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Theorem List for Metamath Proof Explorer - 46101-46200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrngcrescrhm 46101 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 46102 Lemma 1 for rhmsubc 46106. (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 46103 Lemma 2 for rhmsubc 46106. (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 46104* Lemma 3 for rhmsubc 46106. (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 46105* Lemma 4 for rhmsubc 46106. (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 46106 According to df-subc 17630, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17661 and subcss2 17664). 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 46107 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)
 
TheoremsrhmsubcALTVlem1 46108* Lemma 1 for srhmsubcALTV 46110. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
βˆ€π‘Ÿ ∈ 𝑆 π‘Ÿ ∈ Ring    &   πΆ = (π‘ˆ ∩ 𝑆)    β‡’   ((π‘ˆ ∈ 𝑉 ∧ 𝑋 ∈ 𝐢) β†’ 𝑋 ∈ (Baseβ€˜(RingCatALTVβ€˜π‘ˆ)))
 
TheoremsrhmsubcALTVlem2 46109* Lemma 2 for srhmsubcALTV 46110. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
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€˜(RingCatALTVβ€˜π‘ˆ))π‘Œ))
 
TheoremsrhmsubcALTV 46110* According to df-subc 17630, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17661 and subcss2 17664). 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.) (New usage is discouraged.)
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€˜(RingCatALTVβ€˜π‘ˆ)))
 
TheoremsringcatALTV 46111* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
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†’ ((RingCatALTVβ€˜π‘ˆ) β†Ύcat 𝐽) ∈ Cat)
 
TheoremcrhmsubcALTV 46112* According to df-subc 17630, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17661 and subcss2 17664). 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.) (New usage is discouraged.)
𝐢 = (π‘ˆ ∩ CRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatALTVβ€˜π‘ˆ)))
 
TheoremcringcatALTV 46113* 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.) (New usage is discouraged.)
𝐢 = (π‘ˆ ∩ CRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ ((RingCatALTVβ€˜π‘ˆ) β†Ύcat 𝐽) ∈ Cat)
 
TheoremdrhmsubcALTV 46114* According to df-subc 17630, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17661 and subcss2 17664). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐢 = (π‘ˆ ∩ DivRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ 𝐽 ∈ (Subcatβ€˜(RingCatALTVβ€˜π‘ˆ)))
 
TheoremdrngcatALTV 46115* 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.) (New usage is discouraged.)
𝐢 = (π‘ˆ ∩ DivRing)    &   π½ = (π‘Ÿ ∈ 𝐢, 𝑠 ∈ 𝐢 ↦ (π‘Ÿ RingHom 𝑠))    β‡’   (π‘ˆ ∈ 𝑉 β†’ ((RingCatALTVβ€˜π‘ˆ) β†Ύcat 𝐽) ∈ Cat)
 
TheoremfldcatALTV 46116* 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.) (New usage is discouraged.)
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†’ ((RingCatALTVβ€˜π‘ˆ) β†Ύcat 𝐹) ∈ Cat)
 
TheoremfldcALTV 46117* 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.) (New usage is discouraged.)
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†’ (((RingCatALTVβ€˜π‘ˆ) β†Ύcat 𝐽) β†Ύcat 𝐹) ∈ Cat)
 
TheoremfldhmsubcALTV 46118* According to df-subc 17630, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17661 and subcss2 17664). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
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€˜((RingCatALTVβ€˜π‘ˆ) β†Ύcat 𝐽)))
 
TheoremrngcrescrhmALTV 46119 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.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatALTVβ€˜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⟩))
 
TheoremrhmsubcALTVlem1 46120 Lemma 1 for rhmsubcALTV 46124. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatALTVβ€˜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))
 
TheoremrhmsubcALTVlem2 46121 Lemma 2 for rhmsubcALTV 46124. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatALTVβ€˜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))
 
TheoremrhmsubcALTVlem3 46122* Lemma 3 for rhmsubcALTV 46124. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatALTVβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   ((πœ‘ ∧ π‘₯ ∈ 𝑅) β†’ ((Idβ€˜(RngCatALTVβ€˜π‘ˆ))β€˜π‘₯) ∈ (π‘₯𝐻π‘₯))
 
TheoremrhmsubcALTVlem4 46123* Lemma 4 for rhmsubcALTV 46124. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatALTVβ€˜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€˜(RngCatALTVβ€˜π‘ˆ))𝑧)𝑓) ∈ (π‘₯𝐻𝑧))
 
TheoremrhmsubcALTV 46124 According to df-subc 17630, the subcategories (Subcatβ€˜πΆ) of a category 𝐢 are subsets of the homomorphisms of 𝐢 (see subcssc 17661 and subcss2 17664). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatALTVβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   (πœ‘ β†’ 𝐻 ∈ (Subcatβ€˜(RngCatALTVβ€˜π‘ˆ)))
 
