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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvrcan1 20301 A cancellation law for division. (divcan1 11886 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    / = (/rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ) β†’ ((𝑋 / π‘Œ) Β· π‘Œ) = 𝑋)
 
Theoremdvrcan3 20302 A cancellation law for division. (divcan3 11903 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    / = (/rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ) β†’ ((𝑋 Β· π‘Œ) / π‘Œ) = 𝑋)
 
Theoremdvreq1 20303 Equality in terms of ratio equal to ring unity. (diveq1 11910 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    / = (/rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ) β†’ ((𝑋 / π‘Œ) = 1 ↔ 𝑋 = π‘Œ))
 
Theoremdvrdir 20304 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    / = (/rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ π‘ˆ)) β†’ ((𝑋 + π‘Œ) / 𝑍) = ((𝑋 / 𝑍) + (π‘Œ / 𝑍)))
 
Theoremrdivmuldivd 20305 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    / = (/rβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ π‘Š ∈ π‘ˆ)    β‡’   (πœ‘ β†’ ((𝑋 / π‘Œ) Β· (𝑍 / π‘Š)) = ((𝑋 Β· 𝑍) / (π‘Œ Β· π‘Š)))
 
Theoremringinvdv 20306 Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    / = (/rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ) β†’ (πΌβ€˜π‘‹) = ( 1 / 𝑋))
 
Theoremrngidpropd 20307* The ring unity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (1rβ€˜πΎ) = (1rβ€˜πΏ))
 
Theoremdvdsrpropd 20308* The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (βˆ₯rβ€˜πΎ) = (βˆ₯rβ€˜πΏ))
 
Theoremunitpropd 20309* The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (Unitβ€˜πΎ) = (Unitβ€˜πΏ))
 
Theoreminvrpropd 20310* The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(.rβ€˜πΎ)𝑦) = (π‘₯(.rβ€˜πΏ)𝑦))    β‡’   (πœ‘ β†’ (invrβ€˜πΎ) = (invrβ€˜πΏ))
 
Theoremisirred 20311* An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (Irredβ€˜π‘…)    &   π‘ = (𝐡 βˆ– π‘ˆ)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝑁 ∧ βˆ€π‘₯ ∈ 𝑁 βˆ€π‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) β‰  𝑋))
 
Theoremisnirred 20312* The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (Irredβ€˜π‘…)    &   π‘ = (𝐡 βˆ– π‘ˆ)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑋 ∈ 𝐡 β†’ (Β¬ 𝑋 ∈ 𝐼 ↔ (𝑋 ∈ π‘ˆ ∨ βˆƒπ‘₯ ∈ 𝑁 βˆƒπ‘¦ ∈ 𝑁 (π‘₯ Β· 𝑦) = 𝑋)))
 
Theoremisirred2 20313* Expand out the class difference from isirred 20311. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &   πΌ = (Irredβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐡 ∧ Β¬ 𝑋 ∈ π‘ˆ ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 𝑋 β†’ (π‘₯ ∈ π‘ˆ ∨ 𝑦 ∈ π‘ˆ))))
 
Theoremopprirred 20314 Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (opprβ€˜π‘…)    &   πΌ = (Irredβ€˜π‘…)    β‡’   πΌ = (Irredβ€˜π‘†)
 
Theoremirredn0 20315 The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) β†’ 𝑋 β‰  0 )
 
Theoremirredcl 20316 An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝑋 ∈ 𝐼 β†’ 𝑋 ∈ 𝐡)
 
Theoremirrednu 20317 An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    β‡’   (𝑋 ∈ 𝐼 β†’ Β¬ 𝑋 ∈ π‘ˆ)
 
Theoremirredn1 20318 The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) β†’ 𝑋 β‰  1 )
 
Theoremirredrmul 20319 The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼 ∧ π‘Œ ∈ π‘ˆ) β†’ (𝑋 Β· π‘Œ) ∈ 𝐼)
 
Theoremirredlmul 20320 The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ 𝐼) β†’ (𝑋 Β· π‘Œ) ∈ 𝐼)
 
