))

Theoremqusscaval 30929 Value of the scalar multiplication operation on the quotient structure. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑀)    &    = (𝑀 ~QG 𝐺)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))    &   (𝜑𝐾𝑆)    &   𝑁 = (𝑀 /s (𝑀 ~QG 𝐺))    &    = ( ·𝑠𝑁)       ((𝜑𝐾𝑆𝑋𝐵) → (𝐾 [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺))

Theoremimaslmod 30930* The image structure of a left module is a left module. (Contributed by Thierry Arnoux, 15-May-2023.)
(𝜑𝑁 = (𝐹s 𝑀))    &   𝑉 = (Base‘𝑀)    &   𝑆 = (Base‘(Scalar‘𝑀))    &    + = (+g𝑀)    &    · = ( ·𝑠𝑀)    &    0 = (0g𝑀)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ((𝜑 ∧ (𝑘𝑆𝑎𝑉𝑏𝑉)) → ((𝐹𝑎) = (𝐹𝑏) → (𝐹‘(𝑘 · 𝑎)) = (𝐹‘(𝑘 · 𝑏))))    &   (𝜑𝑀 ∈ LMod)       (𝜑𝑁 ∈ LMod)

Theoremquslmod 30931 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))    &   𝑉 = (Base‘𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))       (𝜑𝑁 ∈ LMod)

Theoremquslmhm 30932* 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))    &   𝑉 = (Base‘𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐺 ∈ (LSubSp‘𝑀))    &   𝐹 = (𝑥𝑉 ↦ [𝑥](𝑀 ~QG 𝐺))       (𝜑𝐹 ∈ (𝑀 LMHom 𝑁))

Theoremecxpid 30933 The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
(𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)

Theoremeqg0el 30934 Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
= (𝐺 ~QG 𝐻)       ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] = 𝐻𝑋𝐻))

Theoremqsxpid 30935 The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)
(𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})

Theoremqusxpid 30936 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.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → (𝐺 ~QG 𝐵) = (𝐵 × 𝐵))

Theoremqustriv 30937 The quotient of a group 𝐺 by itself is trivial. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐵))       (𝐺 ∈ Grp → (Base‘𝑄) = {𝐵})

Theoremqustrivr 30938 Converse of qustriv 30937. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐻))       ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → 𝐻 = 𝐵)

20.3.9.22  Univariate Polynomials

Theoremfply1 30939 Conditions for a function to be an univariate polynomial. (Contributed by Thierry Arnoux, 19-Aug-2023.)
0 = (0g𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑃 = (Base‘(Poly1𝑅))    &   (𝜑𝐹:(ℕ0m 1o)⟶𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑𝐹𝑃)

20.3.9.23  Independent sets and families

Theoremislinds5 30940* A set is linearly independent if and only if it has no non-trivial representations of zero. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝑊)    &    0 = (0g𝐹)       ((𝑊 ∈ LMod ∧ 𝑉𝐵) → (𝑉 ∈ (LIndS‘𝑊) ↔ ∀𝑎 ∈ (𝐾m 𝑉)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑣𝑉 ↦ ((𝑎𝑣) · 𝑣))) = 𝑂) → 𝑎 = (𝑉 × { 0 }))))

Theoremellspds 30941* Variation on ellspd 20922. (Contributed by Thierry Arnoux, 18-May-2023.)
𝑁 = (LSpan‘𝑀)    &   𝐵 = (Base‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝑉𝐵)       (𝜑 → (𝑋 ∈ (𝑁𝑉) ↔ ∃𝑎 ∈ (𝐾m 𝑉)(𝑎 finSupp 0𝑋 = (𝑀 Σg (𝑣𝑉 ↦ ((𝑎𝑣) · 𝑣))))))

Theorem0ellsp 30942 Zero is in all spans. (Contributed by Thierry Arnoux, 8-May-2023.)
0 = (0g𝑊)    &   𝐵 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑆𝐵) → 0 ∈ (𝑁𝑆))