TheoremrhmsubcALTVcat 46125 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.) (New usage is discouraged.)
(πœ‘ β†’ π‘ˆ ∈ 𝑉)    &   πΆ = (RngCatALTVβ€˜π‘ˆ)    &   (πœ‘ β†’ 𝑅 = (Ring ∩ π‘ˆ))    &   π» = ( RingHom β†Ύ (𝑅 Γ— 𝑅))    β‡’   (πœ‘ β†’ ((RngCatALTVβ€˜π‘ˆ) β†Ύcat 𝐻) ∈ Cat)
 
21.43.20  Basic algebraic structures (extension)
 
21.43.20.1  Auxiliary theorems
 
Theoremopeliun2xp 46126 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5696. (Contributed by AV, 30-Mar-2019.)
(⟨𝐢, π‘¦βŸ© ∈ βˆͺ 𝑦 ∈ 𝐡 (𝐴 Γ— {𝑦}) ↔ (𝑦 ∈ 𝐡 ∧ 𝐢 ∈ 𝐴))
 
Theoremeliunxp2 46127* Membership in a union of Cartesian products over its second component, analogous to eliunxp 5790. (Contributed by AV, 30-Mar-2019.)
(𝐢 ∈ βˆͺ 𝑦 ∈ 𝐡 (𝐴 Γ— {𝑦}) ↔ βˆƒπ‘₯βˆƒπ‘¦(𝐢 = ⟨π‘₯, π‘¦βŸ© ∧ (π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡)))
 
Theoremmpomptx2 46128* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐴(𝑦) is not assumed to be constant w.r.t 𝑦, analogous to mpomptx 7462. (Contributed by AV, 30-Mar-2019.)
(𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ 𝐢 = 𝐷)    β‡’   (𝑧 ∈ βˆͺ 𝑦 ∈ 𝐡 (𝐴 Γ— {𝑦}) ↦ 𝐢) = (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ 𝐷)
 
Theoremcbvmpox2 46129* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7444 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7443. (Contributed by AV, 30-Mar-2019.)
Ⅎ𝑧𝐴    &   β„²π‘¦π·    &   β„²π‘§πΆ    &   β„²π‘€πΆ    &   β„²π‘₯𝐸    &   β„²π‘¦πΈ    &   (𝑦 = 𝑧 β†’ 𝐴 = 𝐷)    &   ((𝑦 = 𝑧 ∧ π‘₯ = 𝑀) β†’ 𝐢 = 𝐸)    β‡’   (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ 𝐢) = (𝑀 ∈ 𝐷, 𝑧 ∈ 𝐡 ↦ 𝐸)
 
Theoremdmmpossx2 46130* The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpossx 7987. (Contributed by AV, 30-Mar-2019.)
𝐹 = (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ 𝐢)    β‡’   dom 𝐹 βŠ† βˆͺ 𝑦 ∈ 𝐡 (𝐴 Γ— {𝑦})
 
Theoremmpoexxg2 46131* Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpoexxg 7997. (Contributed by AV, 30-Mar-2019.)
𝐹 = (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ 𝐢)    β‡’   ((𝐡 ∈ 𝑅 ∧ βˆ€π‘¦ ∈ 𝐡 𝐴 ∈ 𝑆) β†’ 𝐹 ∈ V)
 
Theoremovmpordxf 46132* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7498. (Contributed by AV, 30-Mar-2019.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐢, 𝑦 ∈ 𝐷 ↦ 𝑅))    &   ((πœ‘ ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐡)) β†’ 𝑅 = 𝑆)    &   ((πœ‘ ∧ 𝑦 = 𝐡) β†’ 𝐢 = 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝐿)    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    &   (πœ‘ β†’ 𝑆 ∈ 𝑋)    &   β„²π‘₯πœ‘    &   β„²π‘¦πœ‘    &   β„²π‘¦π΄    &   β„²π‘₯𝐡    &   β„²π‘₯𝑆    &   β„²π‘¦π‘†    β‡’   (πœ‘ β†’ (𝐴𝐹𝐡) = 𝑆)
 
Theoremovmpordx 46133* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7498. (Contributed by AV, 30-Mar-2019.)
(πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐢, 𝑦 ∈ 𝐷 ↦ 𝑅))    &   ((πœ‘ ∧ (π‘₯ = 𝐴 ∧ 𝑦 = 𝐡)) β†’ 𝑅 = 𝑆)    &   ((πœ‘ ∧ 𝑦 = 𝐡) β†’ 𝐢 = 𝐿)    &   (πœ‘ β†’ 𝐴 ∈ 𝐿)    &   (πœ‘ β†’ 𝐡 ∈ 𝐷)    &   (πœ‘ β†’ 𝑆 ∈ 𝑋)    β‡’   (πœ‘ β†’ (𝐴𝐹𝐡) = 𝑆)
 