Theoremirredmul 20321 If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ (𝑋 Β· π‘Œ) ∈ 𝐼) β†’ (𝑋 ∈ π‘ˆ ∨ π‘Œ ∈ π‘ˆ))
 
Theoremirredneg 20322 The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐼) β†’ (π‘β€˜π‘‹) ∈ 𝐼)
 
Theoremirrednegb 20323 An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irredβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡) β†’ (𝑋 ∈ 𝐼 ↔ (π‘β€˜π‘‹) ∈ 𝐼))
 
10.3.8  Ring primes
 
Syntaxcrpm 20324 Syntax for the ring primes function.
class RPrime
 
Definitiondf-rprm 20325* Define the function associating with a ring its set of prime elements. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 16655. Prime elements are closely related to irreducible elements (see df-irred 20251). (Contributed by Mario Carneiro, 17-Feb-2015.)
RPrime = (𝑀 ∈ V ↦ ⦋(Baseβ€˜π‘€) / π‘β¦Œ{𝑝 ∈ (𝑏 βˆ– ((Unitβ€˜π‘€) βˆͺ {(0gβ€˜π‘€)})) ∣ βˆ€π‘₯ ∈ 𝑏 βˆ€π‘¦ ∈ 𝑏 [(βˆ₯rβ€˜π‘€) / 𝑑](𝑝𝑑(π‘₯(.rβ€˜π‘€)𝑦) β†’ (𝑝𝑑π‘₯ ∨ 𝑝𝑑𝑦))})
 
10.3.9  Homomorphisms of non-unital rings
 
Syntaxcrnghm 20326 non-unital ring homomorphisms.
class RngHom
 
Syntaxcrngim 20327 non-unital ring isomorphisms.
class RngIso
 
Definitiondf-rnghm 20328* Define the set of non-unital ring homomorphisms from π‘Ÿ to 𝑠. (Contributed by AV, 20-Feb-2020.)
RngHom = (π‘Ÿ ∈ Rng, 𝑠 ∈ Rng ↦ ⦋(Baseβ€˜π‘Ÿ) / π‘£β¦Œβ¦‹(Baseβ€˜π‘ ) / π‘€β¦Œ{𝑓 ∈ (𝑀 ↑m 𝑣) ∣ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘¦ ∈ 𝑣 ((π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(+gβ€˜π‘ )(π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(.rβ€˜π‘ )(π‘“β€˜π‘¦)))})
 
Definitiondf-rngim 20329* Define the set of non-unital ring isomorphisms from π‘Ÿ to 𝑠. (Contributed by AV, 20-Feb-2020.)
RngIso = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (π‘Ÿ RngHom 𝑠) ∣ ◑𝑓 ∈ (𝑠 RngHom π‘Ÿ)})
 
Theoremrnghmrcl 20330 Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020.)
(𝐹 ∈ (𝑅 RngHom 𝑆) β†’ (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
 
Theoremrnghmfn 20331 The mapping of two non-unital rings to the non-unital ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
RngHom Fn (Rng Γ— Rng)
 
Theoremrnghmval 20332* The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘†)    &   πΆ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘…)    &    ✚ = (+gβ€˜π‘†)    β‡’   ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) β†’ (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝐢 ↑m 𝐡) ∣ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ + 𝑦)) = ((π‘“β€˜π‘₯) ✚ (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) βˆ— (π‘“β€˜π‘¦)))})
 
Theoremisrnghm 20333* A function is a non-unital ring homomorphism iff it is a group homomorphism and preserves multiplication. (Contributed by AV, 22-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) βˆ— (πΉβ€˜π‘¦)))))
 
Theoremisrnghmmul 20334 A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)
𝑀 = (mulGrpβ€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))
 
Theoremrnghmmgmhm 20335 A non-unital ring homomorphism is a homomorphism of multiplicative magmas. (Contributed by AV, 27-Feb-2020.)
𝑀 = (mulGrpβ€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RngHom 𝑆) β†’ 𝐹 ∈ (𝑀 MgmHom 𝑁))
 
