HomeHome Metamath Proof Explorer
Theorem List (p. 334 of 491)
< 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
(1-30946)
  Hilbert Space Explorer  Hilbert Space Explorer
(30947-32469)
  Users' Mathboxes  Users' Mathboxes
(32470-49035)
 

Theorem List for Metamath Proof Explorer - 33301-33400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfldgenfld 33301 A generated subfield is a field. (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝐹)    &   (𝜑𝐹 ∈ Field)    &   (𝜑𝑆𝐵)       (𝜑 → (𝐹s (𝐹 fldGen 𝑆)) ∈ Field)
 
Theoremprimefldgen1 33302 The prime field of a division ring is the subfield generated by the multiplicative identity element. In general, we should write "prime division ring", but since most later usages are in the case where the ambient ring is commutative, we keep the term "prime field". (Contributed by Thierry Arnoux, 11-Jan-2025.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &   (𝜑𝑅 ∈ DivRing)       (𝜑 (SubDRing‘𝑅) = (𝑅 fldGen { 1 }))
 
Theorem1fldgenq 33303 The field of rational numbers is generated by 1 in fld, that is, is the prime field of fld. (Contributed by Thierry Arnoux, 15-Jan-2025.)
(ℂfld fldGen {1}) = ℚ
 
21.3.9.27  Totally ordered rings and fields
 
Syntaxcorng 33304 Extend class notation with the class of all ordered rings.
class oRing
 
Syntaxcofld 33305 Extend class notation with the class of all ordered fields.
class oField
 
Definitiondf-orng 33306* Define class of all ordered rings. An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
 
Definitiondf-ofld 33307 Define class of all ordered fields. An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
oField = (Field ∩ oRing)
 
Theoremisorng 33308* An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (le‘𝑅)       (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))
 
Theoremorngring 33309 An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ Ring)
 
Theoremorngogrp 33310 An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ oGrp)
 
Theoremisofld 33311 An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))
 
Theoremorngmul 33312 In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝑅)    &    = (le‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))
 
Theoremorngsqr 33313 In an ordered ring, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝑅)    &    = (le‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0 (𝑋 · 𝑋))
 
Theoremornglmulle 33314 In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (le‘𝑅)    &   (𝜑𝑋 𝑌)    &   (𝜑0 𝑍)       (𝜑 → (𝑍 · 𝑋) (𝑍 · 𝑌))
 
Theoremorngrmulle 33315 In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (le‘𝑅)    &   (𝜑𝑋 𝑌)    &   (𝜑0 𝑍)       (𝜑 → (𝑋 · 𝑍) (𝑌 · 𝑍))
 
Theoremornglmullt 33316 In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋 < 𝑌)    &   (𝜑0 < 𝑍)       (𝜑 → (𝑍 · 𝑋) < (𝑍 · 𝑌))
 
Theoremorngrmullt 33317 In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋 < 𝑌)    &   (𝜑0 < 𝑍)       (𝜑 → (𝑋 · 𝑍) < (𝑌 · 𝑍))
 
Theoremorngmullt 33318 In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑0 < 𝑌)       (𝜑0 < (𝑋 · 𝑌))
 
Theoremofldfld 33319 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Field)
 
Theoremofldtos 33320 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Toset)
 
Theoremorng0le1 33321 In an ordered ring, the ring unity is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    = (le‘𝐹)       (𝐹 ∈ oRing → 0 1 )
 
Theoremofldlt1 33322 In an ordered field, the ring unity is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    < = (lt‘𝐹)       (𝐹 ∈ oField → 0 < 1 )
 
Theoremofldchr 33323 The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.) (Proof shortened by AV, 6-Oct-2020.)
(𝐹 ∈ oField → (chr‘𝐹) = 0)
 
Theoremsuborng 33324 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 33325 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 33326* 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)))
 
21.3.9.28  Ring homomorphisms - misc additions
 
Theoremrhmdvd 33327 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))))
 
Theoremkerunit 33328 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 )
 
21.3.9.29  Scalar restriction operation
 
Syntaxcresv 33329 Extend class notation with the scalar restriction operation.
class v
 
Definitiondf-resv 33330* 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 33331 The scalar restriction is a proper operator, so it can be used with ovprc1 7469. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom ↾v
 
