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Theorem List for Metamath Proof Explorer - 19901-20000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcrngringd 19901 A commutative ring is a ring. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Ring)
 
Theoremcrnggrpd 19902 A commutative ring is a group. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ CRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Grp)
 
Theoremmgpf 19903 Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015.)
(mulGrp β†Ύ Ring):Ring⟢Mnd
 
Theoremringdilem 19904 Properties of a unital ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 Β· (π‘Œ + 𝑍)) = ((𝑋 Β· π‘Œ) + (𝑋 Β· 𝑍)) ∧ ((𝑋 + π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) + (π‘Œ Β· 𝑍))))
 
Theoremringcl 19905 Closure of the multiplication operation of a ring. (Contributed by NM, 26-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 Β· π‘Œ) ∈ 𝐡)
 
Theoremcrngcom 19906 A commutative ring's multiplication operation is commutative. (Contributed by Mario Carneiro, 7-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 Β· π‘Œ) = (π‘Œ Β· 𝑋))
 
Theoremiscrng2 19907* A commutative ring is a ring whose multiplication is a commutative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ CRing ↔ (𝑅 ∈ Ring ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯)))
 
Theoremringass 19908 Associative law for multiplication in a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 Β· π‘Œ) Β· 𝑍) = (𝑋 Β· (π‘Œ Β· 𝑍)))
 
Theoremringideu 19909* The unity element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ βˆƒ!𝑒 ∈ 𝐡 βˆ€π‘₯ ∈ 𝐡 ((𝑒 Β· π‘₯) = π‘₯ ∧ (π‘₯ Β· 𝑒) = π‘₯))
 
Theoremringcld 19910 Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· π‘Œ) ∈ 𝐡)
 
Theoremringdi 19911 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ (𝑋 Β· (π‘Œ + 𝑍)) = ((𝑋 Β· π‘Œ) + (𝑋 Β· 𝑍)))
 
Theoremringdir 19912 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Steve Rodriguez, 9-Sep-2007.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 + π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) + (π‘Œ Β· 𝑍)))
 
Theoremringidcl 19913 The unity 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.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 1 ∈ 𝐡)
 
Theoremring0cl 19914 The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 0 ∈ 𝐡)
 
Theoremringidmlem 19915 Lemma for ringlidm 19916 and ringridm 19917. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (( 1 Β· 𝑋) = 𝑋 ∧ (𝑋 Β· 1 ) = 𝑋))
 
Theoremringlidm 19916 The unity element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ ( 1 Β· 𝑋) = 𝑋)
 
Theoremringridm 19917 The unity element of a ring is a right multiplicative identity. (Contributed by NM, 15-Sep-2011.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 Β· 1 ) = 𝑋)
 
Theoremisringid 19918* Properties showing that an element 𝐼 is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ ((𝐼 ∈ 𝐡 ∧ βˆ€π‘₯ ∈ 𝐡 ((𝐼 Β· π‘₯) = π‘₯ ∧ (π‘₯ Β· 𝐼) = π‘₯)) ↔ 1 = 𝐼))
 
Theoremringid 19919* 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.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ βˆƒπ‘’ ∈ 𝐡 ((𝑒 Β· 𝑋) = 𝑋 ∧ (𝑋 Β· 𝑒) = 𝑋))
 
Theoremringadd2 19920* 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.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝑋 + 𝑋) = ((π‘₯ + π‘₯) Β· 𝑋))
 
Theoremrngo2times 19921 A ring element plus itself is two times the element. "Two" in an arbitrary unital ring is the sum of the unity element with itself. (Contributed by AV, 24-Aug-2021.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 + 𝐴) = (( 1 + 1 ) Β· 𝐴))
 
Theoremringidss 19922 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 = ((mulGrpβ€˜π‘…) β†Ύs 𝐴)    &   π΅ = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐴 βŠ† 𝐡 ∧ 1 ∈ 𝐴) β†’ 1 = (0gβ€˜π‘€))
 