Theoremovmpox2 46134* The value of an operation class abstraction. Variant of ovmpoga 7502 which does not require 𝐷 and π‘₯ to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
((π‘₯ = 𝐴 ∧ 𝑦 = 𝐡) β†’ 𝑅 = 𝑆)    &   (𝑦 = 𝐡 β†’ 𝐢 = 𝐿)    &   πΉ = (π‘₯ ∈ 𝐢, 𝑦 ∈ 𝐷 ↦ 𝑅)    β‡’   ((𝐴 ∈ 𝐿 ∧ 𝐡 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) β†’ (𝐴𝐹𝐡) = 𝑆)
 
Theoremfdmdifeqresdif 46135* The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
𝐹 = (π‘₯ ∈ 𝐷 ↦ if(π‘₯ = π‘Œ, 𝑋, (πΊβ€˜π‘₯)))    β‡’   (𝐺:(𝐷 βˆ– {π‘Œ})βŸΆπ‘… β†’ 𝐺 = (𝐹 β†Ύ (𝐷 βˆ– {π‘Œ})))
 
Theoremoffvalfv 46136* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹 Fn 𝐴)    &   (πœ‘ β†’ 𝐺 Fn 𝐴)    β‡’   (πœ‘ β†’ (𝐹 ∘f 𝑅𝐺) = (π‘₯ ∈ 𝐴 ↦ ((πΉβ€˜π‘₯)𝑅(πΊβ€˜π‘₯))))
 
Theoremofaddmndmap 46137 The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.)
𝑅 = (Baseβ€˜π‘€)    &    + = (+gβ€˜π‘€)    β‡’   ((𝑀 ∈ Mnd ∧ 𝑉 ∈ π‘Œ ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ (𝐴 ∘f + 𝐡) ∈ (𝑅 ↑m 𝑉))
 
Theoremmapsnop 46138 A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)
𝐹 = {βŸ¨π‘‹, π‘ŒβŸ©}    β‡’   ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑅 ∧ 𝑅 ∈ π‘Š) β†’ 𝐹 ∈ (𝑅 ↑m {𝑋}))
 
Theoremfprmappr 46139 A function with a domain of two elements as element of the mapping operator applied to a pair. (Contributed by AV, 20-May-2024.)
((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ π‘Š ∧ 𝐴 β‰  𝐡) ∧ (𝐢 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) β†’ {⟨𝐴, 𝐢⟩, ⟨𝐡, 𝐷⟩} ∈ (𝑋 ↑m {𝐴, 𝐡}))
 
Theoremmapprop 46140 An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.) (Proof shortened by AV, 2-Jun-2024.)
𝐹 = {βŸ¨π‘‹, 𝐴⟩, βŸ¨π‘Œ, 𝐡⟩}    β‡’   (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (π‘Œ ∈ 𝑉 ∧ 𝐡 ∈ 𝑅) ∧ (𝑋 β‰  π‘Œ ∧ 𝑅 ∈ π‘Š)) β†’ 𝐹 ∈ (𝑅 ↑m {𝑋, π‘Œ}))
 
Theoremztprmneprm 46141 A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
((𝑍 ∈ β„€ ∧ 𝐴 ∈ β„™ ∧ 𝐡 ∈ β„™) β†’ ((𝑍 Β· 𝐴) = 𝐡 β†’ 𝐴 = 𝐡))
 
Theorem2t6m3t4e0 46142 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
((2 Β· 6) βˆ’ (3 Β· 4)) = 0
 
Theoremssnn0ssfz 46143* For any finite subset of β„•0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 31472. (Contributed by AV, 30-Sep-2019.)
(𝐴 ∈ (𝒫 β„•0 ∩ Fin) β†’ βˆƒπ‘› ∈ β„•0 𝐴 βŠ† (0...𝑛))
 
Theoremnn0sumltlt 46144 If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
((π‘Ž ∈ β„•0 ∧ 𝑏 ∈ β„•0 ∧ 𝑐 ∈ β„•0) β†’ ((π‘Ž + 𝑏) < 𝑐 β†’ 𝑏 < 𝑐))
 
21.43.20.2  The binomial coefficient operation (extension)
 
Theorembcpascm1 46145 Pascal's rule for the binomial coefficient, generalized to all integers 𝐾, shifted down by 1. (Contributed by AV, 8-Sep-2019.)
((𝑁 ∈ β„• ∧ 𝐾 ∈ β„€) β†’ (((𝑁 βˆ’ 1)C𝐾) + ((𝑁 βˆ’ 1)C(𝐾 βˆ’ 1))) = (𝑁C𝐾))
 
Theoremaltgsumbc 46146* The sum of binomial coefficients for a fixed positive 𝑁 with alternating signs is zero. Notice that this is not valid for 𝑁 = 0 (since ((-1↑0) Β· (0C0)) = (1 Β· 1) = 1). For a proof using Pascal's rule (bcpascm1 46145) instead of the binomial theorem (binom 15650) , see altgsumbcALT 46147. (Contributed by AV, 13-Sep-2019.)
(𝑁 ∈ β„• β†’ Ξ£π‘˜ ∈ (0...𝑁)((-1β†‘π‘˜) Β· (𝑁Cπ‘˜)) = 0)
 