Theorem0nellinds 30943 The group identity cannot be an element of an independent set. (Contributed by Thierry Arnoux, 8-May-2023.)
0 = (0g𝑊)       ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) → ¬ 0𝐹)

Theoremrspsnel 30944* Membership in a principal ideal. Analogous to lspsnel 19751. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥𝐵 𝐼 = (𝑥 · 𝑋)))

Theoremrspsnid 30945 A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → 𝐺 ∈ (𝐾‘{𝐺}))

Theorempidlnz 30946 A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑋0 ) → (𝐾‘{𝑋}) ≠ { 0 })

Theoremlbslsp 30947* 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.)
𝐵 = (Base‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝑉 ∈ (LBasis‘𝑀))       (𝜑 → (𝑋𝐵 ↔ ∃𝑎 ∈ (𝐾m 𝑉)(𝑎 finSupp 0𝑋 = (𝑀 Σg (𝑣𝑉 ↦ ((𝑎𝑣) · 𝑣))))))

Theoremlindssn 30948 Any singleton of a nonzero element is an independent set. (Contributed by Thierry Arnoux, 5-Aug-2023.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LVec ∧ 𝑋𝐵𝑋0 ) → {𝑋} ∈ (LIndS‘𝑊))

Theoremlindflbs 30949 Conditions for an independent family to be a basis. (Contributed by Thierry Arnoux, 21-Jul-2023.)
𝐵 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑆 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝑊)    &    0 = (0g𝑆)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑆 ∈ NzRing)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼1-1𝐵)       (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵)))

Theoremlinds2eq 30950 Deduce equality of elements in an independent set. (Contributed by Thierry Arnoux, 18-Jul-2023.)
𝐹 = (Base‘(Scalar‘𝑊))    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g‘(Scalar‘𝑊))    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐵 ∈ (LIndS‘𝑊))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐾𝐹)    &   (𝜑𝐿𝐹)    &   (𝜑𝐾0 )    &   (𝜑 → (𝐾 · 𝑋) = (𝐿 · 𝑌))       (𝜑 → (𝑋 = 𝑌𝐾 = 𝐿))

Theoremlindfpropd 30951* Property deduction for linearly independent families. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))    &   (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)    &   (𝜑𝑋𝐴)       (𝜑 → (𝑋 LIndF 𝐾𝑋 LIndF 𝐿))

Theoremlindspropd 30952* Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))    &   (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)       (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))

20.3.9.24  Subgroup sum / Sumset / Minkowski sum

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

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

Theoremelgrplsmsn 30953* Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))

Theoremlsmsnorb 30954* The sumset of a group with a single element is the element's orbit by the group action. See gaorb 18416. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴 {𝑋}) = [𝑋] )

Theoremelringlsm 30955* Membership in a product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥𝐸𝑦𝐹 𝑍 = (𝑥 · 𝑦)))

Theoremelringlsmd 30956 Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐹)       (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹))

Theoremringlsmss 30957 Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐸 × 𝐹) ⊆ 𝐵)

Theoremlsmsnpridl 30958 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.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐵 × {𝑋}) = (𝐾‘{𝑋}))

Theoremlsmsnidl 30959 The product of the ring with a single element is a principal ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐵 × {𝑋}) ∈ (LPIdeal‘𝑅))

Theoremlsmidllsp 30960 The sum of two ideals is the ideal generated by their union. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 𝐽) = (𝐾‘(𝐼𝐽)))

Theoremlsmidl 30961 The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 𝐽) ∈ (LIdeal‘𝑅))

20.3.9.25  Prime Ideals

Syntaxcprmidl 30962 Extend class notation with the class of prime ideals.
class PrmIdeal

Definitiondf-prmidl 30963* Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵𝐼 for ideals 𝐴 and 𝐵, either 𝐴𝐼 or 𝐵𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 30968 and isprmidlc 30974. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Theoremprmidlval 30964* The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Theoremisprmidl 30965* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))