Theoremresvval 33332 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 33333 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 33334 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 33335 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 33336 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 33337 Obsolete version of resvlem 33336 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 33338 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 33339 Obsolete proof of resvbas 33338 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 33340 +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 33341 Obsolete proof of resvplusg 33340 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 33342 ·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 33343 Obsolete proof of resvvsca 33342 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 33344 .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 33345 Obsolete proof of resvmulr 33344 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 33346 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    0 = (0g𝐺)       (𝐴𝑉0 = (0g𝐻))
 
Theoremresv1r 33347 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    1 = (1r𝐺)       (𝐴𝑉1 = (1r𝐻))
 
Theoremresvcmn 33348 Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)       (𝐴𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
 
21.3.9.30  The commutative ring of gaussian integers
 
Theoremgzcrng 33349 The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
(ℂflds ℤ[i]) ∈ CRing
 
21.3.9.31  The archimedean ordered field of real numbers
 
Theoremcnfldfld 33350 The complex numbers form a field. (Contributed by Thierry Arnoux, 6-Jul-2025.)
fld ∈ Field
 
Theoremreofld 33351 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ oField
 
Theoremnn0omnd 33352 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(ℂflds0) ∈ oMnd
 
Theoremrearchi 33353 The field of the real numbers is Archimedean. See also arch 12520. (Contributed by Thierry Arnoux, 9-Apr-2018.)
fld ∈ Archi
 
Theoremnn0archi 33354 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(ℂflds0) ∈ Archi
 
Theoremxrge0slmod 33355 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
 
21.3.9.32  The quotient map and quotient modules
 
Theoremqusker 33356* 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 33357 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 33358* 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))))
 
Theoremqusvsval 33359 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)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐾 [𝑋](𝑀 ~QG 𝐺)) = [(𝐾 · 𝑋)](𝑀 ~QG 𝐺))
 
Theoremimaslmod 33360* 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)
 
Theoremimasmhm 33361* Given a function 𝐹 with homomorphic properties, build the image of a monoid. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑𝐹:𝐵𝐶)    &    + = (+g𝑊)    &   ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ (𝑝𝐵𝑞𝐵)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑊 ∈ Mnd)       (𝜑 → ((𝐹s 𝑊) ∈ Mnd ∧ 𝐹 ∈ (𝑊 MndHom (𝐹s 𝑊))))
 
Theoremimasghm 33362* Given a function 𝐹 with homomorphic properties, build the image of a group. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑𝐹:𝐵𝐶)    &    + = (+g𝑊)    &   ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ (𝑝𝐵𝑞𝐵)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   (𝜑𝑊 ∈ Grp)       (𝜑 → ((𝐹s 𝑊) ∈ Grp ∧ 𝐹 ∈ (𝑊 GrpHom (𝐹s 𝑊))))
 
Theoremimasrhm 33363* Given a function 𝐹 with homomorphic properties, build the image of a ring. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑𝐹:𝐵𝐶)    &    + = (+g𝑊)    &   ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ (𝑝𝐵𝑞𝐵)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &    · = (.r𝑊)    &   ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ (𝑝𝐵𝑞𝐵)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑊 ∈ Ring)       (𝜑 → ((𝐹s 𝑊) ∈ Ring ∧ 𝐹 ∈ (𝑊 RingHom (𝐹s 𝑊))))
 
Theoremimaslmhm 33364* Given a function 𝐹 with homomorphic properties, build the image of a left module. (Contributed by Thierry Arnoux, 2-Apr-2025.)
𝐵 = (Base‘𝑊)    &   (𝜑𝐹:𝐵𝐶)    &    + = (+g𝑊)    &   ((𝜑 ∧ (𝑎𝐵𝑏𝐵) ∧ (𝑝𝐵𝑞𝐵)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   𝐷 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐷)    &   ((𝜑 ∧ (𝑘𝐾𝑎𝐵𝑏𝐵)) → ((𝐹𝑎) = (𝐹𝑏) → (𝐹‘(𝑘 × 𝑎)) = (𝐹‘(𝑘 × 𝑏))))    &   (𝜑𝑊 ∈ LMod)    &    × = ( ·𝑠𝑊)       (𝜑 → ((𝐹s 𝑊) ∈ LMod ∧ 𝐹 ∈ (𝑊 LMHom (𝐹s 𝑊))))
 
Theoremquslmod 33365 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 33366* 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))
 