TheoremaltgsumbcALT 46147* Alternate proof of altgsumbc 46146, using Pascal's rule (bcpascm1 46145) instead of the binomial theorem (binom 15650). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ β„• β†’ Ξ£π‘˜ ∈ (0...𝑁)((-1β†‘π‘˜) Β· (𝑁Cπ‘˜)) = 0)
 
21.43.20.3  The ` ZZ `-module ` ZZ X. ZZ `
 
Theoremzlmodzxzlmod 46148 The β„€-module β„€ Γ— β„€ is a (left) module with the ring of integers as base set. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (β„€ring freeLMod {0, 1})    β‡’   (𝑍 ∈ LMod ∧ β„€ring = (Scalarβ€˜π‘))
 
Theoremzlmodzxzel 46149 An element of the (base set of the) β„€-module β„€ Γ— β„€. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (β„€ring freeLMod {0, 1})    β‡’   ((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ {⟨0, 𝐴⟩, ⟨1, 𝐡⟩} ∈ (Baseβ€˜π‘))
 
Theoremzlmodzxz0 46150 The 0 of the β„€-module β„€ Γ— β„€. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (β„€ring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    β‡’    0 = (0gβ€˜π‘)
 
Theoremzlmodzxzscm 46151 The scalar multiplication of the β„€-module β„€ Γ— β„€. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (β„€ring freeLMod {0, 1})    &    βˆ™ = ( ·𝑠 β€˜π‘)    β‡’   ((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„€) β†’ (𝐴 βˆ™ {⟨0, 𝐡⟩, ⟨1, 𝐢⟩}) = {⟨0, (𝐴 Β· 𝐡)⟩, ⟨1, (𝐴 Β· 𝐢)⟩})
 
Theoremzlmodzxzadd 46152 The addition of the β„€-module β„€ Γ— β„€. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (β„€ring freeLMod {0, 1})    &    + = (+gβ€˜π‘)    β‡’   (((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) ∧ (𝐢 ∈ β„€ ∧ 𝐷 ∈ β„€)) β†’ ({⟨0, 𝐴⟩, ⟨1, 𝐢⟩} + {⟨0, 𝐡⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐡)⟩, ⟨1, (𝐢 + 𝐷)⟩})
 
Theoremzlmodzxzsubm 46153 The subtraction of the β„€-module β„€ Γ— β„€ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (β„€ring freeLMod {0, 1})    &    βˆ’ = (-gβ€˜π‘)    β‡’   (((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) ∧ (𝐢 ∈ β„€ ∧ 𝐷 ∈ β„€)) β†’ ({⟨0, 𝐴⟩, ⟨1, 𝐢⟩} βˆ’ {⟨0, 𝐡⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐢⟩} (+gβ€˜π‘)(-1( ·𝑠 β€˜π‘){⟨0, 𝐡⟩, ⟨1, 𝐷⟩})))
 
Theoremzlmodzxzsub 46154 The subtraction of the β„€-module β„€ Γ— β„€. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (β„€ring freeLMod {0, 1})    &    βˆ’ = (-gβ€˜π‘)    β‡’   (((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) ∧ (𝐢 ∈ β„€ ∧ 𝐷 ∈ β„€)) β†’ ({⟨0, 𝐴⟩, ⟨1, 𝐢⟩} βˆ’ {⟨0, 𝐡⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 βˆ’ 𝐡)⟩, ⟨1, (𝐢 βˆ’ 𝐷)⟩})
 
21.43.20.4  Group sum operation (extension 2)
 
Theoremmgpsumunsn 46155* Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrpβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑁) β†’ 𝐴 ∈ (Baseβ€˜π‘…))    &   (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜π‘…))    &   (π‘˜ = 𝐼 β†’ 𝐴 = 𝑋)    β‡’   (πœ‘ β†’ (𝑀 Ξ£g (π‘˜ ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Ξ£g (π‘˜ ∈ (𝑁 βˆ– {𝐼}) ↦ 𝐴)) Β· 𝑋))
 
Theoremmgpsumz 46156* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrpβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑁) β†’ 𝐴 ∈ (Baseβ€˜π‘…))    &    0 = (0gβ€˜π‘…)    &   (π‘˜ = 𝐼 β†’ 𝐴 = 0 )    β‡’   (πœ‘ β†’ (𝑀 Ξ£g (π‘˜ ∈ 𝑁 ↦ 𝐴)) = 0 )
 
