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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcfield 20501 Class of fields.
class Field
 
Definitiondf-drng 20502 Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
DivRing = {π‘Ÿ ∈ Ring ∣ (Unitβ€˜π‘Ÿ) = ((Baseβ€˜π‘Ÿ) βˆ– {(0gβ€˜π‘Ÿ)})}
 
Definitiondf-field 20503 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field = (DivRing ∩ CRing)
 
Theoremisdrng 20504 The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ π‘ˆ = (𝐡 βˆ– { 0 })))
 
Theoremdrngunit 20505 Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 )))
 
Theoremdrngui 20506 The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘… ∈ DivRing    β‡’   (𝐡 βˆ– { 0 }) = (Unitβ€˜π‘…)
 
Theoremdrngring 20507 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ Ring)
 
Theoremdrngringd 20508 A division ring is a ring. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ DivRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Ring)
 
Theoremdrnggrpd 20509 A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ DivRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Grp)
 
Theoremdrnggrp 20510 A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ Grp)
 
Theoremisfld 20511 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
 
Theoremflddrngd 20512 A field is a division ring. (Contributed by SN, 17-Jan-2025.)
(πœ‘ β†’ 𝑅 ∈ Field)    β‡’   (πœ‘ β†’ 𝑅 ∈ DivRing)
 
Theoremfldcrngd 20513 A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
(πœ‘ β†’ 𝑅 ∈ Field)    β‡’   (πœ‘ β†’ 𝑅 ∈ CRing)
 
Theoremisdrng2 20514 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΊ = ((mulGrpβ€˜π‘…) β†Ύs (𝐡 βˆ– { 0 }))    β‡’   (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
 
Theoremdrngprop 20515 If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
(Baseβ€˜πΎ) = (Baseβ€˜πΏ)    &   (+gβ€˜πΎ) = (+gβ€˜πΏ)    &   (.rβ€˜πΎ) = (.rβ€˜πΏ)    β‡’   (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)
 
Theoremdrngmgp 20516 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΊ = ((mulGrpβ€˜π‘…) β†Ύs (𝐡 βˆ– { 0 }))    β‡’   (𝑅 ∈ DivRing β†’ 𝐺 ∈ Grp)
 
Theoremdrngmcl 20517 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐡 βˆ– { 0 }) ∧ π‘Œ ∈ (𝐡 βˆ– { 0 })) β†’ (𝑋 Β· π‘Œ) ∈ (𝐡 βˆ– { 0 }))
 
Theoremdrngid 20518 A division ring's unity is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΊ = ((mulGrpβ€˜π‘…) β†Ύs (𝐡 βˆ– { 0 }))    β‡’   (𝑅 ∈ DivRing β†’ 1 = (0gβ€˜πΊ))
 
Theoremdrngunz 20519 A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.)
0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ 1 β‰  0 )
 
Theoremdrngnzr 20520 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ NzRing)
 
Theoremdrngid2 20521 Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)
𝐡 = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ ((𝐼 ∈ 𝐡 ∧ 𝐼 β‰  0 ∧ (𝐼 Β· 𝐼) = 𝐼) ↔ 1 = 𝐼))
 
Theoremdrnginvrcl 20522 Closure of the multiplicative inverse in a division ring. (reccl 11883 analog). (Contributed by NM, 19-Apr-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
 
Theoremdrnginvrn0 20523 The multiplicative inverse in a division ring is nonzero. (recne0 11889 analog). (Contributed by NM, 19-Apr-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ (πΌβ€˜π‘‹) β‰  0 )
 
Theoremdrnginvrcld 20524 Closure of the multiplicative inverse in a division ring. (reccld 11987 analog). (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 β‰  0 )    β‡’   (πœ‘ β†’ (πΌβ€˜π‘‹) ∈ 𝐡)
 
Theoremdrnginvrl 20525 Property of the multiplicative inverse in a division ring. (recid2 11891 analog). (Contributed by NM, 19-Apr-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ ((πΌβ€˜π‘‹) Β· 𝑋) = 1 )
 
