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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremringacl 18401 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 18402 Commutativity of the additive group of a ring. (See also lmodcom 18732.) (Contributed by Gérard Lang, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremringabl 18403 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Abel)

Theoremringcmn 18404 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ CMnd)

Theoremringpropd 18405* 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 18406* 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 18407 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 18408* 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 18409* 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 18410 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 18411 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 18412 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring → 𝑅 ∈ SRing)

Theoremring1eq0 18413 If one and zero are equal, then any two elements of a ring are equal. Alternatively, 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 18414 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 18415* 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 18416* 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 18417 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 32910 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 18418 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 32911 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 18419 Negation of a product in a ring. (mulneg1 10345 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 18420 Negation of a product in a ring. (mulneg2 10346 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 18421 Double negation of a product in a ring. (mul2neg 10348 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) · (𝑁𝑌)) = (𝑋 · 𝑌))

Theoremringsubdi 18422 Ring multiplication distributes over subtraction. (subdi 10342 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 18423 Ring multiplication distributes over subtraction. (subdir 10343 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 18424 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 18425 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 18426 Rings exist. (Contributed by AV, 29-Apr-2019.)
Ring ≠ ∅

Theoremringlghm 18427* 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 18428* 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 18429* 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 18430* 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 18431* 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 18432* 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 18433 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 18434 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 18435 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Ring)       (𝜑𝑌 ∈ Ring)

Theoremprdscrngd 18436 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 18437 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 18438 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 18439 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 18440 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 18441 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 18442* 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 18443* 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 18444* The binomial theorem for commutative rings (special case of csrgbinom 18369): (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))))))

10.4.4  Opposite ring

Syntaxcoppr 18445 The opposite ring operation.
class oppr

Definitiondf-oppr 18446 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 18447 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 18448 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 18449 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 18450 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 18520). (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    = (.r𝑂)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑋 𝑌))

Theoremopprlem 18451 Lemma for opprbas 18452 and oppradd 18453. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 3       (𝐸𝑅) = (𝐸𝑂)

Theoremopprbas 18452 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑂)

Theoremoppradd 18453 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    + = (+g𝑅)        + = (+g𝑂)

Theoremopprring 18454 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 18455 Bidirectional form of opprring 18454. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)

Theoremoppr0 18456 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    0 = (0g𝑅)        0 = (0g𝑂)

Theoremoppr1 18457 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    1 = (1r𝑅)        1 = (1r𝑂)

Theoremopprneg 18458 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 18459 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (SubGrp‘𝑅) = (SubGrp‘𝑂)

Theoremmulgass3 18460 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)))

10.4.5  Divisibility

Syntaxcdsr 18461 Ring divisibility relation.
class r

Syntaxcui 18462 Ring unit.
class Unit

Syntaxcir 18463 Ring irreducibles.
class Irred

Definitiondf-dvdsr 18464* 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 18465 Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))

Definitiondf-irred 18466* 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 18467 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
= (∥r𝑅)       Rel

Theoremdvdsrval 18468* 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)}

Theoremdvdsr 18469* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)       (𝑋 𝑌 ↔ (𝑋𝐵 ∧ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))

Theoremdvdsr2 18470* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)       (𝑋𝐵 → (𝑋 𝑌 ↔ ∃𝑧𝐵 (𝑧 · 𝑋) = 𝑌))

Theoremdvdsrmul 18471 A left-multiple of 𝑋 is divisible by 𝑋. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)       ((𝑋𝐵𝑌𝐵) → 𝑋 (𝑌 · 𝑋))

Theoremdvdsrcl 18472 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       (𝑋 𝑌𝑋𝐵)

Theoremdvdsrcl2 18473 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 𝑌) → 𝑌𝐵)

Theoremdvdsrid 18474 An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → 𝑋 𝑋)

Theoremdvdsrtr 18475 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝑌 𝑍𝑍 𝑋) → 𝑌 𝑋)

Theoremdvdsrmul1 18476 The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑍𝐵𝑋 𝑌) → (𝑋 · 𝑍) (𝑌 · 𝑍))

Theoremdvdsrneg 18477 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → 𝑋 (𝑁𝑋))

Theoremdvdsr01 18478 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 19104.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → 𝑋 0 )

Theoremdvdsr02 18479 Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ( 0 𝑋𝑋 = 0 ))

Theoremisunit 18480 Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)
𝑈 = (Unit‘𝑅)    &    1 = (1r𝑅)    &    = (∥r𝑅)    &   𝑆 = (oppr𝑅)    &   𝐸 = (∥r𝑆)       (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))

Theorem1unit 18481 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑈 = (Unit‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 1𝑈)

Theoremunitcl 18482 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑋𝑈𝑋𝐵)

Theoremunitss 18483 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)       𝑈𝐵

Theoremopprunit 18484 Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝑆 = (oppr𝑅)       𝑈 = (Unit‘𝑆)

Theoremcrngunit 18485 Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝑈 = (Unit‘𝑅)    &    1 = (1r𝑅)    &    = (∥r𝑅)       (𝑅 ∈ CRing → (𝑋𝑈𝑋 1 ))

Theoremdvdsunit 18486 A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝑈 = (Unit‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ CRing ∧ 𝑌 𝑋𝑋𝑈) → 𝑌𝑈)

Theoremunitmulcl 18487 The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝑈) → (𝑋 · 𝑌) ∈ 𝑈)

Theoremunitmulclb 18488 Reversal of unitmulcl 18487 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) ∈ 𝑈 ↔ (𝑋𝑈𝑌𝑈)))

Theoremunitgrpbas 18489 The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       𝑈 = (Base‘𝐺)

Theoremunitgrp 18490 The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Theoremunitabl 18491 The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       (𝑅 ∈ CRing → 𝐺 ∈ Abel)

Theoremunitgrpid 18492 The identity of the multiplicative group is 1r. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 1 = (0g𝐺))

Theoremunitsubm 18493 The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (Unit‘𝑅)    &   𝑀 = (mulGrp‘𝑅)       (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀))

Syntaxcinvr 18494 Extend class notation with multiplicative inverse.
class invr

Definitiondf-invr 18495 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))

Theoreminvrfval 18496 Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    &   𝐼 = (invr𝑅)       𝐼 = (invg𝐺)

Theoremunitinvcl 18497 The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝐼𝑋) ∈ 𝑈)

Theoremunitinvinv 18498 The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝐼‘(𝐼𝑋)) = 𝑋)

Theoremringinvcl 18499 The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝐼𝑋) ∈ 𝐵)

Theoremunitlinv 18500 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝐼𝑋) · 𝑋) = 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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