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Theorem List for Metamath Proof Explorer - 28901-29000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgsummpt2co 28901* Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐸𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐸)    &   𝐹 = (𝑥𝐴𝐷)       (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶)))))
 
Theoremgsummpt2d 28902* Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 18104. (Contributed by Thierry Arnoux, 27-Apr-2020.)
𝑧𝐶    &   𝑦𝜑    &   𝐵 = (Base‘𝑊)    &   (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ CMnd)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)       (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
 
Theoremgsumvsca1 28903* Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐺 = (Scalar‘𝑊)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑃𝐾)    &   ((𝜑𝑘𝐴) → 𝑄𝐵)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘𝐴𝑄))))
 
Theoremgsumvsca2 28904* Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) (Proof shortened by AV, 12-Dec-2019.)
𝐵 = (Base‘𝑊)    &   𝐺 = (Scalar‘𝑊)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑄𝐵)    &   ((𝜑𝑘𝐴) → 𝑃𝐾)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘𝐴𝑃)) · 𝑄))
 
Theoremgsummptres 28905* Extend a finite group sum by padding outside with zeroes. Proof generated using OpenAI's proof assistant. (Contributed by Thierry Arnoux, 11-Jul-2020.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥 ∈ (𝐴𝐷)) → 𝐶 = 0 )       (𝜑 → (𝐺 Σg (𝑥𝐴𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴𝐷) ↦ 𝐶)))
 
Theoremxrge0tsmsd 28906* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   (𝜑𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑠))), ℝ*, < ))       (𝜑 → (𝐺 tsums 𝐹) = {𝑆})
 
Theoremxrge0tsmsbi 28907 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))       (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = (𝐺 tsums 𝐹)))
 
Theoremxrge0tsmseq 28908 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   (𝜑𝐶 ∈ (𝐺 tsums 𝐹))       (𝜑𝐶 = (𝐺 tsums 𝐹))
 
20.3.8.7  Rings - misc additions
 
Theoremrngurd 28909* Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)       (𝜑1 = (1r𝑅))
 
Theoremress1r 28910 1r is unaffected by restriction. This is a bit more generic than subrg1 18523. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 1𝐴𝐴𝐵) → 1 = (1r𝑆))
 
Theoremdvrdir 28911 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝑈)) → ((𝑋 + 𝑌) / 𝑍) = ((𝑋 / 𝑍) + (𝑌 / 𝑍)))
 
Theoremrdivmuldivd 28912 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝑈)       (𝜑 → ((𝑋 / 𝑌) · (𝑍 / 𝑊)) = ((𝑋 · 𝑍) / (𝑌 · 𝑊)))
 
Theoremringinvval 28913* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
𝐵 = (Base‘𝑅)    &    = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑈 = (Unit‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) = (𝑦𝑈 (𝑦 𝑋) = 1 ))
 
Theoremdvrcan5 28914 Cancellation law for common factor in ratio. (divcan5 10479 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝑈𝑍𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌))
 
Theoremsubrgchr 28915 If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
(𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅s 𝐴)) = (chr‘𝑅))
 
20.3.8.8  Ordered rings and fields
 
Syntaxcorng 28916 Extend class notation with the class of all ordered rings.
class oRing
 
Syntaxcofld 28917 Extend class notation with the class of all ordered fields.
class oField
 
Definitiondf-orng 28918* 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 28919 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 28920* 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 28921 An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ Ring)
 
Theoremorngogrp 28922 An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ oGrp)
 
Theoremisofld 28923 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 28924 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 28925 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 28926 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 28927 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 28928 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 28929 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 28930 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 28931 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Field)
 
Theoremofldtos 28932 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Toset)
 
Theoremorng0le1 28933 In an ordered ring, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    = (le‘𝐹)       (𝐹 ∈ oRing → 0 1 )
 
Theoremofldlt1 28934 In an ordered field, the ring unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    < = (lt‘𝐹)       (𝐹 ∈ oField → 0 < 1 )
 
