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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsrhmsubclem3 20601* Lemma 3 for srhmsubc 20602. (Contributed by AV, 19-Feb-2020.)
βˆ€π‘Ÿ ∈ 𝑆 π‘Ÿ ∈ Ring    &   πΆ = (π‘ˆ ∩ 𝑆)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   ((π‘ˆ ∈ 𝑉 ∧ (𝑋 ∈ 𝐢 ∧ π‘Œ ∈ 𝐢)) β†’ (π‘‹π½π‘Œ) = (𝑋(Hom β€˜(RingCatβ€˜π‘ˆ))π‘Œ))
 
Theoremsrhmsubc 20602* According to df-subc 17786, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17817 and subcss2 17820). Therefore, the set of special ring homomorphisms (i.e., ring homomorphisms from a special ring to another ring of that kind) is a subcategory of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
βˆ€π‘Ÿ ∈ 𝑆 π‘Ÿ ∈ Ring    &   πΆ = (π‘ˆ ∩ 𝑆)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)))
 
Theoremsringcat 20603* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.)
βˆ€π‘Ÿ ∈ 𝑆 π‘Ÿ ∈ Ring    &   πΆ = (π‘ˆ ∩ 𝑆)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ ((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽) ∈ Cat)
 
Theoremcrhmsubc 20604* According to df-subc 17786, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17817 and subcss2 17820). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
𝐢 = (π‘ˆ ∩ CRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)))
 
Theoremcringcat 20605* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.)
𝐢 = (π‘ˆ ∩ CRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ ((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽) ∈ Cat)
 
Theoremrngcrescrhm 20606 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   (πœ‘ β†’ (𝐢 β†Ύcat 𝐻) = ((𝐢 β†Ύs 𝑅) sSet ⟨(Hom β€˜ndx), 𝐻⟩))
 
Theoremrhmsubclem1 20607 Lemma 1 for rhmsubc 20611. (Contributed by AV, 2-Mar-2020.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   (πœ‘ β†’ 𝐻 Fn (𝑅 Γ— 𝑅))
 
Theoremrhmsubclem2 20608 Lemma 2 for rhmsubc 20611. (Contributed by AV, 2-Mar-2020.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   ((πœ‘ ∧ 𝑋 ∈ 𝑅 ∧ π‘Œ ∈ 𝑅) β†’ (π‘‹π»π‘Œ) = (𝑋 RingHom π‘Œ))
 
Theoremrhmsubclem3 20609* Lemma 3 for rhmsubc 20611. (Contributed by AV, 2-Mar-2020.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   ((πœ‘ ∧ π‘₯ ∈ 𝑅) β†’ ((Idβ€˜(RngCatβ€˜π‘ˆ))β€˜π‘₯) ∈ (π‘₯𝐻π‘₯))
 
Theoremrhmsubclem4 20610* Lemma 4 for rhmsubc 20611. (Contributed by AV, 2-Mar-2020.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   ((((πœ‘ ∧ π‘₯ ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (π‘₯𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(RngCatβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐻𝑧))
 
Theoremrhmsubc 20611 According to df-subc 17786, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17817 and subcss2 17820). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   (πœ‘ β†’ 𝐻 ∈ (Subcatβ€˜(RngCatβ€˜π‘ˆ)))
 
Theoremrhmsubccat 20612 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   (πœ‘ β†’ ((RngCatβ€˜π‘ˆ) β†Ύcat 𝐻) ∈ Cat)
 
10.4  Division rings and fields
 
10.4.1  Definition and basic properties
 
Syntaxcdr 20613 Extend class notation with class of all division rings.
class DivRing
 
Syntaxcfield 20614 Class of fields.
class Field
 
Definitiondf-drng 20615 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 20616 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field = (DivRing ∩ CRing)
 
Theoremisdrng 20617 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 20618 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 20619 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 20620 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ Ring)
 
Theoremdrngringd 20621 A division ring is a ring. (Contributed by SN, 16-May-2024.)
(πœ‘ β†’ 𝑅 ∈ DivRing)    β‡’   (πœ‘ β†’ 𝑅 ∈ Ring)
 
Theoremdrnggrpd 20622 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 20623 A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ Grp)
 
