HomeHome Metamath Proof Explorer
Theorem List (p. 310 of 451)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-28680)
  Hilbert Space Explorer  Hilbert Space Explorer
(28681-30203)
  Users' Mathboxes  Users' Mathboxes
(30204-45093)
 

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))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45093
  Copyright terms: Public domain < Previous  Next >