Theoremmgpsumn 46157* If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrpβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑁 ∈ Fin)    &   (πœ‘ β†’ 𝐼 ∈ 𝑁)    &   ((πœ‘ ∧ π‘˜ ∈ 𝑁) β†’ 𝐴 ∈ (Baseβ€˜π‘…))    &    1 = (1rβ€˜π‘…)    &   (π‘˜ = 𝐼 β†’ 𝐴 = 1 )    β‡’   (πœ‘ β†’ (𝑀 Ξ£g (π‘˜ ∈ 𝑁 ↦ 𝐴)) = (𝑀 Ξ£g (π‘˜ ∈ (𝑁 βˆ– {𝐼}) ↦ 𝐴)))
 
21.43.20.5  Symmetric groups (extension)
 
Theoremexple2lt6 46158 A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.)
((𝑁 ∈ β„•0 ∧ 𝑁 ≀ 2) β†’ (𝑁↑𝑁) < 6)
 
Theorempgrple2abl 46159 Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
𝐺 = (SymGrpβ€˜π΄)    β‡’   ((𝐴 ∈ 𝑉 ∧ (β™―β€˜π΄) ≀ 2) β†’ 𝐺 ∈ Abel)
 
Theorempgrpgt2nabl 46160 Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
𝐺 = (SymGrpβ€˜π΄)    β‡’   ((𝐴 ∈ 𝑉 ∧ 2 < (β™―β€˜π΄)) β†’ 𝐺 βˆ‰ Abel)
 
21.43.20.6  Divisibility (extension)
 
Theoreminvginvrid 46161 Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
𝐡 = (Baseβ€˜π‘…)    &   π‘ˆ = (Unitβ€˜π‘…)    &   π‘ = (invgβ€˜π‘…)    &   πΌ = (invrβ€˜π‘…)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ) β†’ ((π‘β€˜π‘Œ) Β· ((πΌβ€˜(π‘β€˜π‘Œ)) Β· 𝑋)) = 𝑋)
 
21.43.20.7  The support of functions (extension)
 
Theoremrmsupp0 46162* The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐢 = (0gβ€˜π‘€)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜π‘€)(π΄β€˜π‘£))) supp (0gβ€˜π‘€)) = βˆ…)
 
Theoremdomnmsuppn0 46163* The support of a mapping of a multiplication of a nonzero constant with a function into a (ring theoretic) domain equals the support of the function. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Domn ∧ 𝑉 ∈ 𝑋) ∧ (𝐢 ∈ 𝑅 ∧ 𝐢 β‰  (0gβ€˜π‘€)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜π‘€)(π΄β€˜π‘£))) supp (0gβ€˜π‘€)) = (𝐴 supp (0gβ€˜π‘€)))
 
Theoremrmsuppss 46164* The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐢 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜π‘€)(π΄β€˜π‘£))) supp (0gβ€˜π‘€)) βŠ† (𝐴 supp (0gβ€˜π‘€)))
 
Theoremmndpsuppss 46165 The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉))) β†’ ((𝐴 ∘f (+gβ€˜π‘€)𝐡) supp (0gβ€˜π‘€)) βŠ† ((𝐴 supp (0gβ€˜π‘€)) βˆͺ (𝐡 supp (0gβ€˜π‘€))))
 
Theoremscmsuppss 46166* The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
𝑆 = (Scalarβ€˜π‘€)    &   π‘… = (Baseβ€˜π‘†)    β‡’   ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) β†’ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) supp (0gβ€˜π‘€)) βŠ† (𝐴 supp (0gβ€˜π‘†)))
 
21.43.20.8  Finitely supported functions (extension)
 
Theoremrmsuppfi 46167* The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐢 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0gβ€˜π‘€)) ∈ Fin) β†’ ((𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜π‘€)(π΄β€˜π‘£))) supp (0gβ€˜π‘€)) ∈ Fin)
 
Theoremrmfsupp 46168* A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐢 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0gβ€˜π‘€)) β†’ (𝑣 ∈ 𝑉 ↦ (𝐢(.rβ€˜π‘€)(π΄β€˜π‘£))) finSupp (0gβ€˜π‘€))
 
Theoremmndpsuppfi 46169 The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ ((𝐴 supp (0gβ€˜π‘€)) ∈ Fin ∧ (𝐡 supp (0gβ€˜π‘€)) ∈ Fin)) β†’ ((𝐴 ∘f (+gβ€˜π‘€)𝐡) supp (0gβ€˜π‘€)) ∈ Fin)
 
Theoremmndpfsupp 46170 A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑅 = (Baseβ€˜π‘€)    β‡’   (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑋) ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐡 ∈ (𝑅 ↑m 𝑉)) ∧ (𝐴 finSupp (0gβ€˜π‘€) ∧ 𝐡 finSupp (0gβ€˜π‘€))) β†’ (𝐴 ∘f (+gβ€˜π‘€)𝐡) finSupp (0gβ€˜π‘€))
 
Theoremscmsuppfi 46171* The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
𝑆 = (Scalarβ€˜π‘€)    &   π‘… = (Baseβ€˜π‘†)    β‡’   (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0gβ€˜π‘†)) ∈ Fin) β†’ ((𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) supp (0gβ€˜π‘€)) ∈ Fin)
 