Theoremringacl 19923 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 + π‘Œ) ∈ 𝐡)
 
Theoremringcom 19924 Commutativity of the additive group of a ring. (See also lmodcom 20291.) (Contributed by GΓ©rard Lang, 4-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 + π‘Œ) = (π‘Œ + 𝑋))
 
Theoremringabl 19925 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ Abel)
 
Theoremringcmn 19926 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ CMnd)
 
Theoremringabld 19927 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
(πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑅 ∈ Abel)
 
Theoremringcmnd 19928 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝑅 ∈ CMnd)
 
Theoremringpropd 19929* 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.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring))
 
Theoremcrngpropd 19930* 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.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing))
 
Theoremringprop 19931 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.)
(Baseβ€˜πΎ) = (Baseβ€˜πΏ)    &   (+gβ€˜πΎ) = (+gβ€˜πΏ)    &   (.rβ€˜πΎ) = (.rβ€˜πΏ)    β‡’   (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)
 
Theoremisringd 19932* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ + = (+gβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Grp)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ Β· 𝑦) Β· 𝑧) = (π‘₯ Β· (𝑦 Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))    &   (πœ‘ β†’ 1 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ( 1 Β· π‘₯) = π‘₯)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯ Β· 1 ) = π‘₯)    β‡’   (πœ‘ β†’ 𝑅 ∈ Ring)
 
Theoremiscrngd 19933* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ + = (+gβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Grp)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ Β· 𝑦) Β· 𝑧) = (π‘₯ Β· (𝑦 Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ (π‘₯ Β· (𝑦 + 𝑧)) = ((π‘₯ Β· 𝑦) + (π‘₯ Β· 𝑧)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡)) β†’ ((π‘₯ + 𝑦) Β· 𝑧) = ((π‘₯ Β· 𝑧) + (𝑦 Β· 𝑧)))    &   (πœ‘ β†’ 1 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ ( 1 Β· π‘₯) = π‘₯)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯ Β· 1 ) = π‘₯)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (π‘₯ Β· 𝑦) = (𝑦 Β· π‘₯))    β‡’   (πœ‘ β†’ 𝑅 ∈ CRing)
 
Theoremringlz 19934 The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ ( 0 Β· 𝑋) = 0 )
 
Theoremringrz 19935 The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 Β· 0 ) = 0 )
 
Theoremringsrg 19936 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
 
Theoremring1eq0 19937 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.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ( 1 = 0 β†’ 𝑋 = π‘Œ))
 
Theoremring1ne0 19938 If a ring has at least two elements, its one and zero are different. (Contributed by AV, 13-Apr-2019.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 1 < (β™―β€˜π΅)) β†’ 1 β‰  0 )
 
Theoremringinvnz1ne0 19939* In a unital ring, a left invertible element is different from zero iff 1 β‰  0. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝐡 (π‘Ž Β· 𝑋) = 1 )    β‡’   (πœ‘ β†’ (𝑋 β‰  0 ↔ 1 β‰  0 ))
 
Theoremringinvnzdiv 19940* In a unital 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.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ βˆƒπ‘Ž ∈ 𝐡 (π‘Ž Β· 𝑋) = 1 )    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 Β· π‘Œ) = 0 ↔ π‘Œ = 0 ))
 
Theoremringnegl 19941 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 36286 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((π‘β€˜ 1 ) Β· 𝑋) = (π‘β€˜π‘‹))
 
Theoremrngnegr 19942 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 36287 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· (π‘β€˜ 1 )) = (π‘β€˜π‘‹))
 
Theoremringmneg1 19943 Negation of a product in a ring. (mulneg1 11525 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((π‘β€˜π‘‹) Β· π‘Œ) = (π‘β€˜(𝑋 Β· π‘Œ)))
 
Theoremringmneg2 19944 Negation of a product in a ring. (mulneg2 11526 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· (π‘β€˜π‘Œ)) = (π‘β€˜(𝑋 Β· π‘Œ)))
 
