Home Metamath Proof ExplorerTheorem List (p. 194 of 453) < 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 (1-28710) Hilbert Space Explorer (28711-30233) Users' Mathboxes (30234-45286)

Theorem List for Metamath Proof Explorer - 19301-19400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-ring 19301* Define class of all (unital) rings. A unital ring is a set equipped with two everywhere-defined internal operations, whose first one is an additive group structure and the second one is a multiplicative monoid structure, and where the addition is left- and right-distributive for the multiplication. Definition 1 in [BourbakiAlg1] p. 92 or definition of a ring with identity in part Preliminaries of [Roman] p. 19. So that the additive structure must be abelian (see ringcom 19334), care must be taken that in the case of a non-unital ring, the commutativity of addition must be postulated and cannot be proved from the other conditions. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 27-Dec-2014.)
Ring = {𝑓 ∈ Grp ∣ ((mulGrp‘𝑓) ∈ Mnd ∧ [(Base‘𝑓) / 𝑟][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡]𝑥𝑟𝑦𝑟𝑧𝑟 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}

Definitiondf-cring 19302 Define class of all commutative rings. (Contributed by Mario Carneiro, 7-Jan-2015.)
CRing = {𝑓 ∈ Ring ∣ (mulGrp‘𝑓) ∈ CMnd}

Theoremisring 19303* The predicate "is a (unital) ring." Definition of ring with unit in [Schechter] p. 187. (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))

Theoremringgrp 19304 A ring is a group. (Contributed by NM, 15-Sep-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Grp)

Theoremringmgp 19305 A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ Ring → 𝐺 ∈ Mnd)

Theoremiscrng 19306 A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ CMnd))

Theoremcrngmgp 19307 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

Theoremringgrpd 19308 A ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ Ring)       (𝜑𝑅 ∈ Grp)

Theoremringmnd 19309 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ Mnd)

Theoremringmgm 19310 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
(𝑅 ∈ Ring → 𝑅 ∈ Mgm)

Theoremcrngring 19311 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ CRing → 𝑅 ∈ Ring)

Theoremcrngringd 19312 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ CRing)       (𝜑𝑅 ∈ Ring)

Theoremcrnggrpd 19313 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ CRing)       (𝜑𝑅 ∈ Grp)

Theoremmgpf 19314 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
(mulGrp ↾ Ring):Ring⟶Mnd

Theoremringi 19315 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)) ∧ ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍))))

Theoremringcl 19316 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) ∈ 𝐵)

Theoremcrngcom 19317 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑌 · 𝑋))

Theoremiscrng2 19318* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 · 𝑦) = (𝑦 · 𝑥)))

Theoremringass 19319 Associative law for the multiplication operation of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))

Theoremringideu 19320* The unit element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → ∃!𝑢𝐵𝑥𝐵 ((𝑢 · 𝑥) = 𝑥 ∧ (𝑥 · 𝑢) = 𝑥))

Theoremringdi 19321 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))

Theoremringdir 19322 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

Theoremringidcl 19323 The unit element of a ring belongs to the base set of the ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 1𝐵)

Theoremring0cl 19324 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → 0𝐵)

Theoremringidmlem 19325 Lemma for ringlidm 19326 and ringridm 19327. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (( 1 · 𝑋) = 𝑋 ∧ (𝑋 · 1 ) = 𝑋))

Theoremringlidm 19326 The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ( 1 · 𝑋) = 𝑋)

Theoremringridm 19327 The unit element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋 · 1 ) = 𝑋)

Theoremisringid 19328* Properties showing that an element 𝐼 is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → ((𝐼𝐵 ∧ ∀𝑥𝐵 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥)) ↔ 1 = 𝐼))

Theoremringid 19329* The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ∃𝑢𝐵 ((𝑢 · 𝑋) = 𝑋 ∧ (𝑋 · 𝑢) = 𝑋))

Theoremringadd2 19330* A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ∃𝑥𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋))

Theoremrngo2times 19331 A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unit with itself. (Contributed by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐴𝐵) → (𝐴 + 𝐴) = (( 1 + 1 ) · 𝐴))

Theoremringidss 19332 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑀 = ((mulGrp‘𝑅) ↾s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐴𝐵1𝐴) → 1 = (0g𝑀))

Theoremringacl 19333 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Theoremringcom 19334 Commutativity of the additive group of a ring. (See also lmodcom 19682.) (Contributed by Gérard Lang, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremringabl 19335 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Abel)

Theoremringcmn 19336 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ CMnd)

Theoremringpropd 19337* If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))

Theoremcrngpropd 19338* If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing))

Theoremringprop 19339 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)    &   (.r𝐾) = (.r𝐿)       (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)