Theoremrnghmval2 20336 The non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 1-Mar-2020.)
((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) β†’ (𝑅 RngHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrpβ€˜π‘…) MgmHom (mulGrpβ€˜π‘†))))
 
Theoremisrngim 20337 An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◑𝐹 ∈ (𝑆 RngHom 𝑅))))
 
Theoremrngimrcl 20338 Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)
(𝐹 ∈ (𝑅 RngIso 𝑆) β†’ (𝑅 ∈ V ∧ 𝑆 ∈ V))
 
Theoremrnghmghm 20339 A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.)
(𝐹 ∈ (𝑅 RngHom 𝑆) β†’ 𝐹 ∈ (𝑅 GrpHom 𝑆))
 
Theoremrnghmf 20340 A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RngHom 𝑆) β†’ 𝐹:𝐡⟢𝐢)
 
Theoremrnghmmul 20341 A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020.)
𝑋 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    Γ— = (.rβ€˜π‘†)    β‡’   ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (πΉβ€˜(𝐴 Β· 𝐡)) = ((πΉβ€˜π΄) Γ— (πΉβ€˜π΅)))
 
Theoremisrnghm2d 20342* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    Γ— = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝑆 ∈ Rng)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Γ— (πΉβ€˜π‘¦)))    &   (πœ‘ β†’ 𝐹 ∈ (𝑅 GrpHom 𝑆))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
 
Theoremisrnghmd 20343* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    Γ— = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝑅 ∈ Rng)    &   (πœ‘ β†’ 𝑆 ∈ Rng)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Γ— (πΉβ€˜π‘¦)))    &   πΆ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘†)    &   (πœ‘ β†’ 𝐹:𝐡⟢𝐢)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (πΉβ€˜(π‘₯ + 𝑦)) = ((πΉβ€˜π‘₯) ⨣ (πΉβ€˜π‘¦)))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
 
Theoremrnghmf1o 20344 A non-unital ring homomorphism is bijective iff its converse is also a non-unital ring homomorphism. (Contributed by AV, 27-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RngHom 𝑆) β†’ (𝐹:𝐡–1-1-onto→𝐢 ↔ ◑𝐹 ∈ (𝑆 RngHom 𝑅)))
 
Theoremisrngim2 20345 An isomorphism of non-unital rings is a bijective homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐡–1-1-onto→𝐢)))
 
Theoremrngimf1o 20346 An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RngIso 𝑆) β†’ 𝐹:𝐡–1-1-onto→𝐢)
 
Theoremrngimrnghm 20347 An isomorphism of non-unital rings is a homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RngIso 𝑆) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
 
Theoremrngimcnv 20348 The converse of an isomorphism of non-unital rings is an isomorphism of non-unital rings. (Contributed by AV, 27-Feb-2025.)
(𝐹 ∈ (𝑆 RngIso 𝑇) β†’ ◑𝐹 ∈ (𝑇 RngIso 𝑆))
 
Theoremrnghmco 20349 The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑇 RngHom π‘ˆ) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑆 RngHom π‘ˆ))
 
Theoremidrnghm 20350 The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Rng β†’ ( I β†Ύ 𝐡) ∈ (𝑅 RngHom 𝑅))
 
Theoremc0mgm 20351* The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)
𝐡 = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘‡)    &   π» = (π‘₯ ∈ 𝐡 ↦ 0 )    β‡’   ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) β†’ 𝐻 ∈ (𝑆 MgmHom 𝑇))
 
Theoremc0mhm 20352* The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)
𝐡 = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘‡)    &   π» = (π‘₯ ∈ 𝐡 ↦ 0 )    β‡’   ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) β†’ 𝐻 ∈ (𝑆 MndHom 𝑇))
 
Theoremc0ghm 20353* The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)
𝐡 = (Baseβ€˜π‘†)    &    0 = (0gβ€˜π‘‡)    &   π» = (π‘₯ ∈ 𝐡 ↦ 0 )    β‡’   ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) β†’ 𝐻 ∈ (𝑆 GrpHom 𝑇))
 