Theoremisfld 20624 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
 
Theoremflddrngd 20625 A field is a division ring. (Contributed by SN, 17-Jan-2025.)
(πœ‘ β†’ 𝑅 ∈ Field)    β‡’   (πœ‘ β†’ 𝑅 ∈ DivRing)
 
Theoremfldcrngd 20626 A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
(πœ‘ β†’ 𝑅 ∈ Field)    β‡’   (πœ‘ β†’ 𝑅 ∈ CRing)
 
Theoremisdrng2 20627 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 20628 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 20629 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 20630 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 20631 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 20632 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 20633 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing β†’ 𝑅 ∈ NzRing)
 
Theoremdrngid2 20634 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 20635 Closure of the multiplicative inverse in a division ring. (reccl 11901 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 20636 The multiplicative inverse in a division ring is nonzero. (recne0 11907 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 20637 Closure of the multiplicative inverse in a division ring. (reccld 12005 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 20638 Property of the multiplicative inverse in a division ring. (recid2 11909 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 20639 Property of the multiplicative inverse in a division ring. (recid 11908 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 20640 Property of the multiplicative inverse in a division ring. (recid2d 12008 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 20641 Property of the multiplicative inverse in a division ring. (recidd 12007 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 20642 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 20643 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 20644 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 20645 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 20646* 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 20647 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 20647* 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 20646 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 20648* Obsolete version of isdrngd 20646 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 20649* Obsolete version of isdrngrd 20647 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 20650* 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 20651* 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 20652 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 20653 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 20654 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 20655 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 20656 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 20657* 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))
 
Theoremdrhmsubc 20658* According to df-subc 17786, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17817 and subcss2 17820). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.)
𝐢 = (π‘ˆ ∩ DivRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatβ€˜π‘ˆ)))
 
Theoremdrngcat 20659* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.)
𝐢 = (π‘ˆ ∩ DivRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ ((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽) ∈ Cat)
 
Theoremfldcat 20660* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
𝐢 = (π‘ˆ ∩ DivRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    &   π· = (π‘ˆ ∩ Field)    &   πΉ = (π‘Ÿ ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ ((RingCatβ€˜π‘ˆ) β†Ύcat 𝐹) ∈ Cat)
 
Theoremfldc 20661* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
𝐢 = (π‘ˆ ∩ DivRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    &   π· = (π‘ˆ ∩ Field)    &   πΉ = (π‘Ÿ ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ (((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽) β†Ύcat 𝐹) ∈ Cat)
 
Theoremfldhmsubc 20662* According to df-subc 17786, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17817 and subcss2 17820). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
𝐢 = (π‘ˆ ∩ DivRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    &   π· = (π‘ˆ ∩ Field)    &   πΉ = (π‘Ÿ ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ 𝐹 ∈ (Subcatβ€˜((RingCatβ€˜π‘ˆ) β†Ύcat 𝐽)))
 
10.4.2  Sub-division rings
 
Syntaxcsdrg 20663 Syntax for subfields (sub-division-rings).
class SubDRing
 
Definitiondf-sdrg 20664* 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 20675), 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 20665 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 20666 Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.)
(𝐴 ∈ (SubDRingβ€˜π‘…) β†’ 𝑅 ∈ DivRing)
 
Theoremsdrgdrng 20667 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 20668 A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
(𝐴 ∈ (SubDRingβ€˜π‘…) β†’ 𝐴 ∈ (SubRingβ€˜π‘…))
 
Theoremsdrgid 20669 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 20670 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 20671 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 20672* 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 20673 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 20674 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 20675 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 20676* 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 20677 Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (SubRingβ€˜π‘…) ∈ (ACSβ€˜π΅))
 
Theoremsdrgacs 20678 Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐡 = (Baseβ€˜π‘…)    β‡’   (𝑅 ∈ DivRing β†’ (SubDRingβ€˜π‘…) ∈ (ACSβ€˜π΅))
 
Theoremcntzsdrg 20679 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 20680* 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 20681 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 20682 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 20683 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 20684 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 20685 The set of absolute values on a ring.
class AbsVal
 
Definitiondf-abv 20686* Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 15207 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 20687* 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 20688* 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 20689* 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 20690 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsValβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐴 β†’ 𝑅 ∈ Ring)
 
Theoremabvfge0 20691 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 20692 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 20693 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 20694 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 20695 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 20696 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 20697 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 20698 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 20699 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 20700 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)
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