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Theorem List for Metamath Proof Explorer - 42601-42700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfldc 42601* 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.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)

Theoremfldhmsubc 42602* According to df-subc 16678, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16706 and subcss2 16709). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽)))

Theoremrngcrescrhm 42603 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.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Theoremrhmsubclem1 42604 Lemma 1 for rhmsubc 42608. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))

Theoremrhmsubclem2 42605 Lemma 2 for rhmsubc 42608. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Theoremrhmsubclem3 42606* Lemma 3 for rhmsubc 42608. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubclem4 42607* Lemma 4 for rhmsubc 42608. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))

Theoremrhmsubc 42608 According to df-subc 16678, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16706 and subcss2 16709). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

Theoremrhmsubccat 42609 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.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → ((RngCat‘𝑈) ↾cat 𝐻) ∈ Cat)

TheoremsrhmsubcALTVlem1 42610* Lemma 1 for srhmsubcALTV 42612. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈)))

TheoremsrhmsubcALTVlem2 42611* Lemma 2 for srhmsubcALTV 42612. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       ((𝑈𝑉 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌))

TheoremsrhmsubcALTV 42612* According to df-subc 16678, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16706 and subcss2 16709). 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.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

TheoremsringcatALTV 42613* 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.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)

TheoremcrhmsubcALTV 42614* According to df-subc 16678, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16706 and subcss2 16709). 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.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

TheoremcringcatALTV 42615* 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.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)

TheoremdrhmsubcALTV 42616* According to df-subc 16678, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16706 and subcss2 16709). 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.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

TheoremdrngcatALTV 42617* 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.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)

TheoremfldcatALTV 42618* 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.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐹) ∈ Cat)

TheoremfldcALTV 42619* 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.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → (((RingCatALTV‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)

TheoremfldhmsubcALTV 42620* According to df-subc 16678, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16706 and subcss2 16709). 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.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)))

TheoremrngcrescrhmALTV 42621 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.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))

TheoremrhmsubcALTVlem1 42622 Lemma 1 for rhmsubcALTV 42626. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))

TheoremrhmsubcALTVlem2 42623 Lemma 2 for rhmsubcALTV 42626. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

TheoremrhmsubcALTVlem3 42624* Lemma 3 for rhmsubcALTV 42626. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))

TheoremrhmsubcALTVlem4 42625* Lemma 4 for rhmsubcALTV 42626. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))

TheoremrhmsubcALTV 42626 According to df-subc 16678, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16706 and subcss2 16709). 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.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈)))

TheoremrhmsubcALTVcat 42627 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.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → ((RngCatALTV‘𝑈) ↾cat 𝐻) ∈ Cat)

20.36.15  Basic algebraic structures (extension)

20.36.15.1  Auxiliary theorems

Theoremxpprsng 42628 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
((𝐴𝑉𝐵𝑊𝐶𝑈) → ({𝐴, 𝐵} × {𝐶}) = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐶⟩})

Theoremopeliun2xp 42629 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5310. (Contributed by AV, 30-Mar-2019.)
(⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))

Theoremeliunxp2 42630* Membership in a union of Cartesian products over its second component, analogous to eliunxp 5398. (Contributed by AV, 30-Mar-2019.)
(𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))

Theoremmpt2mptx2 42631* 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 mpt2mptx 6897. (Contributed by AV, 30-Mar-2019.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Theoremcbvmpt2x2 42632* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6880 allows 𝐴 to be a function of 𝑦, analogous to cbvmpt2x 6879. (Contributed by AV, 30-Mar-2019.)
𝑧𝐴    &   𝑦𝐷    &   𝑧𝐶    &   𝑤𝐶    &   𝑥𝐸    &   𝑦𝐸    &   (𝑦 = 𝑧𝐴 = 𝐷)    &   ((𝑦 = 𝑧𝑥 = 𝑤) → 𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤𝐷, 𝑧𝐵𝐸)

Theoremdmmpt2ssx2 42633* The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 7384. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})

Theoremmpt2exxg2 42634* 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 mpt2exxg 7393. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)

Theoremovmpt2rdxf 42635* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6932. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)    &   𝑥𝜑    &   𝑦𝜑    &   𝑦𝐴    &   𝑥𝐵    &   𝑥𝑆    &   𝑦𝑆       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2rdx 42636* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6932. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2x2 42637* The value of an operation class abstraction. Variant of ovmpt2ga 6936 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝑦 = 𝐵𝐶 = 𝐿)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐿𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)