Theoremscmfsupp 46172* A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝑆 = (Scalarβ€˜π‘€)    &   π‘… = (Baseβ€˜π‘†)    β‡’   (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Baseβ€˜π‘€)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0gβ€˜π‘†)) β†’ (𝑣 ∈ 𝑉 ↦ ((π΄β€˜π‘£)( ·𝑠 β€˜π‘€)𝑣)) finSupp (0gβ€˜π‘€))
 
Theoremsuppmptcfin 46173* The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
𝐡 = (Baseβ€˜π‘€)    &   π‘… = (Scalarβ€˜π‘€)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΉ = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑋, 1 , 0 ))    β‡’   ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ (𝐹 supp 0 ) ∈ Fin)
 
Theoremmptcfsupp 46174* A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝐡 = (Baseβ€˜π‘€)    &   π‘… = (Scalarβ€˜π‘€)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΉ = (π‘₯ ∈ 𝑉 ↦ if(π‘₯ = 𝑋, 1 , 0 ))    β‡’   ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐡 ∧ 𝑋 ∈ 𝑉) β†’ 𝐹 finSupp 0 )
 
Theoremfsuppmptdmf 46175* A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
β„²π‘₯πœ‘    &   πΉ = (π‘₯ ∈ 𝐴 ↦ π‘Œ)    &   (πœ‘ β†’ 𝐴 ∈ Fin)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ π‘Š)    β‡’   (πœ‘ β†’ 𝐹 finSupp 𝑍)
 
21.43.20.9  Left modules (extension)
 
Theoremlmodvsmdi 46176 Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
𝑉 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &    ↑ = (.gβ€˜π‘Š)    &   πΈ = (.gβ€˜πΉ)    β‡’   ((π‘Š ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑁 ∈ β„•0 ∧ 𝑋 ∈ 𝑉)) β†’ (𝑅 Β· (𝑁 ↑ 𝑋)) = ((𝑁𝐸𝑅) Β· 𝑋))
 
Theoremgsumlsscl 46177* Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝑆 = (LSubSpβ€˜π‘€)    &   π‘… = (Scalarβ€˜π‘€)    &   π΅ = (Baseβ€˜π‘…)    β‡’   ((𝑀 ∈ LMod ∧ 𝑍 ∈ 𝑆 ∧ 𝑉 βŠ† 𝑍) β†’ ((𝐹 ∈ (𝐡 ↑m 𝑉) ∧ 𝐹 finSupp (0gβ€˜π‘…)) β†’ (𝑀 Ξ£g (𝑣 ∈ 𝑉 ↦ ((πΉβ€˜π‘£)( ·𝑠 β€˜π‘€)𝑣))) ∈ 𝑍))
 
21.43.20.10  Associative algebras (extension)
 
Theoremassaascl0 46178 The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algScβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ AssAlg)    β‡’   (πœ‘ β†’ (π΄β€˜(0gβ€˜πΉ)) = (0gβ€˜π‘Š))
 
Theoremassaascl1 46179 The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algScβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ AssAlg)    β‡’   (πœ‘ β†’ (π΄β€˜(1rβ€˜πΉ)) = (1rβ€˜π‘Š))
 
21.43.20.11  Univariate polynomials (extension)
 
Theoremply1vr1smo 46180 The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &    1 = (1rβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &   πΊ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜πΊ)    &   π‘‹ = (var1β€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ ( 1 Β· (1 ↑ 𝑋)) = 𝑋)
 
Theoremply1ass23l 46181 Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &    Γ— = (.rβ€˜π‘ƒ)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)))
 
Theoremply1sclrmsm 46182 The ring multiplication of a polynomial with a scalar polynomial is equal to the scalar multiplication of the polynomial with the corresponding scalar. (Contributed by AV, 14-Aug-2019.)
𝐾 = (Baseβ€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &   πΈ = (Baseβ€˜π‘ƒ)    &   π‘‹ = (var1β€˜π‘…)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &    Γ— = (.rβ€˜π‘ƒ)    &   π‘ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘)    &   π΄ = (algScβ€˜π‘ƒ)    β‡’   ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐾 ∧ 𝑍 ∈ 𝐸) β†’ ((π΄β€˜πΉ) Γ— 𝑍) = (𝐹 Β· 𝑍))
 
Theoremcoe1id 46183* Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &   πΌ = (1rβ€˜π‘ƒ)    &    0 = (0gβ€˜π‘…)    &    1 = (1rβ€˜π‘…)    β‡’   (𝑅 ∈ Ring β†’ (coe1β€˜πΌ) = (π‘₯ ∈ β„•0 ↦ if(π‘₯ = 0, 1 , 0 )))
 
