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Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcrngmgp 18601 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ CRing → 𝐺 ∈ CMnd)

Theoremringmnd 18602 A ring is a monoid under addition. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ Mnd)

Theoremringmgm 18603 A ring is a magma. (Contributed by AV, 31-Jan-2020.)
(𝑅 ∈ Ring → 𝑅 ∈ Mgm)

Theoremcrngring 18604 A commutative ring is a ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ CRing → 𝑅 ∈ Ring)

Theoremmgpf 18605 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
(mulGrp ↾ Ring):Ring⟶Mnd

Theoremringi 18606 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 18607 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 18608 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 18609* 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 18610 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 18611* 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 18612 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 18613 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 18614 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 18615 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 18616 Lemma for ringlidm 18617 and ringridm 18618. (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 18617 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 18618 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 18619* 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 18620* 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 18621* 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 18622 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 18623 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 18624 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 18625 Commutativity of the additive group of a ring. (See also lmodcom 18957.) (Contributed by Gérard Lang, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremringabl 18626 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring → 𝑅 ∈ Abel)

Theoremringcmn 18627 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring → 𝑅 ∈ CMnd)

Theoremringpropd 18628* 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 18629* 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 18630 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 18631* 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 18632* 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 18633 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 18634 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 18635 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring → 𝑅 ∈ SRing)

Theoremring1eq0 18636 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 18637 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 18638* 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 18639* 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 18640 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 33870 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 18641 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 33871 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 18642 Negation of a product in a ring. (mulneg1 10504 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 18643 Negation of a product in a ring. (mulneg2 10505 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 18644 Double negation of a product in a ring. (mul2neg 10507 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑁 = (invg𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑁𝑋) · (𝑁𝑌)) = (𝑋 · 𝑌))

Theoremringsubdi 18645 Ring multiplication distributes over subtraction. (subdi 10501 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 18646 Ring multiplication distributes over subtraction. (subdir 10502 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 18647 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 18648 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 18649 Rings exist. (Contributed by AV, 29-Apr-2019.)
Ring ≠ ∅

Theoremringlghm 18650* 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 18651* 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 18652* 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 18653* 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 18654* 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 18655* 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 18656 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 18657 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 18658 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Ring)       (𝜑𝑌 ∈ Ring)

Theoremprdscrngd 18659 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 18660 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 18661 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 18662 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 18663 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 18664 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 18665* 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 18666* 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 18667* The binomial theorem for commutative rings (special case of csrgbinom 18592): (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 18668 The opposite ring operation.
class oppr

Definitiondf-oppr 18669 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 18670 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 18671 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 18672 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 18673 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 18743). (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑂 = (oppr𝑅)    &    = (.r𝑂)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) = (𝑋 𝑌))

Theoremopprlem 18674 Lemma for opprbas 18675 and oppradd 18676. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 < 3       (𝐸𝑅) = (𝐸𝑂)

Theoremopprbas 18675 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑂)

Theoremoppradd 18676 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    + = (+g𝑅)        + = (+g𝑂)

Theoremopprring 18677 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 18678 Bidirectional form of opprring 18677. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)

Theoremoppr0 18679 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    0 = (0g𝑅)        0 = (0g𝑂)

Theoremoppr1 18680 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (oppr𝑅)    &    1 = (1r𝑅)        1 = (1r𝑂)

Theoremopprneg 18681 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 18682 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (SubGrp‘𝑅) = (SubGrp‘𝑂)

Theoremmulgass3 18683 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 18684 Ring divisibility relation.
class r

Syntaxcui 18685 Ring unit.
class Unit

Syntaxcir 18686 Ring irreducibles.
class Irred

Definitiondf-dvdsr 18687* 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 18688 Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit = (𝑤 ∈ V ↦ (((∥r𝑤) ∩ (∥r‘(oppr𝑤))) “ {(1r𝑤)}))

Definitiondf-irred 18689* 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 18690 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
= (∥r𝑅)       Rel

Theoremdvdsrval 18691* 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 18692* 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 18693* 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 18694 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 18695 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       (𝑋 𝑌𝑋𝐵)

Theoremdvdsrcl2 18696 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 𝑌) → 𝑌𝐵)

Theoremdvdsrid 18697 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 18698 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝑌 𝑍𝑍 𝑋) → 𝑌 𝑋)

Theoremdvdsrmul1 18699 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 18700 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → 𝑋 (𝑁𝑋))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 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