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Theorem List for Metamath Proof Explorer - 29101-29200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremslmdvsass 29101 Associative law for scalar product. (ax-hvmulass 27248 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))

Theoremslmd0cl 29102 The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)       (𝑊 ∈ SLMod → 0𝐾)

Theoremslmd1cl 29103 The ring unit in a semiring left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    1 = (1r𝐹)       (𝑊 ∈ SLMod → 1𝐾)

Theoremslmdvs1 29104 Scalar product with ring unit. (ax-hvmulid 27247 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 1 · 𝑋) = 𝑋)

Theoremslmd0vcl 29105 The zero vector is a vector. (ax-hv0cl 27244 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ SLMod → 0𝑉)

Theoremslmd0vlid 29106 Left identity law for the zero vector. (hvaddid2 27264 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)

Theoremslmd0vrid 29107 Right identity law for the zero vector. (ax-hvaddid 27245 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)

Theoremslmd0vs 29108 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27251 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )

Theoremslmdvs0 29109 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 27265 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )

21.3.8.6  Finitely supported group sums - misc additions

Theoremgsumle 29110 A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐹𝑟 𝐺)       (𝜑 → (𝑀 Σg 𝐹) (𝑀 Σg 𝐺))

Theoremgsummpt2co 29111* 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 29112* Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 18194. (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 29113* 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 29114* 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 29115* 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 29116* 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 29117 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 29118 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))

Theoremrngurd 29119* 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 29120 1r is unaffected by restriction. This is a bit more generic than subrg1 18613. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 1𝐴𝐴𝐵) → 1 = (1r𝑆))

Theoremdvrdir 29121 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 29122 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 29123* 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 29124 Cancellation law for common factor in ratio. (divcan5 10606 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 29125 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))

21.3.8.8  Ordered rings and fields

Syntaxcorng 29126 Extend class notation with the class of all ordered rings.
class oRing

Syntaxcofld 29127 Extend class notation with the class of all ordered fields.
class oField

Definitiondf-orng 29128* 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 29129 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 29130* 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 29131 An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ Ring)

Theoremorngogrp 29132 An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Theoremisofld 29133 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 29134 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 29135 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 29136 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 29137 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 29138 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 29139 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 29140 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 29141 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Field)

Theoremofldtos 29142 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Toset)

Theoremorng0le1 29143 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 29144 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 29145 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 29146 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 29147 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 29148* 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)))

21.3.8.9  Ring homomorphisms - misc additions

Theoremrhmdvdsr 29149 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 29150 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 29151 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹𝐴) ∈ (Unit‘𝑆))

Theoremrhmdvd 29152 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 29153 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 29154 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 )

21.3.8.10  Scalar restriction operation

Syntaxcresv 29155 Extend class notation with the scalar restriction operation.
class v

Definitiondf-resv 29156* 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 29157 The scalar restriction is a proper operator, so it can be used with ovprc1 6582. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom ↾v

Theoremresvval 29158 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 29159 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 29160 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 29161 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 29162 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 29163 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))

Theoremresvplusg 29164 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))

Theoremresvvsca 29165 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))

Theoremresvmulr 29166 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = (.r𝐺)       (𝐴𝑉· = (.r𝐻))

Theoremresv0g 29167 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    0 = (0g𝐺)       (𝐴𝑉0 = (0g𝐻))

Theoremresv1r 29168 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    1 = (1r𝐺)       (𝐴𝑉1 = (1r𝐻))

Theoremresvcmn 29169 Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)       (𝐴𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd))

21.3.8.11  The commutative ring of gaussian integers

Theoremgzcrng 29170 The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.)
(ℂflds ℤ[i]) ∈ CRing

21.3.8.12  The archimedean ordered field of real numbers

Theoremreofld 29171 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
fld ∈ oField

Theoremnn0omnd 29172 The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(ℂflds0) ∈ oMnd

Theoremrearchi 29173 The field of the real numbers is Archimedean. See also arch 11166. (Contributed by Thierry Arnoux, 9-Apr-2018.)
fld ∈ Archi

Theoremnn0archi 29174 The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(ℂflds0) ∈ Archi

Theoremxrge0slmod 29175 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  Matrices

21.3.9.1  The symmetric group

Theoremsymgfcoeu 29176* Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
𝐺 = (Base‘(SymGrp‘𝐷))       ((𝐷𝑉𝑃𝐺𝑄𝐺) → ∃!𝑝𝐺 𝑄 = (𝑃𝑝))

21.3.9.2  Permutation Signs

Theorempsgndmfi 29177 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 29178 Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.)
𝑆 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1)

Theorempmtrprfv2 29179 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 29180 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 29181* Lemma for psgnfzto1st 29186. 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 29182* 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 29183* Special case where the permutation defined in psgnfzto1st 29186 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 29184* 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 29185* 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 29186* 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)))

21.3.9.3  Submatrices

Syntaxcsmat 29187 Syntax for a function generating submatrixes.
class subMat1

Definitiondf-smat 29188* 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 20202. (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 29189* 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 29190 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 29191 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 29192 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 29193 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 29194 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 29195 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 29196 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 29197* 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 29198 The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20208. (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 29199 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 29200 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)))))

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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 42301-42360
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