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Theorem List for Metamath Proof Explorer - 31501-31600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsuborng 31501 Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)
 
Theoremsubofld 31502 Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝐹 ∈ oField ∧ (𝐹s 𝐴) ∈ Field) → (𝐹s 𝐴) ∈ oField)
 
Theoremisarchiofld 31503* Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑊)    &   𝐻 = (ℤRHom‘𝑊)    &    < = (lt‘𝑊)       (𝑊 ∈ oField → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑛 ∈ ℕ 𝑥 < (𝐻𝑛)))
 
20.3.9.17  Ring homomorphisms - misc additions
 
Theoremrhmdvdsr 31504 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝑋 = (Base‘𝑅)    &    = (∥r𝑅)    &    / = (∥r𝑆)       (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 𝐵) → (𝐹𝐴) / (𝐹𝐵))
 
Theoremrhmopp 31505 A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))
 
Theoremelrhmunit 31506 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹𝐴) ∈ (Unit‘𝑆))
 
Theoremrhmdvd 31507 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 31508 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 31509 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 31510 Extend class notation with the scalar restriction operation.
class v
 
Definitiondf-resv 31511* 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 31512 The scalar restriction is a proper operator, so it can be used with ovprc1 7308. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom ↾v
 
Theoremresvval 31513 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 31514 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 31515 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 31516 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 31517 Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝑅 = (𝑊v 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (Scalar‘ndx)       (𝐴𝑉𝐶 = (𝐸𝑅))
 
TheoremresvlemOLD 31518 Obsolete version of resvlem 31517 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 = (𝑊v 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 5       (𝐴𝑉𝐶 = (𝐸𝑅))
 
Theoremresvbas 31519 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))
 
TheoremresvbasOLD 31520 Obsolete proof of resvbas 31519 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 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))
 
Theoremresvplusg 31521 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))
 
TheoremresvplusgOLD 31522 Obsolete proof of resvplusg 31521 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 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))
 
Theoremresvvsca 31523 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Proof shortened by AV, 31-Oct-2024.)
𝐻 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))
 
TheoremresvvscaOLD 31524 Obsolete proof of resvvsca 31523 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 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))
 
Theoremresvmulr 31525 .r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
𝐻 = (𝐺v 𝐴)    &    · = (.r𝐺)       (𝐴𝑉· = (.r𝐻))
 
TheoremresvmulrOLD 31526 Obsolete proof of resvmulr 31525 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 = (𝐺v 𝐴)    &    · = (.r𝐺)       (𝐴𝑉· = (.r𝐻))
 
Theoremresv0g 31527 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    0 = (0g𝐺)       (𝐴𝑉0 = (0g𝐻))
 
Theoremresv1r 31528 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    1 = (1r𝐺)       (𝐴𝑉1 = (1r𝐻))
 
Theoremresvcmn 31529 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 31530 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 31531 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ oField
 
Theoremnn0omnd 31532 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(ℂflds0) ∈ oMnd
 
Theoremrearchi 31533 The field of the real numbers is Archimedean. See also arch 12219. (Contributed by Thierry Arnoux, 9-Apr-2018.)
fld ∈ Archi
 
Theoremnn0archi 31534 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(ℂflds0) ∈ Archi
 
Theoremxrge0slmod 31535 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 31536* 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 31537 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 31538* 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 31539 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 31540* 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 31541 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 31542* 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 31543 The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
(𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
 
Theoremeqg0el 31544 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 31545 The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)
(𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})
 
Theoremqusxpid 31546 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 31547 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 31548 Converse of qustriv 31547. (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  The ring of integers modulo ` N `
 
Theoremznfermltl 31549 Fermat's little theorem in ℤ/n. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑍 = (ℤ/nℤ‘𝑃)    &   𝐵 = (Base‘𝑍)    &    = (.g‘(mulGrp‘𝑍))       ((𝑃 ∈ ℙ ∧ 𝐴𝐵) → (𝑃 𝐴) = 𝐴)
 
20.3.9.23  Independent sets and families
 
Theoremislinds5 31550* 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 31551* Variation on ellspd 20998. (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 31552 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 31553 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 31554* Membership in a principal ideal. Analogous to lspsnel 20254. (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 31555 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}))
 
Theoremelrsp 31556* Write the elements of a ring span as finite linear combinations. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝐵)       (𝜑 → (𝑋 ∈ (𝑁𝐼) ↔ ∃𝑎 ∈ (𝐵m 𝐼)(𝑎 finSupp 0𝑋 = (𝑅 Σg (𝑖𝐼 ↦ ((𝑎𝑖) · 𝑖))))))
 
Theoremrspidlid 31557 The ideal span of an ideal is the ideal itself. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝐾 = (RSpan‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝐾𝐼) = 𝐼)
 