Theoremdrnginvrr 20526 Property of the multiplicative inverse in a division ring. (recid 11890 analog). (Contributed by NM, 19-Apr-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ (𝑋 Β· (πΌβ€˜π‘‹)) = 1 )
 
Theoremdrnginvrld 20527 Property of the multiplicative inverse in a division ring. (recid2d 11990 analog). (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 β‰  0 )    β‡’   (πœ‘ β†’ ((πΌβ€˜π‘‹) Β· 𝑋) = 1 )
 
Theoremdrnginvrrd 20528 Property of the multiplicative inverse in a division ring. (recidd 11989 analog). (Contributed by SN, 14-Aug-2024.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ 𝑋 β‰  0 )    β‡’   (πœ‘ β†’ (𝑋 Β· (πΌβ€˜π‘‹)) = 1 )
 
Theoremdrngmul0or 20529 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 Β· π‘Œ) = 0 ↔ (𝑋 = 0 ∨ π‘Œ = 0 )))
 
Theoremdrngmulne0 20530 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ ((𝑋 Β· π‘Œ) β‰  0 ↔ (𝑋 β‰  0 ∧ π‘Œ β‰  0 )))
 
Theoremdrngmuleq0 20531 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)
𝐡 = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑋 ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    &   (πœ‘ β†’ π‘Œ β‰  0 )    β‡’   (πœ‘ β†’ ((𝑋 Β· π‘Œ) = 0 ↔ 𝑋 = 0 ))
 
Theoremopprdrng 20532 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
𝑂 = (opprβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)
 
Theoremisdrngd 20533* Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element π‘₯ should have a left-inverse 𝐼(π‘₯). See isdrngrd 20534 for the characterization using right-inverses. (Contributed by NM, 2-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 0 = (0gβ€˜π‘…))    &   (πœ‘ β†’ 1 = (1rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) ∧ (𝑦 ∈ 𝐡 ∧ 𝑦 β‰  0 )) β†’ (π‘₯ Β· 𝑦) β‰  0 )    &   (πœ‘ β†’ 1 β‰  0 )    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ 𝐼 ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ (𝐼 Β· π‘₯) = 1 )    β‡’   (πœ‘ β†’ 𝑅 ∈ DivRing)
 
Theoremisdrngrd 20534* Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element π‘₯ should have a right-inverse 𝐼(π‘₯). See isdrngd 20533 for the characterization using left-inverses. (Contributed by NM, 10-Aug-2013.) Remove hypothesis. (Revised by SN, 19-Feb-2025.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 0 = (0gβ€˜π‘…))    &   (πœ‘ β†’ 1 = (1rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) ∧ (𝑦 ∈ 𝐡 ∧ 𝑦 β‰  0 )) β†’ (π‘₯ Β· 𝑦) β‰  0 )    &   (πœ‘ β†’ 1 β‰  0 )    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ 𝐼 ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ (π‘₯ Β· 𝐼) = 1 )    β‡’   (πœ‘ β†’ 𝑅 ∈ DivRing)
 
TheoremisdrngdOLD 20535* Obsolete version of isdrngd 20533 as of 19-Feb-2025. (Contributed by NM, 2-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 0 = (0gβ€˜π‘…))    &   (πœ‘ β†’ 1 = (1rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) ∧ (𝑦 ∈ 𝐡 ∧ 𝑦 β‰  0 )) β†’ (π‘₯ Β· 𝑦) β‰  0 )    &   (πœ‘ β†’ 1 β‰  0 )    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ 𝐼 ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ 𝐼 β‰  0 )    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ (𝐼 Β· π‘₯) = 1 )    β‡’   (πœ‘ β†’ 𝑅 ∈ DivRing)
 