Theoremfdmdifeqresdif 42638* 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 = (𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)))       (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Theoremoffvalfv 42639* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))

Theoremofaddmndmap 42640 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 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → (𝐴𝑓 + 𝐵) ∈ (𝑅𝑚 𝑉))

Theoremmapsnop 42641 A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)
𝐹 = {⟨𝑋, 𝑌⟩}       ((𝑋𝑉𝑌𝑅𝑅𝑊) → 𝐹 ∈ (𝑅𝑚 {𝑋}))

Theoremmapprop 42642 An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.)
𝐹 = {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩}       (((𝑋𝑉𝐴𝑅) ∧ (𝑌𝑉𝐵𝑅) ∧ (𝑋𝑌𝑅𝑊)) → 𝐹 ∈ (𝑅𝑚 {𝑋, 𝑌}))

Theoremztprmneprm 42643 A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵𝐴 = 𝐵))

Theorem2t6m3t4e0 42644 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
((2 · 6) − (3 · 4)) = 0

Theoremssnn0ssfz 42645* For any finite subset of 0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 29883. (Contributed by AV, 30-Sep-2019.)
(𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛))

Theoremnn0sumltlt 42646 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.)
((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑐 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑐𝑏 < 𝑐))

20.36.15.2  The binomial coefficient operation (extension)

Theorembcpascm1 42647 Pascal's rule for the binomial coefficient, generalized to all integers 𝐾, shifted down by 1. (Contributed by AV, 8-Sep-2019.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (((𝑁 − 1)C𝐾) + ((𝑁 − 1)C(𝐾 − 1))) = (𝑁C𝐾))

Theoremaltgsumbc 42648* 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) · (0C0)) = (1 · 1) = 1). For a proof using Pascal's rule (bcpascm1 42647) instead of the binomial theorem (binom 14768) , see altgsumbcALT 42649. (Contributed by AV, 13-Sep-2019.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)

TheoremaltgsumbcALT 42649* Alternate proof of altgsumbc 42648, using Pascal's rule (bcpascm1 42647) instead of the binomial theorem (binom 14768). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)

20.36.15.3  The ` ZZ `-module ` ZZ X. ZZ `

Theoremzlmodzxzlmod 42650 The -module ℤ × ℤ 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 = (ℤring freeLMod {0, 1})       (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍))

Theoremzlmodzxzel 42651 An element of the (base set of the) -module ℤ × ℤ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ (Base‘𝑍))

Theoremzlmodzxz0 42652 The 0 of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}        0 = (0g𝑍)

Theoremzlmodzxzscm 42653 The scalar multiplication of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = ( ·𝑠𝑍)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 {⟨0, 𝐵⟩, ⟨1, 𝐶⟩}) = {⟨0, (𝐴 · 𝐵)⟩, ⟨1, (𝐴 · 𝐶)⟩})

Theoremzlmodzxzadd 42654 The addition of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    + = (+g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} + {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩})

Theoremzlmodzxzsubm 42655 The subtraction of the -module ℤ × ℤ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} (+g𝑍)(-1( ·𝑠𝑍){⟨0, 𝐵⟩, ⟨1, 𝐷⟩})))

Theoremzlmodzxzsub 42656 The subtraction of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴𝐵)⟩, ⟨1, (𝐶𝐷)⟩})

20.36.15.4  Ordered group sum operation (extension)

Theoremgsumpr 42657* Group sum of a pair. (Contributed by AV, 6-Dec-2018.) (Proof shortened by AV, 28-Jul-2019.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝑘 = 𝑀𝐴 = 𝐶)    &   (𝑘 = 𝑁𝐴 = 𝐷)       ((𝐺 ∈ CMnd ∧ (𝑀𝑉𝑁𝑊𝑀𝑁) ∧ (𝐶𝐵𝐷𝐵)) → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁} ↦ 𝐴)) = (𝐶 + 𝐷))

Theoremmgpsumunsn 42658* Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
𝑀 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &   (𝜑𝑋 ∈ (Base‘𝑅))    &   (𝑘 = 𝐼𝐴 = 𝑋)       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋))

Theoremmgpsumz 42659* 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 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &    0 = (0g𝑅)    &   (𝑘 = 𝐼𝐴 = 0 )       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = 0 )

Theoremmgpsumn 42660* 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 = (mulGrp‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐼𝑁)    &   ((𝜑𝑘𝑁) → 𝐴 ∈ (Base‘𝑅))    &    1 = (1r𝑅)    &   (𝑘 = 𝐼𝐴 = 1 )       (𝜑 → (𝑀 Σg (𝑘𝑁𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)))