Theoremringm2neg 19945 Double negation of a product in a ring. (mul2neg 11528 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((π‘β€˜π‘‹) Β· (π‘β€˜π‘Œ)) = (𝑋 Β· π‘Œ))
 
Theoremringsubdi 19946 Ring multiplication distributes over subtraction. (subdi 11522 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 Β· (π‘Œ βˆ’ 𝑍)) = ((𝑋 Β· π‘Œ) βˆ’ (𝑋 Β· 𝑍)))
 
Theoremrngsubdir 19947 Ring multiplication distributes over subtraction. (subdir 11523 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 βˆ’ π‘Œ) Β· 𝑍) = ((𝑋 Β· 𝑍) βˆ’ (π‘Œ Β· 𝑍)))
 
Theoremmulgass2 19948 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑁 ∈ β„€ ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((𝑁 Β· 𝑋) Γ— π‘Œ) = (𝑁 Β· (𝑋 Γ— π‘Œ)))
 
Theoremring1 19949 The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)
𝑀 = {⟨(Baseβ€˜ndx), {𝑍}⟩, ⟨(+gβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩, ⟨(.rβ€˜ndx), {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©}⟩}    β‡’   (𝑍 ∈ 𝑉 β†’ 𝑀 ∈ Ring)
 
Theoremringn0 19950 Rings exist. (Contributed by AV, 29-Apr-2019.)
Ring β‰  βˆ…
 
Theoremringlghm 19951* 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.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (π‘₯ ∈ 𝐡 ↦ (𝑋 Β· π‘₯)) ∈ (𝑅 GrpHom 𝑅))
 
Theoremringrghm 19952* 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.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (π‘₯ ∈ 𝐡 ↦ (π‘₯ Β· 𝑋)) ∈ (𝑅 GrpHom 𝑅))
 
Theoremgsummulc1 19953* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) finSupp 0 )    β‡’   (πœ‘ β†’ (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ (𝑋 Β· π‘Œ))) = ((𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋)) Β· π‘Œ))
 
Theoremgsummulc2 19954* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝑋) finSupp 0 )    β‡’   (πœ‘ β†’ (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ (π‘Œ Β· 𝑋))) = (π‘Œ Β· (𝑅 Ξ£g (π‘˜ ∈ 𝐴 ↦ 𝑋))))
 
Theoremgsummgp0 19955* 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 = (mulGrpβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   ((πœ‘ ∧ 𝑛 ∈ 𝑁) β†’ 𝐴 ∈ (Baseβ€˜π‘…))    &   ((πœ‘ ∧ 𝑛 = 𝑖) β†’ 𝐴 = 𝐡)    &   (πœ‘ β†’ βˆƒπ‘– ∈ 𝑁 𝐡 = 0 )    β‡’   (πœ‘ β†’ (𝐺 Ξ£g (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 )
 
Theoremgsumdixp 19956* 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.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐽 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ 𝑋 ∈ 𝐡)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐽) β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ (π‘₯ ∈ 𝐼 ↦ 𝑋) finSupp 0 )    &   (πœ‘ β†’ (𝑦 ∈ 𝐽 ↦ π‘Œ) finSupp 0 )    β‡’   (πœ‘ β†’ ((𝑅 Ξ£g (π‘₯ ∈ 𝐼 ↦ 𝑋)) Β· (𝑅 Ξ£g (𝑦 ∈ 𝐽 ↦ π‘Œ))) = (𝑅 Ξ£g (π‘₯ ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 Β· π‘Œ))))
 
Theoremprdsmgp 19957 The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
π‘Œ = (𝑆Xs𝑅)    &   π‘€ = (mulGrpβ€˜π‘Œ)    &   π‘ = (𝑆Xs(mulGrp ∘ 𝑅))    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑆 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅 Fn 𝐼)    β‡’   (πœ‘ β†’ ((Baseβ€˜π‘€) = (Baseβ€˜π‘) ∧ (+gβ€˜π‘€) = (+gβ€˜π‘)))
 