TheoremisdrngrdOLD 20536* Obsolete version of isdrngrd 20534 as of 19-Feb-2025. (Contributed by NM, 10-Aug-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 0 = (0gβ€˜π‘…))    &   (πœ‘ β†’ 1 = (1rβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) ∧ (𝑦 ∈ 𝐡 ∧ 𝑦 β‰  0 )) β†’ (π‘₯ Β· 𝑦) β‰  0 )    &   (πœ‘ β†’ 1 β‰  0 )    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ 𝐼 ∈ 𝐡)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ 𝐼 β‰  0 )    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 )) β†’ (π‘₯ Β· 𝐼) = 1 )    β‡’   (πœ‘ β†’ 𝑅 ∈ DivRing)
 
Theoremdrngpropd 20537* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-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 ∈ DivRing ↔ 𝐿 ∈ DivRing))
 
Theoremfldpropd 20538* If two structures have the same group components (properties), one is a field 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 ∈ Field ↔ 𝐿 ∈ Field))
 
Theoremrng1nnzr 20539 The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-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 βˆ‰ NzRing)
 
Theoremring1zr 20540 The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) ∧ 𝑍 ∈ 𝐡) β†’ (𝐡 = {𝑍} ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremrngen1zr 20541 The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    β‡’   (((𝑅 ∈ Ring ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) ∧ 𝑍 ∈ 𝐡) β†’ (𝐡 β‰ˆ 1o ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremringen1zr 20542 The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐡 = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    βˆ— = (.rβ€˜π‘…)    &   π‘ = (0gβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ + Fn (𝐡 Γ— 𝐡) ∧ βˆ— Fn (𝐡 Γ— 𝐡)) β†’ (𝐡 β‰ˆ 1o ↔ ( + = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©} ∧ βˆ— = {βŸ¨βŸ¨π‘, π‘βŸ©, π‘βŸ©})))
 
Theoremrng1nfld 20543 The zero ring is not a field. (Contributed by AV, 29-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 βˆ‰ Field)
 
Theoremissubdrg 20544* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝑆 = (𝑅 β†Ύs 𝐴)    &    0 = (0gβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRingβ€˜π‘…)) β†’ (𝑆 ∈ DivRing ↔ βˆ€π‘₯ ∈ (𝐴 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝐴))
 
10.4.2  Sub-division rings
 
Syntaxcsdrg 20545 Syntax for subfields (sub-division-rings).
class SubDRing
 
Definitiondf-sdrg 20546* Define the function associating with a ring the set of its sub-division-rings. A sub-division-ring of a ring is a subset of its base set which is a division ring when equipped with the induced structure (sum, multiplication, zero, and unity). If a ring is commutative (resp., a field), then its sub-division-rings are commutative (resp., are fields) (fldsdrgfld 20557), so we do not make a specific definition for subfields. (Contributed by Stefan O'Rear, 3-Oct-2015.) TODO: extend this definition to a function with domain V or at least Ring and not only DivRing.
SubDRing = (𝑀 ∈ DivRing ↦ {𝑠 ∈ (SubRingβ€˜π‘€) ∣ (𝑀 β†Ύs 𝑠) ∈ DivRing})
 
Theoremissdrg 20547 Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
(𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRingβ€˜π‘…) ∧ (𝑅 β†Ύs 𝑆) ∈ DivRing))
 
Theoremsdrgrcl 20548 Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.)
(𝐴 ∈ (SubDRingβ€˜π‘…) β†’ 𝑅 ∈ DivRing)
 
Theoremsdrgdrng 20549 A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.)
𝑆 = (𝑅 β†Ύs 𝐴)    β‡’   (𝐴 ∈ (SubDRingβ€˜π‘…) β†’ 𝑆 ∈ DivRing)
 
Theoremsdrgsubrg 20550 A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
(𝐴 ∈ (SubDRingβ€˜π‘…) β†’ 𝐴 ∈ (SubRingβ€˜π‘…))
 
Theoremsdrgid 20551 Every division ring is a division subring of itself. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ 𝐡 ∈ (SubDRingβ€˜π‘…))
 