Theoremprmidlnr 30966 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃𝐵)

Theoremprmidl 30967* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼𝑃𝐽𝑃))

Theoremprmidl2 30968* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 35384 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))

Theoremidlmulssprm 30969 Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼𝑃 or 𝐽𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
× = (LSSum‘(mulGrp‘𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑃 ∈ (PrmIdeal‘𝑅))    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))    &   (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃)       (𝜑 → (𝐼𝑃𝐽𝑃))

Theorempridln1 30970 A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ¬ 1𝐼)

Theoremprmidlidl 30971 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))

Theoremlidlnsg 30972 An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))

Theoremcringm4 30973 Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))

Theoremisprmidlc 30974* 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.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))))

Theoremprmidlc 30975 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼𝐵𝐽𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼𝑃𝐽𝑃))

Theoremqsidomlem1 30976 If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))

Theoremqsidomlem2 30977 A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)

Theoremqsidom 30978 An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅)))

20.3.9.26  Maximal Ideals

Syntaxcmxidl 30979 Extend class notation with the class of maximal ideals.
class MaxIdeal

Definitiondf-mxidl 30980* Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})

Theoremmxidlval 30981* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})

Theoremismxidl 30982* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))

Theoremmxidlidl 30983 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))

Theoremmxidlnr 30984 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀𝐵)

Theoremmxidlmax 30985 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝐵))

Theoremmxidln1 30986 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1𝑀)

Theoremmxidlnzr 30987 A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing)

Theoremmxidlprm 30988 Every maximal ideal is prime. Statement in [Lang] p. 92 (Contributed by Thierry Arnoux, 21-Jan-2024.)
× = (LSSum‘(mulGrp‘𝑅))       ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))

Theoremssmxidllem 30989* The set 𝑃 used in the proof of ssmxidl 30990 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐼𝐵)    &   (𝜑𝑍𝑃)    &   (𝜑𝑍 ≠ ∅)    &   (𝜑 → [] Or 𝑍)       (𝜑 𝑍𝑃)

Theoremssmxidl 30990* Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚)

Theoremkrull 30991* Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))

Theoremmxidlnzrb 30992* A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)))

20.3.9.27  The semiring of ideals of a ring

Syntaxcidlsrg 30993 Extend class notation with the semiring of ideals of a ring.
class IDLsrg

Definitiondf-idlsrg 30994* Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
IDLsrg = (𝑟 ∈ Ring ↦ (LIdeal‘𝑟) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((+𝑓𝑟) “ (𝑖 × 𝑗)))⟩, ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘((+𝑓‘(mulGrp‘𝑟)) “ (𝑖 × 𝑗))))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ (𝑏 ∖ {𝑗 ∈ (PrmIdeal‘𝑟) ∣ 𝑗𝑖}))⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩}))

Syntaxcrspec 30995 Extend class notation with the spectrum of a ring.
class Spec

Definitiondf-rspec 30996 Define the spectrum of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
Spec = (𝑟 ∈ Ring ↦ ((IDLsrg‘𝑟) ↾s (PrmIdeal‘𝑟)))

20.3.9.28  The subring algebra

Theoremsra1r 30997 The multiplicative neutral element of a subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑1 = (1r𝑊))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑1 = (1r𝐴))

Theoremsraring 30998 Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.)
𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑉𝐵) → 𝐴 ∈ Ring)

Theoremsradrng 30999 Condition for a subring algebra to be a division ring. (Contributed by Thierry Arnoux, 29-Jul-2023.)
𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ DivRing ∧ 𝑉𝐵) → 𝐴 ∈ DivRing)

Theoremsrasubrg 31000 A subring of the original structure is also a subring of the constructed subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑈 ∈ (SubRing‘𝑊))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑𝑈 ∈ (SubRing‘𝐴))

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