Theoremprdsmulrcl 19958 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.)
π‘Œ = (𝑆Xs𝑅)    &   π΅ = (Baseβ€˜π‘Œ)    &    Β· = (.rβ€˜π‘Œ)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅:𝐼⟢Ring)    &   (πœ‘ β†’ 𝐹 ∈ 𝐡)    &   (πœ‘ β†’ 𝐺 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝐹 Β· 𝐺) ∈ 𝐡)
 
Theoremprdsringd 19959 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
π‘Œ = (𝑆Xs𝑅)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅:𝐼⟢Ring)    β‡’   (πœ‘ β†’ π‘Œ ∈ Ring)
 
Theoremprdscrngd 19960 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.)
π‘Œ = (𝑆Xs𝑅)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅:𝐼⟢CRing)    β‡’   (πœ‘ β†’ π‘Œ ∈ CRing)
 
Theoremprds1 19961 Value of the ring unity in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.)
π‘Œ = (𝑆Xs𝑅)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑆 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅:𝐼⟢Ring)    β‡’   (πœ‘ β†’ (1r ∘ 𝑅) = (1rβ€˜π‘Œ))
 
Theorempwsring 19962 A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
π‘Œ = (𝑅 ↑s 𝐼)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ π‘Œ ∈ Ring)
 
Theorempws1 19963 Value of the ring unity in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.)
π‘Œ = (𝑅 ↑s 𝐼)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) β†’ (𝐼 Γ— { 1 }) = (1rβ€˜π‘Œ))
 
Theorempwscrng 19964 A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
π‘Œ = (𝑅 ↑s 𝐼)    β‡’   ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑉) β†’ π‘Œ ∈ CRing)
 
Theorempwsmgp 19965 The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
π‘Œ = (𝑅 ↑s 𝐼)    &   π‘€ = (mulGrpβ€˜π‘…)    &   π‘ = (𝑀 ↑s 𝐼)    &   π‘ = (mulGrpβ€˜π‘Œ)    &   π΅ = (Baseβ€˜π‘)    &   πΆ = (Baseβ€˜π‘)    &    + = (+gβ€˜π‘)    &    ✚ = (+gβ€˜π‘)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ π‘Š) β†’ (𝐡 = 𝐢 ∧ + = ✚ ))
 
Theorempwspjmhmmgpd 19966* The projection given by pwspjmhm 18575 is also a monoid homomorphism between the respective multiplicative groups. (Contributed by SN, 30-Jul-2024.)
π‘Œ = (𝑅 ↑s 𝐼)    &   π΅ = (Baseβ€˜π‘Œ)    &   π‘€ = (mulGrpβ€˜π‘Œ)    &   π‘‡ = (mulGrpβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝐴 ∈ 𝐼)    β‡’   (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ (π‘₯β€˜π΄)) ∈ (𝑀 MndHom 𝑇))
 
Theorempwsexpg 19967 Value of a group exponentiation in a structure power. Compare pwsmulg 18854. (Contributed by SN, 30-Jul-2024.)
π‘Œ = (𝑅 ↑s 𝐼)    &   π΅ = (Baseβ€˜π‘Œ)    &   π‘€ = (mulGrpβ€˜π‘Œ)    &   π‘‡ = (mulGrpβ€˜π‘…)    &    βˆ™ = (.gβ€˜π‘€)    &    Β· = (.gβ€˜π‘‡)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ 𝐼)    β‡’   (πœ‘ β†’ ((𝑁 βˆ™ 𝑋)β€˜π΄) = (𝑁 Β· (π‘‹β€˜π΄)))
 