Theoremc0snmgmhm 20354* The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)
𝐡 = (Baseβ€˜π‘‡)    &    0 = (0gβ€˜π‘†)    &   π» = (π‘₯ ∈ 𝐡 ↦ 0 )    β‡’   ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (β™―β€˜π΅) = 1) β†’ 𝐻 ∈ (𝑇 MgmHom 𝑆))
 
Theoremc0snmhm 20355* The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.)
𝐡 = (Baseβ€˜π‘‡)    &    0 = (0gβ€˜π‘†)    &   π» = (π‘₯ ∈ 𝐡 ↦ 0 )    &   π‘ = (0gβ€˜π‘‡)    β‡’   ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐡 = {𝑍}) β†’ 𝐻 ∈ (𝑇 MndHom 𝑆))
 
Theoremc0snghm 20356* The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.)
𝐡 = (Baseβ€˜π‘‡)    &    0 = (0gβ€˜π‘†)    &   π» = (π‘₯ ∈ 𝐡 ↦ 0 )    &   π‘ = (0gβ€˜π‘‡)    β‡’   ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐡 = {𝑍}) β†’ 𝐻 ∈ (𝑇 GrpHom 𝑆))
 
Theoremrngisomfv1 20357 If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is an element of the base set of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
1 = (1rβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘†)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ (πΉβ€˜ 1 ) ∈ 𝐡)
 
Theoremrngisom1 20358* If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is a ring unity of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
1 = (1rβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘†)    &    Β· = (.rβ€˜π‘†)    β‡’   ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜ 1 ) Β· π‘₯) = π‘₯ ∧ (π‘₯ Β· (πΉβ€˜ 1 )) = π‘₯))
 
Theoremrngisomring 20359 If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then both rings are unital. (Contributed by AV, 27-Feb-2025.)
((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ 𝑆 ∈ Ring)
 
Theoremrngisomring1 20360 If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the ring unity of the second ring is the function value of the ring unity of the first ring for the isomorphism. (Contributed by AV, 16-Mar-2025.)
((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) β†’ (1rβ€˜π‘†) = (πΉβ€˜(1rβ€˜π‘…)))
 
10.3.10  Ring homomorphisms
 
Syntaxcrh 20361 Extend class notation with the ring homomorphisms.
class RingHom
 
Syntaxcrs 20362 Extend class notation with the ring isomorphisms.
class RingIso
 
Syntaxcric 20363 Extend class notation with the ring isomorphism relation.
class β‰ƒπ‘Ÿ
 
Definitiondf-rhm 20364* Define the set of ring homomorphisms from π‘Ÿ to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom = (π‘Ÿ ∈ Ring, 𝑠 ∈ Ring ↦ ⦋(Baseβ€˜π‘Ÿ) / π‘£β¦Œβ¦‹(Baseβ€˜π‘ ) / π‘€β¦Œ{𝑓 ∈ (𝑀 ↑m 𝑣) ∣ ((π‘“β€˜(1rβ€˜π‘Ÿ)) = (1rβ€˜π‘ ) ∧ βˆ€π‘₯ ∈ 𝑣 βˆ€π‘¦ ∈ 𝑣 ((π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(+gβ€˜π‘ )(π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯)(.rβ€˜π‘ )(π‘“β€˜π‘¦))))})
 
Definitiondf-rim 20365* Define the set of ring isomorphisms from π‘Ÿ to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingIso = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (π‘Ÿ RingHom 𝑠) ∣ ◑𝑓 ∈ (𝑠 RingHom π‘Ÿ)})
 
Theoremdfrhm2 20366* The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom = (π‘Ÿ ∈ Ring, 𝑠 ∈ Ring ↦ ((π‘Ÿ GrpHom 𝑠) ∩ ((mulGrpβ€˜π‘Ÿ) MndHom (mulGrpβ€˜π‘ ))))
 