Theoremcoe1sclmulval 46184 The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π΄ = (algScβ€˜π‘ƒ)    &   π‘† = ( ·𝑠 β€˜π‘ƒ)    &    βˆ™ = (.rβ€˜π‘ƒ)    &    Β· = (.rβ€˜π‘…)    β‡’   ((𝑅 ∈ Ring ∧ (π‘Œ ∈ 𝐾 ∧ 𝑍 ∈ 𝐡) ∧ 𝑁 ∈ β„•0) β†’ ((coe1β€˜(π‘Œπ‘†π‘))β€˜π‘) = (π‘Œ Β· ((coe1β€˜π‘)β€˜π‘)))
 
Theoremply1mulgsumlem1 46185* Lemma 1 for ply1mulgsum 46189. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (coe1β€˜πΎ)    &   πΆ = (coe1β€˜πΏ)    &   π‘‹ = (var1β€˜π‘…)    &    Γ— = (.rβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &    βˆ— = (.rβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘€)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡 ∧ 𝐿 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ ((π΄β€˜π‘›) = (0gβ€˜π‘…) ∧ (πΆβ€˜π‘›) = (0gβ€˜π‘…))))
 
Theoremply1mulgsumlem2 46186* Lemma 2 for ply1mulgsum 46189. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (coe1β€˜πΎ)    &   πΆ = (coe1β€˜πΏ)    &   π‘‹ = (var1β€˜π‘…)    &    Γ— = (.rβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &    βˆ— = (.rβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘€)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡 ∧ 𝐿 ∈ 𝐡) β†’ βˆƒπ‘  ∈ β„•0 βˆ€π‘› ∈ β„•0 (𝑠 < 𝑛 β†’ (𝑅 Ξ£g (𝑙 ∈ (0...𝑛) ↦ ((π΄β€˜π‘™) βˆ— (πΆβ€˜(𝑛 βˆ’ 𝑙))))) = (0gβ€˜π‘…)))
 
Theoremply1mulgsumlem3 46187* Lemma 3 for ply1mulgsum 46189. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (coe1β€˜πΎ)    &   πΆ = (coe1β€˜πΏ)    &   π‘‹ = (var1β€˜π‘…)    &    Γ— = (.rβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &    βˆ— = (.rβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘€)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡 ∧ 𝐿 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ (𝑅 Ξ£g (𝑙 ∈ (0...π‘˜) ↦ ((π΄β€˜π‘™) βˆ— (πΆβ€˜(π‘˜ βˆ’ 𝑙)))))) finSupp (0gβ€˜π‘…))
 
Theoremply1mulgsumlem4 46188* Lemma 4 for ply1mulgsum 46189. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (coe1β€˜πΎ)    &   πΆ = (coe1β€˜πΏ)    &   π‘‹ = (var1β€˜π‘…)    &    Γ— = (.rβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &    βˆ— = (.rβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘€)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡 ∧ 𝐿 ∈ 𝐡) β†’ (π‘˜ ∈ β„•0 ↦ ((𝑅 Ξ£g (𝑙 ∈ (0...π‘˜) ↦ ((π΄β€˜π‘™) βˆ— (πΆβ€˜(π‘˜ βˆ’ 𝑙))))) Β· (π‘˜ ↑ 𝑋))) finSupp (0gβ€˜π‘ƒ))
 
Theoremply1mulgsum 46189* The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π΄ = (coe1β€˜πΎ)    &   πΆ = (coe1β€˜πΏ)    &   π‘‹ = (var1β€˜π‘…)    &    Γ— = (.rβ€˜π‘ƒ)    &    Β· = ( ·𝑠 β€˜π‘ƒ)    &    βˆ— = (.rβ€˜π‘…)    &   π‘€ = (mulGrpβ€˜π‘ƒ)    &    ↑ = (.gβ€˜π‘€)    β‡’   ((𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐡 ∧ 𝐿 ∈ 𝐡) β†’ (𝐾 Γ— 𝐿) = (𝑃 Ξ£g (π‘˜ ∈ β„•0 ↦ ((𝑅 Ξ£g (𝑙 ∈ (0...π‘˜) ↦ ((π΄β€˜π‘™) βˆ— (πΆβ€˜(π‘˜ βˆ’ 𝑙))))) Β· (π‘˜ ↑ 𝑋)))))
 
Theoremevl1at0 46190 Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1β€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &    0 = (0gβ€˜π‘…)    &   π‘ = (0gβ€˜π‘ƒ)    β‡’   (𝑅 ∈ CRing β†’ ((π‘‚β€˜π‘)β€˜ 0 ) = 0 )
 
Theoremevl1at1 46191 Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1β€˜π‘…)    &   π‘ƒ = (Poly1β€˜π‘…)    &    1 = (1rβ€˜π‘…)    &   πΌ = (1rβ€˜π‘ƒ)    β‡’   (𝑅 ∈ CRing β†’ ((π‘‚β€˜πΌ)β€˜ 1 ) = 1 )
 