Theoremimasring 19968* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
(πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)    &   ((πœ‘ ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ π‘ž ∈ 𝑉)) β†’ (((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ∧ (πΉβ€˜π‘) = (πΉβ€˜π‘ž)) β†’ (πΉβ€˜(π‘Ž + 𝑏)) = (πΉβ€˜(𝑝 + π‘ž))))    &   ((πœ‘ ∧ (π‘Ž ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ π‘ž ∈ 𝑉)) β†’ (((πΉβ€˜π‘Ž) = (πΉβ€˜π‘) ∧ (πΉβ€˜π‘) = (πΉβ€˜π‘ž)) β†’ (πΉβ€˜(π‘Ž Β· 𝑏)) = (πΉβ€˜(𝑝 Β· π‘ž))))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ (π‘ˆ ∈ Ring ∧ (πΉβ€˜ 1 ) = (1rβ€˜π‘ˆ)))
 
Theoremqusring2 19969* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
(πœ‘ β†’ π‘ˆ = (𝑅 /s ∼ ))    &   (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   (πœ‘ β†’ ∼ Er 𝑉)    &   (πœ‘ β†’ ((π‘Ž ∼ 𝑝 ∧ 𝑏 ∼ π‘ž) β†’ (π‘Ž + 𝑏) ∼ (𝑝 + π‘ž)))    &   (πœ‘ β†’ ((π‘Ž ∼ 𝑝 ∧ 𝑏 ∼ π‘ž) β†’ (π‘Ž Β· 𝑏) ∼ (𝑝 Β· π‘ž)))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ (π‘ˆ ∈ Ring ∧ [ 1 ] ∼ = (1rβ€˜π‘ˆ)))
 
Theoremcrngbinom 19970* The binomial theorem for commutative rings (special case of csrgbinom 19887): (𝐴 + 𝐡)↑𝑁 is the sum from π‘˜ = 0 to 𝑁 of (𝑁Cπ‘˜) Β· ((π΄β†‘π‘˜) Β· (𝐡↑(𝑁 βˆ’ π‘˜)). (Contributed by AV, 24-Aug-2019.)
𝑆 = (Baseβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &   πΊ = (mulGrpβ€˜π‘…)    &    ↑ = (.gβ€˜πΊ)    β‡’   (((𝑅 ∈ CRing ∧ 𝑁 ∈ β„•0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆)) β†’ (𝑁 ↑ (𝐴 + 𝐡)) = (𝑅 Ξ£g (π‘˜ ∈ (0...𝑁) ↦ ((𝑁Cπ‘˜) Β· (((𝑁 βˆ’ π‘˜) ↑ 𝐴) Γ— (π‘˜ ↑ 𝐡))))))
 
10.3.5  Opposite ring
 
Syntaxcoppr 19971 The opposite ring operation.
class oppr
 
Definitiondf-oppr 19972 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 ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘“)⟩))
 
Theoremopprval 19973 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    β‡’   π‘‚ = (𝑅 sSet ⟨(.rβ€˜ndx), tpos Β· ⟩)
 
Theoremopprmulfval 19974 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    &    βˆ™ = (.rβ€˜π‘‚)    β‡’    βˆ™ = tpos Β·
 
Theoremopprmul 19975 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.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    &    βˆ™ = (.rβ€˜π‘‚)    β‡’   (𝑋 βˆ™ π‘Œ) = (π‘Œ Β· 𝑋)
 
Theoremcrngoppr 19976 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 20049). (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    &    βˆ™ = (.rβ€˜π‘‚)    β‡’   ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 Β· π‘Œ) = (𝑋 βˆ™ π‘Œ))
 
Theoremopprlem 19977 Lemma for opprbas 19979 and oppradd 19981. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (opprβ€˜π‘…)    &   πΈ = Slot (πΈβ€˜ndx)    &   (πΈβ€˜ndx) β‰  (.rβ€˜ndx)    β‡’   (πΈβ€˜π‘…) = (πΈβ€˜π‘‚)
 