Theoremsdrgss 20552 A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑆 ∈ (SubDRingβ€˜π‘…) β†’ 𝑆 βŠ† 𝐡)
 
Theoremsdrgbas 20553 Base set of a sub-division-ring structure. (Contributed by SN, 19-Feb-2025.)
𝑆 = (𝑅 β†Ύs 𝐴)    β‡’   (𝐴 ∈ (SubDRingβ€˜π‘…) β†’ 𝐴 = (Baseβ€˜π‘†))
 
Theoremissdrg2 20554* Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐼 = (invrβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑆 ∈ (SubDRingβ€˜π‘…) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRingβ€˜π‘…) ∧ βˆ€π‘₯ ∈ (𝑆 βˆ– { 0 })(πΌβ€˜π‘₯) ∈ 𝑆))
 
Theoremsdrgunit 20555 A unit of a sub-division-ring is a nonzero element of the subring. (Contributed by SN, 19-Feb-2025.)
𝑆 = (𝑅 β†Ύs 𝐴)    &    0 = (0gβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘†)    β‡’   (𝐴 ∈ (SubDRingβ€˜π‘…) β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 β‰  0 )))
 
Theoremimadrhmcl 20556 The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.)
𝑅 = (𝑁 β†Ύs (𝐹 β€œ 𝑆))    &    0 = (0gβ€˜π‘)    &   (πœ‘ β†’ 𝐹 ∈ (𝑀 RingHom 𝑁))    &   (πœ‘ β†’ 𝑆 ∈ (SubDRingβ€˜π‘€))    &   (πœ‘ β†’ ran 𝐹 β‰  { 0 })    β‡’   (πœ‘ β†’ 𝑅 ∈ DivRing)
 
Theoremfldsdrgfld 20557 A sub-division-ring of a field is itself a field, so it is a subfield. We can therefore use SubDRing to express subfields. (Contributed by Thierry Arnoux, 11-Jan-2025.)
((𝐹 ∈ Field ∧ 𝐴 ∈ (SubDRingβ€˜πΉ)) β†’ (𝐹 β†Ύs 𝐴) ∈ Field)
 
Theoremacsfn1p 20558* Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ βˆ€π‘ ∈ π‘Œ 𝐸 ∈ 𝑋) β†’ {π‘Ž ∈ 𝒫 𝑋 ∣ βˆ€π‘ ∈ (π‘Ž ∩ π‘Œ)𝐸 ∈ π‘Ž} ∈ (ACSβ€˜π‘‹))
 
Theoremsubrgacs 20559 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (SubRingβ€˜π‘…) ∈ (ACSβ€˜π΅))
 
Theoremsdrgacs 20560 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) ∈ (ACSβ€˜π΅))
 
Theoremcntzsdrg 20561 Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐡 = (Baseβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘…)    &   π‘ = (Cntzβ€˜π‘€)    β‡’   ((𝑅 ∈ DivRing ∧ 𝑆 βŠ† 𝐡) β†’ (π‘β€˜π‘†) ∈ (SubDRingβ€˜π‘…))
 
Theoremsubdrgint 20562* The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝐿 = (𝑅 β†Ύs ∩ 𝑆)    &   (πœ‘ β†’ 𝑅 ∈ DivRing)    &   (πœ‘ β†’ 𝑆 βŠ† (SubRingβ€˜π‘…))    &   (πœ‘ β†’ 𝑆 β‰  βˆ…)    &   ((πœ‘ ∧ 𝑠 ∈ 𝑆) β†’ (𝑅 β†Ύs 𝑠) ∈ DivRing)    β‡’   (πœ‘ β†’ 𝐿 ∈ DivRing)
 
Theoremsdrgint 20563 The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
((𝑅 ∈ DivRing ∧ 𝑆 βŠ† (SubDRingβ€˜π‘…) ∧ 𝑆 β‰  βˆ…) β†’ ∩ 𝑆 ∈ (SubDRingβ€˜π‘…))
 