21.43.20.12  Univariate polynomials (examples)
 
Theoremlinply1 46192 A term of the form π‘₯ βˆ’ 𝐢 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 25449). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΊ = (𝑋 βˆ’ (π΄β€˜πΆ))    &   (πœ‘ β†’ 𝐢 ∈ 𝐾)    &   (πœ‘ β†’ 𝑅 ∈ Ring)    β‡’   (πœ‘ β†’ 𝐺 ∈ 𝐡)
 
Theoremlineval 46193 A term of the form π‘₯ βˆ’ 𝐢 evaluated for π‘₯ = 𝑉 results in 𝑉 βˆ’ 𝐢 (part of ply1remlem 25449). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1β€˜π‘…)    &   π΅ = (Baseβ€˜π‘ƒ)    &   πΎ = (Baseβ€˜π‘…)    &   π‘‹ = (var1β€˜π‘…)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΊ = (𝑋 βˆ’ (π΄β€˜πΆ))    &   (πœ‘ β†’ 𝐢 ∈ 𝐾)    &   π‘‚ = (eval1β€˜π‘…)    &   (πœ‘ β†’ 𝑅 ∈ CRing)    &   (πœ‘ β†’ 𝑉 ∈ 𝐾)    β‡’   (πœ‘ β†’ ((π‘‚β€˜πΊ)β€˜π‘‰) = (𝑉(-gβ€˜π‘…)𝐢))
 
Theoremlinevalexample 46194 The polynomial π‘₯ βˆ’ 3 over β„€ evaluated for π‘₯ = 5 results in 2. (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1β€˜β„€ring)    &   π΅ = (Baseβ€˜π‘ƒ)    &   π‘‹ = (var1β€˜β„€ring)    &    βˆ’ = (-gβ€˜π‘ƒ)    &   π΄ = (algScβ€˜π‘ƒ)    &   πΊ = (𝑋 βˆ’ (π΄β€˜3))    &   π‘‚ = (eval1β€˜β„€ring)    β‡’   ((π‘‚β€˜(𝑋 βˆ’ (π΄β€˜3)))β€˜5) = 2
 
21.43.21  Linear algebra (extension)
 
21.43.21.1  The subalgebras of diagonal and scalar matrices (extension)

In the following, alternative definitions for diagonal and scalar matrices are provided. These definitions define diagonal and scalar matrices as extensible structures, whereas Definitions df-dmat 21761 and df-scmat 21762 define diagonal and scalar matrices as sets.

 
Syntaxcdmatalt 46195 Alternative notation for the algebra of diagonal matrices.
class DMatALT
 
Syntaxcscmatalt 46196 Alternative notation for the algebra of scalar matrices.
class ScMatALT
 
Definitiondf-dmatalt 46197* Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
DMatALT = (𝑛 ∈ Fin, π‘Ÿ ∈ V ↦ ⦋(𝑛 Mat π‘Ÿ) / π‘Žβ¦Œ(π‘Ž β†Ύs {π‘š ∈ (Baseβ€˜π‘Ž) ∣ βˆ€π‘– ∈ 𝑛 βˆ€π‘— ∈ 𝑛 (𝑖 β‰  𝑗 β†’ (π‘–π‘šπ‘—) = (0gβ€˜π‘Ÿ))}))
 
Definitiondf-scmatalt 46198* Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
ScMatALT = (𝑛 ∈ Fin, π‘Ÿ ∈ V ↦ ⦋(𝑛 Mat π‘Ÿ) / π‘Žβ¦Œ(π‘Ž β†Ύs {π‘š ∈ (Baseβ€˜π‘Ž) ∣ βˆƒπ‘ ∈ (Baseβ€˜π‘Ÿ)βˆ€π‘– ∈ 𝑛 βˆ€π‘— ∈ 𝑛 (π‘–π‘šπ‘—) = if(𝑖 = 𝑗, 𝑐, (0gβ€˜π‘Ÿ))}))
 
TheoremdmatALTval 46199* The algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    0 = (0gβ€˜π‘…)    &   π· = (𝑁 DMatALT 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) β†’ 𝐷 = (𝐴 β†Ύs {π‘š ∈ 𝐡 ∣ βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (𝑖 β‰  𝑗 β†’ (π‘–π‘šπ‘—) = 0 )}))
 
TheoremdmatALTbas 46200* The base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. the set of all 𝑁 x 𝑁 diagonal matrices over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   π΅ = (Baseβ€˜π΄)    &    0 = (0gβ€˜π‘…)    &   π· = (𝑁 DMatALT 𝑅)    β‡’   ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) β†’ (Baseβ€˜π·) = {π‘š ∈ 𝐡 ∣ βˆ€π‘– ∈ 𝑁 βˆ€π‘— ∈ 𝑁 (𝑖 β‰  𝑗 β†’ (π‘–π‘šπ‘—) = 0 )})
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
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