TheoremopprlemOLD 19978 Obsolete version of opprlem 19977 as of 6-Nov-2024. Lemma for opprbas 19979 and oppradd 19981. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑂 = (opprβ€˜π‘…)    &   πΈ = Slot 𝑁    &   π‘ ∈ β„•    &   π‘ < 3    β‡’   (πΈβ€˜π‘…) = (πΈβ€˜π‘‚)
 
Theoremopprbas 19979 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (opprβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   π΅ = (Baseβ€˜π‘‚)
 
TheoremopprbasOLD 19980 Obsolete proof of opprbas 19979 as of 6-Nov-2024. Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑂 = (opprβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   π΅ = (Baseβ€˜π‘‚)
 
Theoremoppradd 19981 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
𝑂 = (opprβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’    + = (+gβ€˜π‘‚)
 
TheoremoppraddOLD 19982 Obsolete proof of opprbas 19979 as of 6-Nov-2024. Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑂 = (opprβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’    + = (+gβ€˜π‘‚)
 
Theoremopprring 19983 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝑂 ∈ Ring)
 
Theoremopprringb 19984 Bidirectional form of opprring 19983. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring)
 
Theoremoppr0 19985 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (opprβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’    0 = (0gβ€˜π‘‚)
 
Theoremoppr1 19986 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (opprβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’    1 = (1rβ€˜π‘‚)
 
Theoremopprneg 19987 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑂 = (opprβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    β‡’   π‘ = (invgβ€˜π‘‚)
 
Theoremopprsubg 19988 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (opprβ€˜π‘…)    β‡’   (SubGrpβ€˜π‘…) = (SubGrpβ€˜π‘‚)
 
Theoremmulgass3 19989 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.gβ€˜π‘…)    &    Γ— = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑁 ∈ β„€ ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋 Γ— (𝑁 Β· π‘Œ)) = (𝑁 Β· (𝑋 Γ— π‘Œ)))
 
10.3.6  Divisibility
 
Syntaxcdsr 19990 Ring divisibility relation.
class βˆ₯r
 
Syntaxcui 19991 Units in a ring.
class Unit
 
Syntaxcir 19992 Ring irreducibles.
class Irred
 
Definitiondf-dvdsr 19993* Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through (βˆ₯rβ€˜(opprβ€˜π‘…)). (Contributed by Mario Carneiro, 1-Dec-2014.)
βˆ₯r = (𝑀 ∈ V ↦ {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (Baseβ€˜π‘€) ∧ βˆƒπ‘§ ∈ (Baseβ€˜π‘€)(𝑧(.rβ€˜π‘€)π‘₯) = 𝑦)})
 
Definitiondf-unit 19994 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 = (𝑀 ∈ V ↦ (β—‘((βˆ₯rβ€˜π‘€) ∩ (βˆ₯rβ€˜(opprβ€˜π‘€))) β€œ {(1rβ€˜π‘€)}))
 
Definitiondf-irred 19995* Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred = (𝑀 ∈ V ↦ ⦋((Baseβ€˜π‘€) βˆ– (Unitβ€˜π‘€)) / π‘β¦Œ{𝑧 ∈ 𝑏 ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 (π‘₯(.rβ€˜π‘€)𝑦) β‰  𝑧})
 
Theoremreldvdsr 19996 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   Rel βˆ₯
 
Theoremdvdsrval 19997* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’    βˆ₯ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ βˆƒπ‘§ ∈ 𝐡 (𝑧 Β· π‘₯) = 𝑦)}
 
Theoremdvdsr 19998* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑋 βˆ₯ π‘Œ ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘§ ∈ 𝐡 (𝑧 Β· 𝑋) = π‘Œ))
 
Theoremdvdsr2 19999* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑋 ∈ 𝐡 β†’ (𝑋 βˆ₯ π‘Œ ↔ βˆƒπ‘§ ∈ 𝐡 (𝑧 Β· 𝑋) = π‘Œ))
 
Theoremdvdsrmul 20000 A left-multiple of 𝑋 is divisible by 𝑋. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 βˆ₯ (π‘Œ Β· 𝑋))
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