Theoremprimefld 20564 The smallest sub division ring of a division ring, here named 𝑃, is a field, called the Prime Field of 𝑅. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝑃 = (𝑅 β†Ύs ∩ (SubDRingβ€˜π‘…))    β‡’   (𝑅 ∈ DivRing β†’ 𝑃 ∈ Field)
 
Theoremprimefld0cl 20565 The prime field contains the zero element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ 0 ∈ ∩ (SubDRingβ€˜π‘…))
 
Theoremprimefld1cl 20566 The prime field contains the unity element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ 1 ∈ ∩ (SubDRingβ€˜π‘…))
 
10.4.3  Absolute value (abstract algebra)
 
Syntaxcabv 20567 The set of absolute values on a ring.
class AbsVal
 
Definitiondf-abv 20568* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 15187 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal = (π‘Ÿ ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Baseβ€˜π‘Ÿ)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘Ÿ)(((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘Ÿ)) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))})
 
Theoremabvfval 20569* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐡) ∣ βˆ€π‘₯ ∈ 𝐡 (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))})
 
Theoremisabv 20570* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
 
Theoremisabvd 20571* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
(πœ‘ β†’ 𝐴 = (AbsValβ€˜π‘…))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘…))    &   (πœ‘ β†’ + = (+gβ€˜π‘…))    &   (πœ‘ β†’ Β· = (.rβ€˜π‘…))    &   (πœ‘ β†’ 0 = (0gβ€˜π‘…))    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   (πœ‘ β†’ 𝐹:π΅βŸΆβ„)    &   (πœ‘ β†’ (πΉβ€˜ 0 ) = 0)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) β†’ 0 < (πΉβ€˜π‘₯))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) ∧ (𝑦 ∈ 𝐡 ∧ 𝑦 β‰  0 )) β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ π‘₯ β‰  0 ) ∧ (𝑦 ∈ 𝐡 ∧ 𝑦 β‰  0 )) β†’ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))    β‡’   (πœ‘ β†’ 𝐹 ∈ 𝐴)
 
Theoremabvrcl 20572 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐴 β†’ 𝑅 ∈ Ring)
 
Theoremabvfge0 20573 An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐴 β†’ 𝐹:𝐡⟢(0[,)+∞))
 
Theoremabvf 20574 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐴 β†’ 𝐹:π΅βŸΆβ„)
 
Theoremabvcl 20575 An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) ∈ ℝ)
 
Theoremabvge0 20576 The absolute value of a number is greater than or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 0 ≀ (πΉβ€˜π‘‹))
 
Theoremabveq0 20577 The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ ((πΉβ€˜π‘‹) = 0 ↔ 𝑋 = 0 ))
 
Theoremabvne0 20578 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ (πΉβ€˜π‘‹) β‰  0)
 
Theoremabvgt0 20579 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ 0 < (πΉβ€˜π‘‹))
 
Theoremabvmul 20580 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = ((πΉβ€˜π‘‹) Β· (πΉβ€˜π‘Œ)))
 
Theoremabvtri 20581 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    + = (+gβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 + π‘Œ)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ)))
 
Theoremabv0 20582 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐴 β†’ (πΉβ€˜ 0 ) = 0)
 
Theoremabv1z 20583 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 1 β‰  0 ) β†’ (πΉβ€˜ 1 ) = 1)
 
Theoremabv1 20584 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) β†’ (πΉβ€˜ 1 ) = 1)
 
Theoremabvneg 20585 The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜(π‘β€˜π‘‹)) = (πΉβ€˜π‘‹))
 
Theoremabvsubtri 20586 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜(𝑋 βˆ’ π‘Œ)) ≀ ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ)))
 
Theoremabvrec 20587 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    β‡’   (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 )) β†’ (πΉβ€˜(πΌβ€˜π‘‹)) = (1 / (πΉβ€˜π‘‹)))
 
