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Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremringlidm 18301 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 18302 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 18303* 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 18304* 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 18305* 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 18306 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 18307 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 18308 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 18309 Commutativity of the additive group of a ring. (See also lmodcom 18636.) (Contributed by Gérard Lang, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theoremringabl 18310 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Abel)
 
Theoremringcmn 18311 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ CMnd)
 
Theoremringpropd 18312* 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 18313* 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 18314 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 18315* 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 18316* 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 18317 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 18318 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 18319 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring → 𝑅 ∈ SRing)
 
Theoremring1eq0 18320 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 18321 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 18322* 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 18323* 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 18324 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 32800 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 18325 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 32801 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 18326 Negation of a product in a ring. (mulneg1 10217 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 18327 Negation of a product in a ring. (mulneg2 10218 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 18328 Double negation of a product in a ring. (mul2neg 10220 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) · (𝑁𝑌)) = (𝑋 · 𝑌))
 
Theoremringsubdi 18329 Ring multiplication distributes over subtraction. (subdi 10214 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 18330 Ring multiplication distributes over subtraction. (subdir 10215 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 18331 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 18332 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 18333 Rings exist. (Contributed by AV, 29-Apr-2019.)
Ring ≠ ∅
 
Theoremringlghm 18334* 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 18335* 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 18336* 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 18337* 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 18338* 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 18339* 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 18340 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 18341 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 18342 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Ring)       (𝜑𝑌 ∈ Ring)
 
Theoremprdscrngd 18343 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 18344 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 18345 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 18346 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 18347 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 18348 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 18349* 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 18350* 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 18351* The binomial theorem for commutative rings (special case of csrgbinom 18276): (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 18352 The opposite ring operation.
class oppr
 
Definitiondf-oppr 18353 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 18354 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 18355 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 18356 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 18357 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 18427). (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    = (.r𝑂)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑋 𝑌))
 
Theoremopprlem 18358 Lemma for opprbas 18359 and oppradd 18360. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 3       (𝐸𝑅) = (𝐸𝑂)
 
Theoremopprbas 18359 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑂)
 
Theoremoppradd 18360 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    + = (+g𝑅)        + = (+g𝑂)
 
Theoremopprring 18361 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 18362 Bidirectional form of opprring 18361. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)
 
Theoremoppr0 18363 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    0 = (0g𝑅)        0 = (0g𝑂)
 
Theoremoppr1 18364 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    1 = (1r𝑅)        1 = (1r𝑂)
 
Theoremopprneg 18365 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 18366 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (SubGrp‘𝑅) = (SubGrp‘𝑂)
 
Theoremmulgass3 18367 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 18368 Ring divisibility relation.
class r
 
Syntaxcui 18369 Ring unit.
class Unit
 
Syntaxcir 18370 Ring irreducibles.
class Irred
 
Definitiondf-dvdsr 18371* 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 18372 Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))
 
Definitiondf-irred 18373* 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 18374 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
= (∥r𝑅)       Rel
 
Theoremdvdsrval 18375* 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 18376* 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 18377* 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 18378 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 18379 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       (𝑋 𝑌𝑋𝐵)
 
Theoremdvdsrcl2 18380 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 𝑌) → 𝑌𝐵)
 
Theoremdvdsrid 18381 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 18382 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝑌 𝑍𝑍 𝑋) → 𝑌 𝑋)
 
Theoremdvdsrmul1 18383 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 18384 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → 𝑋 (𝑁𝑋))
 
Theoremdvdsr01 18385 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 19008.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → 𝑋 0 )
 
Theoremdvdsr02 18386 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 18387 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 18388 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑈 = (Unit‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 1𝑈)
 
Theoremunitcl 18389 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑋𝑈𝑋𝐵)
 
Theoremunitss 18390 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 18391 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 18392 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 18393 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 18394 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 18395 Reversal of unitmulcl 18394 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 18396 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 18397 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 18398 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 18399 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 18400 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))
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