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Theorem List for Intuitionistic Logic Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremringidcl 13201 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 13202 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 13203 Lemma for ringlidm 13204 and ringridm 13205. (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 13204 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 13205 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 13206* 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 13207* 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 13208* 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 13209 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 13210 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 13211 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 13212 Commutativity of the additive group of a ring. (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 13213 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ Abel)
 
Theoremringcmn 13214 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ CMnd)
 
Theoremringpropd 13215* 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 13216* 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 13217 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 13218* 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 13219* 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 13220 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 13221 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 13222 Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑅 ∈ Ring β†’ 𝑅 ∈ SRing)
 
Theoremring1eq0 13223 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))
 
Theoremringinvnz1ne0 13224* 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 13225* 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 13226 Negation in a ring is the same as left multiplication by -1. (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 13227 Negation in a ring is the same as right multiplication by -1. (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 13228 Negation of a product in a ring. (mulneg1 8351 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 13229 Negation of a product in a ring. (mulneg2 8352 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 13230 Double negation of a product in a ring. (mul2neg 8354 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 13231 Ring multiplication distributes over subtraction. (subdi 8341 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 13232 Ring multiplication distributes over subtraction. (subdir 8342 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 13233 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 13234 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 13235 The class of rings is not empty (it is also inhabited, as shown at ring1 13234). (Contributed by AV, 29-Apr-2019.)
Ring β‰  βˆ…
 
Theoremringressid 13236 A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 12529. (Contributed by Jim Kingdon, 28-Feb-2025.)
𝐡 = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ Ring β†’ (𝐺 β†Ύs 𝐡) ∈ Ring)
 
7.3.5  Opposite ring
 
Syntaxcoppr 13237 The opposite ring operation.
class oppr
 
Definitiondf-oppr 13238 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‘“)⟩))
 
Theoremopprvalg 13239 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π‘‚ = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑂 = (𝑅 sSet ⟨(.rβ€˜ndx), tpos Β· ⟩))
 
Theoremopprmulfvalg 13240 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‡’   (𝑅 ∈ 𝑉 β†’ βˆ™ = tpos Β· )
 
Theoremopprmulg 13241 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‡’   ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ π‘Š ∧ π‘Œ ∈ π‘ˆ) β†’ (𝑋 βˆ™ π‘Œ) = (π‘Œ Β· 𝑋))
 
Theoremcrngoppr 13242 In a commutative ring, the opposite ring is equivalent to the original ring. (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))
 
Theoremopprex 13243 Existence of the opposite ring. If you know that 𝑅 is a ring, see opprring 13247. (Contributed by Jim Kingdon, 10-Jan-2025.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 𝑂 ∈ V)
 
Theoremopprsllem 13244 Lemma for opprbasg 13245 and oppraddg 13246. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
𝑂 = (opprβ€˜π‘…)    &   (𝐸 = Slot (πΈβ€˜ndx) ∧ (πΈβ€˜ndx) ∈ β„•)    &   (πΈβ€˜ndx) β‰  (.rβ€˜ndx)    β‡’   (𝑅 ∈ 𝑉 β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘‚))
 
Theoremopprbasg 13245 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‡’   (𝑅 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜π‘‚))
 
Theoremoppraddg 13246 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‡’   (𝑅 ∈ 𝑉 β†’ + = (+gβ€˜π‘‚))
 
Theoremopprring 13247 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)
 
Theoremopprringbg 13248 Bidirectional form of opprring 13247. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring))
 
Theoremoppr0g 13249 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (opprβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 0 = (0gβ€˜π‘‚))
 
Theoremoppr1g 13250 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑂 = (opprβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ 𝑉 β†’ 1 = (1rβ€˜π‘‚))
 
Theoremopprnegg 13251 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‡’   (𝑅 ∈ 𝑉 β†’ 𝑁 = (invgβ€˜π‘‚))
 
Theoremmulgass3 13252 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)))
 