Theoremabvdiv 20588 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    / = (/rβ€˜π‘…)    β‡’   (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  0 )) β†’ (πΉβ€˜(𝑋 / π‘Œ)) = ((πΉβ€˜π‘‹) / (πΉβ€˜π‘Œ)))
 
Theoremabvdom 20589 Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝐹 ∈ 𝐴 ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ β‰  0 )) β†’ (𝑋 Β· π‘Œ) β‰  0 )
 
Theoremabvres 20590 The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐴 = (AbsValβ€˜π‘…)    &   π‘† = (𝑅 β†Ύs 𝐢)    &   π΅ = (AbsValβ€˜π‘†)    β‡’   ((𝐹 ∈ 𝐴 ∧ 𝐢 ∈ (SubRingβ€˜π‘…)) β†’ (𝐹 β†Ύ 𝐢) ∈ 𝐡)
 
Theoremabvtrivd 20591* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ if(π‘₯ = 0 , 0, 1))    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    &   ((πœ‘ ∧ (𝑦 ∈ 𝐡 ∧ 𝑦 β‰  0 ) ∧ (𝑧 ∈ 𝐡 ∧ 𝑧 β‰  0 )) β†’ (𝑦 Β· 𝑧) β‰  0 )    β‡’   (πœ‘ β†’ 𝐹 ∈ 𝐴)
 
Theoremabvtriv 20592* The trivial absolute value. (This theorem is true as long as 𝑅 is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 20589 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsValβ€˜π‘…)    &   π΅ = (Baseβ€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   πΉ = (π‘₯ ∈ 𝐡 ↦ if(π‘₯ = 0 , 0, 1))    β‡’   (𝑅 ∈ DivRing β†’ 𝐹 ∈ 𝐴)
 
Theoremabvpropd 20593* If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-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†’ (AbsValβ€˜πΎ) = (AbsValβ€˜πΏ))
 
10.4.4  Star rings
 
Syntaxcstf 20594 Extend class notation with the functionalization of the *-ring involution.
class *rf
 
Syntaxcsr 20595 Extend class notation with class of all *-rings.
class *-Ring
 
Definitiondf-staf 20596* Define the functionalization of the involution in a star ring. This is not strictly necessary but by having *π‘Ÿ as an actual function we can state the principal properties of an involution much more cleanly. (Contributed by Mario Carneiro, 6-Oct-2015.)
*rf = (𝑓 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘“) ↦ ((*π‘Ÿβ€˜π‘“)β€˜π‘₯)))
 
Definitiondf-srng 20597* Define class of all star rings. A star ring is a ring with an involution (conjugation) function. Involution (unlike say the ring zero) is not unique and therefore must be added as a new component to the ring. For example, two possible involutions for complex numbers are the identity function and complex conjugation. Definition of involution in [Holland95] p. 204. (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
*-Ring = {𝑓 ∣ [(*rfβ€˜π‘“) / 𝑖](𝑖 ∈ (𝑓 RingHom (opprβ€˜π‘“)) ∧ 𝑖 = ◑𝑖)}
 
Theoremstaffval 20598* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ— = (*π‘Ÿβ€˜π‘…)    &    βˆ™ = (*rfβ€˜π‘…)    β‡’    βˆ™ = (π‘₯ ∈ 𝐡 ↦ ( βˆ— β€˜π‘₯))
 
Theoremstafval 20599 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ— = (*π‘Ÿβ€˜π‘…)    &    βˆ™ = (*rfβ€˜π‘…)    β‡’   (𝐴 ∈ 𝐡 β†’ ( βˆ™ β€˜π΄) = ( βˆ— β€˜π΄))
 
Theoremstaffn 20600 The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐡 = (Baseβ€˜π‘…)    &    βˆ— = (*π‘Ÿβ€˜π‘…)    &    βˆ™ = (*rfβ€˜π‘…)    β‡’   ( βˆ— Fn 𝐡 β†’ βˆ™ = βˆ— )
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