Theoremgsumsplit2f 42661* Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
𝑘𝜑    &   𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑 → (𝑘𝐴𝑋) finSupp 0 )    &   (𝜑 → (𝐶𝐷) = ∅)    &   (𝜑𝐴 = (𝐶𝐷))       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘𝐶𝑋)) + (𝐺 Σg (𝑘𝐷𝑋))))

Theoremgsumdifsndf 42662* Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
𝑘𝑌    &   𝑘𝜑    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑊)    &   (𝜑 → (𝑘𝐴𝑋) finSupp (0g𝐺))    &   ((𝜑𝑘𝐴) → 𝑋𝐵)    &   (𝜑𝑀𝐴)    &   (𝜑𝑌𝐵)    &   ((𝜑𝑘 = 𝑀) → 𝑋 = 𝑌)       (𝜑 → (𝐺 Σg (𝑘𝐴𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))

20.36.15.5  Symmetric groups (extension)

Theoremexple2lt6 42663 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 ∈ ℕ0𝑁 ≤ 2) → (𝑁𝑁) < 6)

Theorempgrple2abl 42664 Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       ((𝐴𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel)

Theorempgrpgt2nabl 42665 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 = (SymGrp‘𝐴)       ((𝐴𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)

20.36.15.6  Divisibility (extension)

Theoreminvginvrid 42666 Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝑈) → ((𝑁𝑌) · ((𝐼‘(𝑁𝑌)) · 𝑋)) = 𝑋)

20.36.15.7  The support of functions (extension)

Theoremrmsupp0 42667* 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 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶 = (0g𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) = ∅)

Theoremdomnmsuppn0 42668* 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 = (Base‘𝑀)       (((𝑀 ∈ Domn ∧ 𝑉𝑋) ∧ (𝐶𝑅𝐶 ≠ (0g𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) = (𝐴 supp (0g𝑀)))

Theoremrmsuppss 42669* 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 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑀)))

Theoremmndpsuppss 42670 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 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉))) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) ⊆ ((𝐴 supp (0g𝑀)) ∪ (𝐵 supp (0g𝑀))))

Theoremscmsuppss 42671* 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 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅𝑚 𝑉)) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ⊆ (𝐴 supp (0g𝑆)))

20.36.15.8  Finitely supported functions (extension)

Theoremrmsuppfi 42672* 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 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ (𝐴 supp (0g𝑀)) ∈ Fin) → ((𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) supp (0g𝑀)) ∈ Fin)

Theoremrmfsupp 42673* 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 = (Base‘𝑀)       (((𝑀 ∈ Ring ∧ 𝑉𝑋𝐶𝑅) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑀)) → (𝑣𝑉 ↦ (𝐶(.r𝑀)(𝐴𝑣))) finSupp (0g𝑀))

Theoremmndpsuppfi 42674 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 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ ((𝐴 supp (0g𝑀)) ∈ Fin ∧ (𝐵 supp (0g𝑀)) ∈ Fin)) → ((𝐴𝑓 (+g𝑀)𝐵) supp (0g𝑀)) ∈ Fin)

Theoremmndpfsupp 42675 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 = (Base‘𝑀)       (((𝑀 ∈ Mnd ∧ 𝑉𝑋) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑀) ∧ 𝐵 finSupp (0g𝑀))) → (𝐴𝑓 (+g𝑀)𝐵) finSupp (0g𝑀))

Theoremscmsuppfi 42676* 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 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ (𝐴 supp (0g𝑆)) ∈ Fin) → ((𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) supp (0g𝑀)) ∈ Fin)

Theoremscmfsupp 42677* 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 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐴 finSupp (0g𝑆)) → (𝑣𝑉 ↦ ((𝐴𝑣)( ·𝑠𝑀)𝑣)) finSupp (0g𝑀))

Theoremsuppmptcfin 42678* The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹 supp 0 ) ∈ Fin)

Theoremmptcfsupp 42679* A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → 𝐹 finSupp 0 )

Theoremfsuppmptdmf 42680* A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
𝑥𝜑    &   𝐹 = (𝑥𝐴𝑌)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝑌𝑉)    &   (𝜑𝑍𝑊)       (𝜑𝐹 finSupp 𝑍)