Theoremofldchr 28935 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 28936 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 28937 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 28938* 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)))
 
20.3.8.9  Ring homomorphisms - misc additions
 
Theoremrhmdvdsr 28939 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 28940 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 28941 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹𝐴) ∈ (Unit‘𝑆))
 
Theoremrhmdvd 28942 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 28943 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 28944 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 )
 
20.3.8.10  Scalar restriction operation
 
Syntaxcresv 28945 Extend class notation with the scalar restriction operation.
class v
 
Definitiondf-resv 28946* 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 28947 The scalar restriction is a proper operator, so it can be used with ovprc1 6464. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom ↾v
 
Theoremresvval 28948 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 28949 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 28950 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 28951 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 28952 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 28953 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))
 
Theoremresvplusg 28954 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))
 
Theoremresvvsca 28955 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))
 
Theoremresvmulr 28956 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = (.r𝐺)       (𝐴𝑉· = (.r𝐻))
 
Theoremresv0g 28957 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    0 = (0g𝐺)       (𝐴𝑉0 = (0g𝐻))
 
Theoremresv1r 28958 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    1 = (1r𝐺)       (𝐴𝑉1 = (1r𝐻))
 
Theoremresvcmn 28959 Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)       (𝐴𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))
 
20.3.8.11  The commutative ring of gaussian integers
 
Theoremgzcrng 28960 The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
(ℂflds ℤ[i]) ∈ CRing
 
20.3.8.12  The archimedean ordered field of real numbers
 
Theoremreofld 28961 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ oField
 
Theoremnn0omnd 28962 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(ℂflds0) ∈ oMnd
 
Theoremrearchi 28963 The field of the real numbers is Archimedean. See also arch 11047. (Contributed by Thierry Arnoux, 9-Apr-2018.)
fld ∈ Archi
 
Theoremnn0archi 28964 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(ℂflds0) ∈ Archi
 
Theoremxrge0slmod 28965 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  Matrices
 
20.3.9.1  The symmetric group
 
Theoremsymgfcoeu 28966* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Base‘(SymGrp‘𝐷))       ((𝐷𝑉𝑃𝐺𝑄𝐺) → ∃!𝑝𝐺 𝑄 = (𝑃𝑝))
 
20.3.9.2  Permutation Signs
 
Theorempsgndmfi 28967 For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑆 = (pmSgn‘𝐷)    &   𝐺 = (Base‘(SymGrp‘𝐷))       (𝐷 ∈ Fin → 𝑆 Fn 𝐺)
 
Theorempsgnid 28968 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)
 
Theorempmtrprfv2 28969 In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋)
 
Theorempmtrto1cl 28970 Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑇 = (pmTrsp‘𝐷)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇)
 
Theorempsgnfzto1stlem 28971* Lemma for psgnfzto1st 28976. Our permutation of rank (𝑛 + 1) can be written as a permutation of rank 𝑛 composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)       ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑖𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖𝐾, (𝑖 − 1), 𝑖)))))
 
Theoremfzto1stfv1 28972* Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼𝐷 → (𝑃‘1) = 𝐼)
 
Theoremfzto1st1 28973* Special case where the permutation defined in psgnfzto1st 28976 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))       (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷))
 
Theoremfzto1st 28974* The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷𝑃𝐵)
 
Theoremfzto1stinvn 28975* Value of the inverse of our permutation 𝑃 at 𝐼 (Contributed by Thierry Arnoux, 23-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐼𝐷 → (𝑃𝐼) = 1)
 
Theorempsgnfzto1st 28976* The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝐷 = (1...𝑁)    &   𝑃 = (𝑖𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑆 = (pmSgn‘𝐷)       (𝐼𝐷 → (𝑆𝑃) = (-1↑(𝐼 + 1)))
 