Theoremquslvec 33367 If 𝑆 is a vector subspace in 𝑊, then 𝑄 = 𝑊 / 𝑆 is a vector space, called the quotient space of 𝑊 by 𝑆. (Contributed by Thierry Arnoux, 18-May-2023.)
𝑄 = (𝑊 /s (𝑊 ~QG 𝑆))    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑆 ∈ (LSubSp‘𝑊))       (𝜑𝑄 ∈ LVec)
 
Theoremecxpid 33368 The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
(𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
 
Theoremqsxpid 33369 The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)
(𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})
 
Theoremqusxpid 33370 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 33371 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 33372 Converse of qustriv 33371. (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)
 
21.3.9.33  The ring of integers modulo ` N `
 
Theoremznfermltl 33373 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)
 
21.3.9.34  Independent sets and families
 
Theoremislinds5 33374* 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 33375* Variation on ellspd 21839. (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 33376 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 33377 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)
 
Theoremrspsnid 33378 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 33379* 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))))))
 
Theoremellpi 33380 Elementhood in a left principal ideal in terms of the "divides" relation. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ 𝑋 𝑌))
 
Theoremlpirlidllpi 33381* In a principal ideal ring, ideals are principal. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ LPIR)    &   (𝜑𝐽𝐼)       (𝜑 → ∃𝑥𝐵 𝐽 = (𝐾‘{𝑥}))
 
Theoremrspidlid 33382 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 33383 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 33384* 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 33385 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 33386 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)))
 
Theoremislbs5 33387* An equivalent formulation of the basis predicate in a vector space, using a function 𝐹 for generating the base. (Contributed by Thierry Arnoux, 20-Feb-2025.)
𝐵 = (Base‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝑊)    &    0 = (0g𝑆)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑆 ∈ NzRing)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼1-1𝐵)       (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (∀𝑎 ∈ (𝐾m 𝐼)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑎f · 𝐹)) = 𝑂) → 𝑎 = (𝐼 × { 0 })) ∧ (𝑁‘ran 𝐹) = 𝐵)))
 
Theoremlinds2eq 33388 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 33389* 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 33390* 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))
 
21.3.9.35  Ring associates, ring units
 
Theoremdvdsruassoi 33391 If two elements 𝑋 and 𝑌 of a ring 𝑅 are unit multiples, then they are associates, i.e. each divides the other. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑉𝑈)    &   (𝜑 → (𝑉 · 𝑋) = 𝑌)       (𝜑 → (𝑋 𝑌𝑌 𝑋))
 
Theoremdvdsruasso 33392* Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff they are unit multiples of each other. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ IDomn)       (𝜑 → ((𝑋 𝑌𝑌 𝑋) ↔ ∃𝑢𝑈 (𝑢 · 𝑋) = 𝑌))
 
Theoremdvdsruasso2 33393* A reformulation of dvdsruasso 33392. (Proposed by Gerard Lang, 28-May-2025.) (Contributed by Thiery Arnoux, 29-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ IDomn)    &    1 = (1r𝑅)       (𝜑 → ((𝑋 𝑌𝑌 𝑋) ↔ ∃𝑢𝑈𝑣𝑈 ((𝑢 · 𝑋) = 𝑌 ∧ (𝑣 · 𝑌) = 𝑋 ∧ (𝑢 · 𝑣) = 1 )))
 
Theoremdvdsrspss 33394 In a ring, an element 𝑋 divides 𝑌 iff the ideal generated by 𝑌 is a subset of the ideal generated by 𝑋 (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝑋 𝑌 ↔ (𝐾‘{𝑌}) ⊆ (𝐾‘{𝑋})))
 
Theoremrspsnasso 33395 Two elements 𝑋 and 𝑌 of a ring 𝑅 are associates, i.e. each divides the other, iff the ideals they generate are equal. (Contributed by Thierry Arnoux, 22-Mar-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑅 ∈ Ring)       (𝜑 → ((𝑋 𝑌𝑌 𝑋) ↔ (𝐾‘{𝑌}) = (𝐾‘{𝑋})))
 
Theoremunitprodclb 33396 A finite product is a unit iff all factors are units. (Contributed by Thierry Arnoux, 27-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐹 ∈ Word 𝐵)       (𝜑 → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹𝑈))
 
21.3.9.36  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 33397* 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 33398* The sumset of a group with a single element is the element's orbit by the group action. See gaorb 19337. (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 33399* The sumset of a single element with a group is the element's orbit by the group action. See gaorb 19337. (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 33400* 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)))
    < 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 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49035
  Copyright terms: Public domain < Previous  Next >