7.3.6  Divisibility
 
Syntaxcdsr 13253 Ring divisibility relation.
class βˆ₯r
 
Syntaxcui 13254 Units in a ring.
class Unit
 
Syntaxcir 13255 Ring irreducibles.
class Irred
 
Definitiondf-dvdsr 13256* 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 13257 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 13258* 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})
 
Theoremreldvdsrsrg 13259 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
(𝑅 ∈ SRing β†’ Rel (βˆ₯rβ€˜π‘…))
 
Theoremdvdsrvald 13260* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ βˆ₯ = (βˆ₯rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    β‡’   (πœ‘ β†’ βˆ₯ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ 𝐡 ∧ βˆƒπ‘§ ∈ 𝐡 (𝑧 Β· π‘₯) = 𝑦)})
 
Theoremdvdsrd 13261* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ βˆ₯ = (βˆ₯rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    β‡’   (πœ‘ β†’ (𝑋 βˆ₯ π‘Œ ↔ (𝑋 ∈ 𝐡 ∧ βˆƒπ‘§ ∈ 𝐡 (𝑧 Β· 𝑋) = π‘Œ)))
 
Theoremdvdsr2d 13262* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ βˆ₯ = (βˆ₯rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    β‡’   (πœ‘ β†’ (𝑋 βˆ₯ π‘Œ ↔ βˆƒπ‘§ ∈ 𝐡 (𝑧 Β· 𝑋) = π‘Œ))
 
Theoremdvdsrmuld 13263 A left-multiple of 𝑋 is divisible by 𝑋. (Contributed by Mario Carneiro, 1-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ βˆ₯ = (βˆ₯rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ 𝑋 βˆ₯ (π‘Œ Β· 𝑋))
 
Theoremdvdsrcld 13264 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ βˆ₯ = (βˆ₯rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝑋 βˆ₯ π‘Œ)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝐡)
 
Theoremdvdsrex 13265 Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
(𝑅 ∈ SRing β†’ (βˆ₯rβ€˜π‘…) ∈ V)
 
Theoremdvdsrcl2 13266 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 βˆ₯ π‘Œ) β†’ π‘Œ ∈ 𝐡)
 
Theoremdvdsrid 13267 An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 βˆ₯ 𝑋)
 
Theoremdvdsrtr 13268 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ π‘Œ βˆ₯ 𝑍 ∧ 𝑍 βˆ₯ 𝑋) β†’ π‘Œ βˆ₯ 𝑋)
 
Theoremdvdsrmul1 13269 The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐡 ∧ 𝑋 βˆ₯ π‘Œ) β†’ (𝑋 Β· 𝑍) βˆ₯ (π‘Œ Β· 𝑍))
 
Theoremdvdsrneg 13270 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 βˆ₯ (π‘β€˜π‘‹))
 
Theoremdvdsr01 13271 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 βˆ₯ 0 )
 
Theoremdvdsr02 13272 Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ ( 0 βˆ₯ 𝑋 ↔ 𝑋 = 0 ))
 
Theoremisunitd 13273 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.)
(πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 1 = (1rβ€˜π‘…))    &   (πœ‘ β†’ βˆ₯ = (βˆ₯rβ€˜π‘…))    &   (πœ‘ β†’ 𝑆 = (opprβ€˜π‘…))    &   (πœ‘ β†’ 𝐸 = (βˆ₯rβ€˜π‘†))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    β‡’   (πœ‘ β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 βˆ₯ 1 ∧ 𝑋𝐸 1 )))
 
Theorem1unit 13274 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 1 ∈ π‘ˆ)
 
Theoremunitcld 13275 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 ∈ 𝐡)
 
Theoremunitssd 13276 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    β‡’   (πœ‘ β†’ π‘ˆ βŠ† 𝐡)
 
Theoremopprunitd 13277 Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
(πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝑆 = (opprβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘†))
 
Theoremcrngunit 13278 Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
π‘ˆ = (Unitβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   (𝑅 ∈ CRing β†’ (𝑋 ∈ π‘ˆ ↔ 𝑋 βˆ₯ 1 ))
 