20.3.9.3  Submatrices
 
Syntaxcsmat 28977 Syntax for a function generating submatrixes.
class subMat1
 
Definitiondf-smat 28978* Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) × (1...𝑁)) into a new index in ((1...(𝑀 − 1)) × (1...(𝑁 − 1))). A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma 20108. (Contributed by Thierry Arnoux, 18-Aug-2020.)
subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
 
Theoremsmatfval 28979* Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
 
Theoremsmatrcl 28980 Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))       (𝜑𝑆 ∈ (𝐵𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
 
Theoremsmatlem 28981 Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ ℕ)    &   (𝜑𝐽 ∈ ℕ)    &   (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)       (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))
 
Theoremsmattl 28982 Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽))
 
Theoremsmattr 28983 Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (1..^𝐿))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))
 
Theoremsmatbl 28984 Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (1..^𝐾))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))
 
Theoremsmatbr 28985 Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑀))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))    &   (𝜑𝐼 ∈ (𝐾...𝑀))    &   (𝜑𝐽 ∈ (𝐿...𝑁))       (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))
 
Theoremsmatcl 28986 Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indexes in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅))    &   𝑆 = (𝐾(subMat1‘𝑀)𝐿)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝐿 ∈ (1...𝑁))    &   (𝜑𝑀𝐵)       (𝜑𝑆𝐶)
 
Theoremmatmpt2 28987* Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑖𝑀𝑗)))
 
Theorem1smat1 28988 The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20114. (Contributed by Thierry Arnoux, 19-Aug-2020.)
1 = (1r‘((1...𝑁) Mat 𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))       (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))
 
Theoremsubmat1n 28989 One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ ℕ ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁))
 
Theoremsubmatres 28990 Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁 ∈ ℕ ∧ 𝑀𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
 
Theoremsubmateqlem1 28991 Lemma for submateq 28993. (Contributed by Thierry Arnoux, 25-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝐾𝑀)       (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})))
 
Theoremsubmateqlem2 28992 Lemma for submateq 28993. (Contributed by Thierry Arnoux, 26-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ (1...𝑁))    &   (𝜑𝑀 ∈ (1...(𝑁 − 1)))    &   (𝜑𝑀 < 𝐾)       (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾})))
 
Theoremsubmateq 28993* Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   ((𝜑𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))       (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))
 
Theoremsubmatminr1 28994 If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.)
𝐴 = ((1...𝑁) Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀𝐵)    &   𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)       (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽))
 
20.3.9.4  Matrix literals
 
Syntaxclmat 28995 Extend class notation with the literal matrix conversion function.
class litMat
 
Definitiondf-lmat 28996* Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.)
litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(#‘𝑚)), 𝑗 ∈ (1...(#‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
 
Theoremlmatval 28997* Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
(𝑀𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(#‘𝑀)), 𝑗 ∈ (1...(#‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
 
Theoremlmatfval 28998* Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   (𝜑𝐼 ∈ (1...𝑁))    &   (𝜑𝐽 ∈ (1...𝑁))       (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))
 
Theoremlmatfvlem 28999* Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   𝐾 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   𝐼𝑁    &   𝐽𝑁    &   (𝐾 + 1) = 𝐼    &   (𝐿 + 1) = 𝐽    &   (𝑊𝐾) = 𝑋    &   (𝜑 → (𝑋𝐿) = 𝑌)       (𝜑 → (𝐼𝑀𝐽) = 𝑌)
 
Theoremlmatcl 29000* Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
𝑀 = (litMat‘𝑊)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝑊 ∈ Word Word 𝑉)    &   (𝜑 → (#‘𝑊) = 𝑁)    &   ((𝜑𝑖 ∈ (0..^𝑁)) → (#‘(𝑊𝑖)) = 𝑁)    &   𝑉 = (Base‘𝑅)    &   𝑂 = ((1...𝑁) Mat 𝑅)    &   𝑃 = (Base‘𝑂)    &   (𝜑𝑅𝑋)       (𝜑𝑀𝑃)
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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 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