Definitiondf-ric 20367 Define the ring isomorphism relation, analogous to df-gic 19175: Two (unital) rings are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic rings share all global ring properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by AV, 24-Dec-2019.)
β‰ƒπ‘Ÿ = (β—‘ RingIso β€œ (V βˆ– 1o))
 
Theoremrhmrcl1 20368 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) β†’ 𝑅 ∈ Ring)
 
Theoremrhmrcl2 20369 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) β†’ 𝑆 ∈ Ring)
 
Theoremisrhm 20370 A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑀 = (mulGrpβ€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁))))
 
Theoremrhmmhm 20371 A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑀 = (mulGrpβ€˜π‘…)    &   π‘ = (mulGrpβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingHom 𝑆) β†’ 𝐹 ∈ (𝑀 MndHom 𝑁))
 
Theoremrhmisrnghm 20372 Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.)
(𝐹 ∈ (𝑅 RingHom 𝑆) β†’ 𝐹 ∈ (𝑅 RngHom 𝑆))
 
Theoremisrim0OLD 20373 Obsolete version of isrim0 20375 as of 12-Jan-2025. (Contributed by AV, 22-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◑𝐹 ∈ (𝑆 RingHom 𝑅))))
 
Theoremrimrcl 20374 Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
(𝐹 ∈ (𝑅 RingIso 𝑆) β†’ (𝑅 ∈ V ∧ 𝑆 ∈ V))
 
Theoremisrim0 20375 A ring isomorphism is a homomorphism whose converse is also a homomorphism. Compare isgim2 19180. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ◑𝐹 ∈ (𝑆 RingHom 𝑅)))
 
Theoremrhmghm 20376 A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) β†’ 𝐹 ∈ (𝑅 GrpHom 𝑆))
 
Theoremrhmf 20377 A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingHom 𝑆) β†’ 𝐹:𝐡⟢𝐢)
 
Theoremrhmmul 20378 A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑋 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    Γ— = (.rβ€˜π‘†)    β‡’   ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (πΉβ€˜(𝐴 Β· 𝐡)) = ((πΉβ€˜π΄) Γ— (πΉβ€˜π΅)))
 
Theoremisrhm2d 20379* Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ = (1rβ€˜π‘†)    &    Β· = (.rβ€˜π‘…)    &    Γ— = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑆 ∈ Ring)    &   (πœ‘ β†’ (πΉβ€˜ 1 ) = 𝑁)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Γ— (πΉβ€˜π‘¦)))    &   (πœ‘ β†’ 𝐹 ∈ (𝑅 GrpHom 𝑆))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
 
Theoremisrhmd 20380* Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝐡 = (Baseβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   π‘ = (1rβ€˜π‘†)    &    Β· = (.rβ€˜π‘…)    &    Γ— = (.rβ€˜π‘†)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝑆 ∈ Ring)    &   (πœ‘ β†’ (πΉβ€˜ 1 ) = 𝑁)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Γ— (πΉβ€˜π‘¦)))    &   πΆ = (Baseβ€˜π‘†)    &    + = (+gβ€˜π‘…)    &    ⨣ = (+gβ€˜π‘†)    &   (πœ‘ β†’ 𝐹:𝐡⟢𝐢)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (πΉβ€˜(π‘₯ + 𝑦)) = ((πΉβ€˜π‘₯) ⨣ (πΉβ€˜π‘¦)))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
 
Theoremrhm1 20381 Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
1 = (1rβ€˜π‘…)    &   π‘ = (1rβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingHom 𝑆) β†’ (πΉβ€˜ 1 ) = 𝑁)
 
Theoremidrhm 20382 The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ ( I β†Ύ 𝐡) ∈ (𝑅 RingHom 𝑅))
 
Theoremrhmf1o 20383 A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingHom 𝑆) β†’ (𝐹:𝐡–1-1-onto→𝐢 ↔ ◑𝐹 ∈ (𝑆 RingHom 𝑅)))
 