Theoremisringd 19340* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑𝑅 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)       (𝜑𝑅 ∈ Ring)

Theoremiscrngd 19341* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑𝑅 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 · 𝑦) = (𝑦 · 𝑥))       (𝜑𝑅 ∈ CRing)

Theoremringlz 19342 The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )

Theoremringrz 19343 The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )

Theoremringsrg 19344 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring → 𝑅 ∈ SRing)

Theoremring1eq0 19345 If one and zero are equal, then any two elements of a ring are equal. Alternately, every ring has one distinct from zero except the zero ring containing the single element {0}. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → ( 1 = 0𝑋 = 𝑌))

Theoremring1ne0 19346 If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)) → 10 )

Theoremringinvnz1ne0 19347* In a unitary ring, a left invertible element is different from zero iff 10. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑 → ∃𝑎𝐵 (𝑎 · 𝑋) = 1 )       (𝜑 → (𝑋010 ))

Theoremringinvnzdiv 19348* In a unitary ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑 → ∃𝑎𝐵 (𝑎 · 𝑋) = 1 )    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑋 · 𝑌) = 0𝑌 = 0 ))

Theoremringnegl 19349 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 35351 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → ((𝑁1 ) · 𝑋) = (𝑁𝑋))

Theoremrngnegr 19350 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 35352 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 · (𝑁1 )) = (𝑁𝑋))

Theoremringmneg1 19351 Negation of a product in a ring. (mulneg1 11076 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌)))

Theoremringmneg2 19352 Negation of a product in a ring. (mulneg2 11077 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 · (𝑁𝑌)) = (𝑁‘(𝑋 · 𝑌)))

Theoremringm2neg 19353 Double negation of a product in a ring. (mul2neg 11079 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) · (𝑁𝑌)) = (𝑋 · 𝑌))

Theoremringsubdi 19354 Ring multiplication distributes over subtraction. (subdi 11073 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 · (𝑌 𝑍)) = ((𝑋 · 𝑌) (𝑋 · 𝑍)))

Theoremrngsubdir 19355 Ring multiplication distributes over subtraction. (subdir 11074 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    = (-g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 𝑌) · 𝑍) = ((𝑋 · 𝑍) (𝑌 · 𝑍)))

Theoremmulgass2 19356 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.g𝑅)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌)))

Theoremring1 19357 The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∈ Ring)

Theoremringn0 19358 Rings exist. (Contributed by AV, 29-Apr-2019.)
Ring ≠ ∅

Theoremringlghm 19359* Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 GrpHom 𝑅))

Theoremringrghm 19360* Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑥𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 GrpHom 𝑅))

Theoremgsummulc1 19361* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘𝐴𝑋)) · 𝑌))

Theoremgsummulc2 19362* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘𝐴𝑋))))

Theoremgsummgp0 19363* If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.)
𝐺 = (mulGrp‘𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   ((𝜑𝑛𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ((𝜑𝑛 = 𝑖) → 𝐴 = 𝐵)    &   (𝜑 → ∃𝑖𝑁 𝐵 = 0 )       (𝜑 → (𝐺 Σg (𝑛𝑁𝐴)) = 0 )

Theoremgsumdixp 19364* Distribute a binary product of sums to a sum of binary products in a ring. (Contributed by Mario Carneiro, 8-Mar-2015.) (Revised by AV, 10-Jul-2019.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑥𝐼) → 𝑋𝐵)    &   ((𝜑𝑦𝐽) → 𝑌𝐵)    &   (𝜑 → (𝑥𝐼𝑋) finSupp 0 )    &   (𝜑 → (𝑦𝐽𝑌) finSupp 0 )       (𝜑 → ((𝑅 Σg (𝑥𝐼𝑋)) · (𝑅 Σg (𝑦𝐽𝑌))) = (𝑅 Σg (𝑥𝐼, 𝑦𝐽 ↦ (𝑋 · 𝑌))))

Theoremprdsmgp 19365 The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝑀 = (mulGrp‘𝑌)    &   𝑍 = (𝑆Xs(mulGrp ∘ 𝑅))    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝑊)    &   (𝜑𝑅 Fn 𝐼)       (𝜑 → ((Base‘𝑀) = (Base‘𝑍) ∧ (+g𝑀) = (+g𝑍)))

Theoremprdsmulrcl 19366 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = (.r𝑌)    &   (𝜑𝑆𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶Ring)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)       (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)

Theoremprdsringd 19367 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Ring)       (𝜑𝑌 ∈ Ring)

Theoremprdscrngd 19368 A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶CRing)       (𝜑𝑌 ∈ CRing)

Theoremprds1 19369 Value of the ring unit in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Ring)       (𝜑 → (1r𝑅) = (1r𝑌))