Theorempidlnz 31558 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 31559* 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 31560 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 31561 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 31562 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 31563* 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 31564* 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 31565* 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 31566* The sumset of a group with a single element is the element's orbit by the group action. See gaorb 18902. (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] )
 
Theoremlsmsnorb2 31567* The sumset of a single element with a group is the element's orbit by the group action. See gaorb 18902. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑥 + 𝑔) = 𝑦)}    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → ({𝑋} 𝐴) = [𝑋] )
 
Theoremelringlsm 31568* 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 31569 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 31570 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)
 
Theoremringlsmss1 31571 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.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐸𝐵)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼)
 
Theoremringlsmss2 31572 The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐸𝐵)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐸 × 𝐼) ⊆ 𝐼)
 
Theoremlsmsnpridl 31573 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 31574 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 31575 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 31576 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))
 
Theoremlsmssass 31577 Group sum is associative, subset version (see lsmass 19264). (Contributed by Thierry Arnoux, 1-Jun-2024.)
= (LSSum‘𝐺)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑅𝐵)    &   (𝜑𝑇𝐵)    &   (𝜑𝑈𝐵)       (𝜑 → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))
 
Theoremgrplsm0l 31578 Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝐵𝐴 ≠ ∅) → ({ 0 } 𝐴) = 𝐴)
 
Theoremgrplsmid 31579 The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
= (LSSum‘𝐺)       ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ({𝑋} 𝐴) = 𝐴)
 
20.3.9.25  The quotient map
 
Theoremquslsm 31580 Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋𝐵)       (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} 𝑆))
 
Theoremqusima 31581* The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑥𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑𝐻𝑆)    &   (𝜑𝑆 ⊆ (SubGrp‘𝐺))       (𝜑 → (𝐸𝐻) = (𝐹𝐻))
 
Theoremnsgqus0 31582 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))       ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁𝐹)
 
Theoremnsgmgclem 31583* Lemma for nsgmgc 31584. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑𝐹 ∈ (SubGrp‘𝑄))       (𝜑 → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝐹} ∈ (SubGrp‘𝐺))
 
Theoremnsgmgc 31584* 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.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &   𝐽 = (𝑉MGalConn𝑊)    &   𝑉 = (toInc‘𝑆)    &   𝑊 = (toInc‘𝑇)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑𝐸𝐽𝐹)
 
Theoremnsgqusf1olem1 31585* Lemma for nsgqusf1o 31588. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
 
Theoremnsgqusf1olem2 31586* Lemma for nsgqusf1o 31588. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑 → ran 𝐸 = 𝑇)
 
Theoremnsgqusf1olem3 31587* Lemma for nsgqusf1o 31588. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑 → ran 𝐹 = 𝑆)
 
Theoremnsgqusf1o 31588* 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.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑𝐸:𝑆1-1-onto𝑇)
 
20.3.9.26  Ideals
 
Theoremintlidl 31589 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → 𝐶 ∈ (LIdeal‘𝑅))
 
Theoremrhmpreimaidl 31590 The preimage of an ideal by a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝐼 = (LIdeal‘𝑅)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹𝐽) ∈ 𝐼)
 
Theoremkerlidl 31591 The kernel of a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐼 = (LIdeal‘𝑅)    &    0 = (0g𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ { 0 }) ∈ 𝐼)
 
Theorem0ringidl 31592 The zero ideal is the only ideal of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) = {{ 0 }})
 
Theoremelrspunidl 31593* Elementhood to the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑆 ⊆ (LIdeal‘𝑅))       (𝜑 → (𝑋 ∈ (𝑁 𝑆) ↔ ∃𝑎 ∈ (𝐵m 𝑆)(𝑎 finSupp 0𝑋 = (𝑅 Σg 𝑎) ∧ ∀𝑘𝑆 (𝑎𝑘) ∈ 𝑘)))
 
Theoremlidlincl 31594 Ideals are closed under intersection. (Contributed by Thierry Arnoux, 5-Jul-2024.)
𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐽𝑈) → (𝐼𝐽) ∈ 𝑈)
 
Theoremidlinsubrg 31595 The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (LIdeal‘𝑅)    &   𝑉 = (LIdeal‘𝑆)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ∈ 𝑉)
 
Theoremrhmimaidl 31596 The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝐵 = (Base‘𝑆)    &   𝑇 = (LIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑆)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
 
20.3.9.27  Prime Ideals
 
Syntaxcprmidl 31597 Extend class notation with the class of prime ideals.
class PrmIdeal
 
Definitiondf-prmidl 31598* 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 31603 and isprmidlc 31610. (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 31599* 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 31600* 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)))))
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