Theoremdvdsunit 13279 A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
π‘ˆ = (Unitβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ π‘Œ βˆ₯ 𝑋 ∧ 𝑋 ∈ π‘ˆ) β†’ π‘Œ ∈ π‘ˆ)
 
Theoremunitmulcl 13280 The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ) β†’ (𝑋 Β· π‘Œ) ∈ π‘ˆ)
 
Theoremunitmulclb 13281 Reversal of unitmulcl 13280 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
π‘ˆ = (Unitβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((𝑋 Β· π‘Œ) ∈ π‘ˆ ↔ (𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ)))
 
Theoremunitgrpbasd 13282 The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
(πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝐺 = ((mulGrpβ€˜π‘…) β†Ύs π‘ˆ))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    β‡’   (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΊ))
 
Theoremunitgrp 13283 The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΊ = ((mulGrpβ€˜π‘…) β†Ύs π‘ˆ)    β‡’   (𝑅 ∈ Ring β†’ 𝐺 ∈ Grp)
 
Theoremunitabl 13284 The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΊ = ((mulGrpβ€˜π‘…) β†Ύs π‘ˆ)    β‡’   (𝑅 ∈ CRing β†’ 𝐺 ∈ Abel)
 
Theoremunitgrpid 13285 The identity of the group of units of a ring is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΊ = ((mulGrpβ€˜π‘…) β†Ύs π‘ˆ)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 1 = (0gβ€˜πΊ))
 
Theoremunitsubm 13286 The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.)
π‘ˆ = (Unitβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ π‘ˆ ∈ (SubMndβ€˜π‘€))
 
Syntaxcinvr 13287 Extend class notation with multiplicative inverse.
class invr
 
Definitiondf-invr 13288 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
invr = (π‘Ÿ ∈ V ↦ (invgβ€˜((mulGrpβ€˜π‘Ÿ) β†Ύs (Unitβ€˜π‘Ÿ))))
 
Theoreminvrfvald 13289 Multiplicative inverse function for a ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
(πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝐺 = ((mulGrpβ€˜π‘…) β†Ύs π‘ˆ))    &   (πœ‘ β†’ 𝐼 = (invrβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝐼 = (invgβ€˜πΊ))
 
Theoremunitinvcl 13290 The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (πΌβ€˜π‘‹) ∈ π‘ˆ)
 
Theoremunitinvinv 13291 The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (πΌβ€˜(πΌβ€˜π‘‹)) = 𝑋)
 
Theoremringinvcl 13292 The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
 
Theoremunitlinv 13293 A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ ((πΌβ€˜π‘‹) Β· 𝑋) = 1 )
 
Theoremunitrinv 13294 A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (𝑋 Β· (πΌβ€˜π‘‹)) = 1 )
 
Theorem1rinv 13295 The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝐼 = (invrβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (πΌβ€˜ 1 ) = 1 )
 
Theorem0unit 13296 The additive identity is a unit if and only if 1 = 0, i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ ( 0 ∈ π‘ˆ ↔ 1 = 0 ))
 
Theoremunitnegcl 13297 The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
π‘ˆ = (Unitβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (π‘β€˜π‘‹) ∈ π‘ˆ)
 
Syntaxcdvr 13298 Extend class notation with ring division.
class /r
 
Definitiondf-dvr 13299* Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
/r = (π‘Ÿ ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Unitβ€˜π‘Ÿ) ↦ (π‘₯(.rβ€˜π‘Ÿ)((invrβ€˜π‘Ÿ)β€˜π‘¦))))
 
Theoremdvrfvald 13300* Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ π‘ˆ = (Unitβ€˜π‘…))    &   (πœ‘ β†’ 𝐼 = (invrβ€˜π‘…))    &   (πœ‘ β†’ / = (/rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ SRing)    β‡’   (πœ‘ β†’ / = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))))
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