Theorempwsring 19370 A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ Ring ∧ 𝐼𝑉) → 𝑌 ∈ Ring)

Theorempws1 19371 Value of the ring unit in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑅s 𝐼)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑉) → (𝐼 × { 1 }) = (1r𝑌))

Theorempwscrng 19372 A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ CRing ∧ 𝐼𝑉) → 𝑌 ∈ CRing)

Theorempwsmgp 19373 The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
𝑌 = (𝑅s 𝐼)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (𝑀s 𝐼)    &   𝑁 = (mulGrp‘𝑌)    &   𝐵 = (Base‘𝑁)    &   𝐶 = (Base‘𝑍)    &    + = (+g𝑁)    &    = (+g𝑍)       ((𝑅𝑉𝐼𝑊) → (𝐵 = 𝐶+ = ))

Theoremimasring 19374* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝑈 = (𝐹s 𝑅))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   (𝜑𝐹:𝑉onto𝐵)    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))    &   ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → (((𝐹𝑎) = (𝐹𝑝) ∧ (𝐹𝑏) = (𝐹𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞))))    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝑈 ∈ Ring ∧ (𝐹1 ) = (1r𝑈)))

Theoremqusring2 19375* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝑈 = (𝑅 /s ))    &   (𝜑𝑉 = (Base‘𝑅))    &    + = (+g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   (𝜑 Er 𝑉)    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 + 𝑏) (𝑝 + 𝑞)))    &   (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 · 𝑏) (𝑝 · 𝑞)))    &   (𝜑𝑅 ∈ Ring)       (𝜑 → (𝑈 ∈ Ring ∧ [ 1 ] = (1r𝑈)))

Theoremcrngbinom 19376* The binomial theorem for commutative rings (special case of csrgbinom 19298): (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴𝑘) · (𝐵↑(𝑁𝑘)). (Contributed by AV, 24-Aug-2019.)
𝑆 = (Base‘𝑅)    &    × = (.r𝑅)    &    · = (.g𝑅)    &    + = (+g𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (.g𝐺)       (((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴𝑆𝐵𝑆)) → (𝑁 (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁𝑘) 𝐴) × (𝑘 𝐵))))))

10.3.4  Opposite ring

Syntaxcoppr 19377 The opposite ring operation.
class oppr

Definitiondf-oppr 19378 Define an opposite ring, which is the same as the original ring but with multiplication written the other way around. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(.r‘ndx), tpos (.r𝑓)⟩))

Theoremopprval 19379 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)       𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩)

Theoremopprmulfval 19380 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    = (.r𝑂)        = tpos ·

Theoremopprmul 19381 Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    = (.r𝑂)       (𝑋 𝑌) = (𝑌 · 𝑋)

Theoremcrngoppr 19382 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 19452). (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    = (.r𝑂)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑋 𝑌))

Theoremopprlem 19383 Lemma for opprbas 19384 and oppradd 19385. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 3       (𝐸𝑅) = (𝐸𝑂)

Theoremopprbas 19384 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑂)

Theoremoppradd 19385 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    + = (+g𝑅)        + = (+g𝑂)

Theoremopprring 19386 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Ring → 𝑂 ∈ Ring)

Theoremopprringb 19387 Bidirectional form of opprring 19386. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)

Theoremoppr0 19388 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    0 = (0g𝑅)        0 = (0g𝑂)

Theoremoppr1 19389 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    1 = (1r𝑅)        1 = (1r𝑂)

Theoremopprneg 19390 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑂 = (oppr𝑅)    &   𝑁 = (invg𝑅)       𝑁 = (invg𝑂)

Theoremopprsubg 19391 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (SubGrp‘𝑅) = (SubGrp‘𝑂)

Theoremmulgass3 19392 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.g𝑅)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌)))

10.3.5  Divisibility

Syntaxcdsr 19393 Ring divisibility relation.
class r

Syntaxcui 19394 Ring unit.
class Unit

Syntaxcir 19395 Ring irreducibles.
class Irred

Definitiondf-dvdsr 19396* Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through (∥r‘(oppr𝑅)). (Contributed by Mario Carneiro, 1-Dec-2014.)
r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r𝑤)𝑥) = 𝑦)})

Definitiondf-unit 19397 Define the set of units in a ring, that is, all elements with a left and right multiplicative inverse. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))

Definitiondf-irred 19398* Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred = (𝑤 ∈ V ↦ ((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏{𝑧𝑏 ∣ ∀𝑥𝑏𝑦𝑏 (𝑥(.r𝑤)𝑦) ≠ 𝑧})

Theoremreldvdsr 19399 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
= (∥r𝑅)       Rel

Theoremdvdsrval 19400* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)        = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑥) = 𝑦)}

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 45201-45286
 Copyright terms: Public domain < Previous  Next >