Theoremisrim 20384 An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.) Remove sethood antecedent. (Revised by SN, 12-Jan-2025.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐡–1-1-onto→𝐢))
 
TheoremisrimOLD 20385 Obsolete version of isrim 20384 as of 12-Jan-2025. (Contributed by AV, 22-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐡–1-1-onto→𝐢)))
 
Theoremrimf1o 20386 An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingIso 𝑆) β†’ 𝐹:𝐡–1-1-onto→𝐢)
 
TheoremrimrhmOLD 20387 Obsolete version of rimrhm 20388 as of 12-Jan-2025. (Contributed by AV, 22-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐡 = (Baseβ€˜π‘…)    &   πΆ = (Baseβ€˜π‘†)    β‡’   (𝐹 ∈ (𝑅 RingIso 𝑆) β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
 
Theoremrimrhm 20388 A ring isomorphism is a homomorphism. Compare gimghm 19179. (Contributed by AV, 22-Oct-2019.) Remove hypotheses. (Revised by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))
 
Theoremrimgim 20389 An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019.)
(𝐹 ∈ (𝑅 RingIso 𝑆) β†’ 𝐹 ∈ (𝑅 GrpIso 𝑆))
 
Theoremrimisrngim 20390 Each unital ring isomorphism is a non-unital ring isomorphism. (Contributed by AV, 30-Mar-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) β†’ 𝐹 ∈ (𝑅 RngIso 𝑆))
 
Theoremrhmfn 20391 The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
RingHom Fn (Ring Γ— Ring)
 
Theoremrhmval 20392 The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)
((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) β†’ (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrpβ€˜π‘…) MndHom (mulGrpβ€˜π‘†))))
 
Theoremrhmco 20393 The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐹 ∈ (𝑇 RingHom π‘ˆ) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑆 RingHom π‘ˆ))
 
Theorempwsco1rhm 20394* Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
π‘Œ = (𝑅 ↑s 𝐴)    &   π‘ = (𝑅 ↑s 𝐡)    &   πΆ = (Baseβ€˜π‘)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    β‡’   (πœ‘ β†’ (𝑔 ∈ 𝐢 ↦ (𝑔 ∘ 𝐹)) ∈ (𝑍 RingHom π‘Œ))
 
Theorempwsco2rhm 20395* Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
π‘Œ = (𝑅 ↑s 𝐴)    &   π‘ = (𝑆 ↑s 𝐴)    &   π΅ = (Baseβ€˜π‘Œ)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 ∈ (𝑅 RingHom 𝑆))    β‡’   (πœ‘ β†’ (𝑔 ∈ 𝐡 ↦ (𝐹 ∘ 𝑔)) ∈ (π‘Œ RingHom 𝑍))
 
Theorembrric 20396 The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
(𝑅 β‰ƒπ‘Ÿ 𝑆 ↔ (𝑅 RingIso 𝑆) β‰  βˆ…)
 
Theorembrrici 20397 Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025.)
(𝐹 ∈ (𝑅 RingIso 𝑆) β†’ 𝑅 β‰ƒπ‘Ÿ 𝑆)
 
Theorembrric2 20398* The relation "is isomorphic to" for (unital) rings. This theorem corresponds to Definition df-risc 37155 of the ring isomorphism relation in JM's mathbox. (Contributed by AV, 24-Dec-2019.)
(𝑅 β‰ƒπ‘Ÿ 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ βˆƒπ‘“ 𝑓 ∈ (𝑅 RingIso 𝑆)))
 
Theoremricgic 20399 If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.)
(𝑅 β‰ƒπ‘Ÿ 𝑆 β†’ 𝑅 ≃𝑔 𝑆)
 
Theoremrhmdvdsr 20400 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝑋 = (Baseβ€˜π‘…)    &    βˆ₯ = (βˆ₯rβ€˜π‘…)    &    / = (βˆ₯rβ€˜π‘†)    β‡’   (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) ∧ 𝐴 βˆ₯ 𝐡) β†’ (πΉβ€˜π΄) / (πΉβ€˜π΅))
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