20.36.15.9  Left modules (extension)

Theoremlmodvsmdi 42681 Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (.g𝑊)    &   𝐸 = (.g𝐹)       ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑁 ∈ ℕ0𝑋𝑉)) → (𝑅 · (𝑁 𝑋)) = ((𝑁𝐸𝑅) · 𝑋))

Theoremgsumlsscl 42682* 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 = (LSubSp‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)       ((𝑀 ∈ LMod ∧ 𝑍𝑆𝑉𝑍) → ((𝐹 ∈ (𝐵𝑚 𝑉) ∧ 𝐹 finSupp (0g𝑅)) → (𝑀 Σg (𝑣𝑉 ↦ ((𝐹𝑣)( ·𝑠𝑀)𝑣))) ∈ 𝑍))

20.36.15.10  Associative algebras (extension)

Theoremascl0 42683 The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑊 ∈ Ring)       (𝜑 → (𝐴‘(0g𝐹)) = (0g𝑊))

Theoremascl1 42684 The scalar 1 embedded into a left module corresponds to the 1 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑊 ∈ Ring)       (𝜑 → (𝐴‘(1r𝐹)) = (1r𝑊))

Theoremassaascl0 42685 The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → (𝐴‘(0g𝐹)) = (0g𝑊))

Theoremassaascl1 42686 The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → (𝐴‘(1r𝐹)) = (1r𝑊))

20.36.15.11  Univariate polynomials (extension)

Theoremply1vr1smo 42687 The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
𝑃 = (Poly1𝑅)    &    1 = (1r𝑅)    &    · = ( ·𝑠𝑃)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝑋 = (var1𝑅)       (𝑅 ∈ Ring → ( 1 · (1 𝑋)) = 𝑋)

Theoremply1ass23l 42688 Associative identity with scalar and ring multiplication for the polynomial ring. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &    × = (.r𝑃)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝑃)       ((𝑅 ∈ Ring ∧ (𝐴𝐾𝑋𝐵𝑌𝐵)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Theoremply1sclrmsm 42689 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 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑃)    &    × = (.r𝑃)    &   𝑁 = (mulGrp‘𝑃)    &    = (.g𝑁)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ Ring ∧ 𝐹𝐾𝑍𝐸) → ((𝐴𝐹) × 𝑍) = (𝐹 · 𝑍))

Theoremcoe1id 42690* Coefficient vector of the unit polynomial. (Contributed by AV, 9-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝐼 = (1r𝑃)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (coe1𝐼) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, 1 , 0 )))

Theoremcoe1sclmulval 42691 The value of the coefficient vector of a polynomial multiplied on the left by a scalar. (Contributed by AV, 14-Aug-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝑆 = ( ·𝑠𝑃)    &    = (.r𝑃)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑌𝐾𝑍𝐵) ∧ 𝑁 ∈ ℕ0) → ((coe1‘(𝑌𝑆𝑍))‘𝑁) = (𝑌 · ((coe1𝑍)‘𝑁)))

Theoremply1mulgsumlem1 42692* Lemma 1 for ply1mulgsum 42696. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))

Theoremply1mulgsumlem2 42693* Lemma 2 for ply1mulgsum 42696. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (𝑅 Σg (𝑙 ∈ (0...𝑛) ↦ ((𝐴𝑙) (𝐶‘(𝑛𝑙))))) = (0g𝑅)))

Theoremply1mulgsumlem3 42694* Lemma 3 for ply1mulgsum 42696. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙)))))) finSupp (0g𝑅))

Theoremply1mulgsumlem4 42695* Lemma 4 for ply1mulgsum 42696. (Contributed by AV, 19-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙))))) · (𝑘 𝑋))) finSupp (0g𝑃))

Theoremply1mulgsum 42696* The product of two polynomials expressed as group sum of scaled monomials. (Contributed by AV, 20-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (coe1𝐾)    &   𝐶 = (coe1𝐿)    &   𝑋 = (var1𝑅)    &    × = (.r𝑃)    &    · = ( ·𝑠𝑃)    &    = (.r𝑅)    &   𝑀 = (mulGrp‘𝑃)    &    = (.g𝑀)       ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (𝐾 × 𝐿) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑅 Σg (𝑙 ∈ (0...𝑘) ↦ ((𝐴𝑙) (𝐶‘(𝑘𝑙))))) · (𝑘 𝑋)))))

Theoremevl1at0 42697 Polynomial evaluation for the 0 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑃)       (𝑅 ∈ CRing → ((𝑂𝑍)‘ 0 ) = 0 )

Theoremevl1at1 42698 Polynomial evaluation for the 1 scalar. (Contributed by AV, 10-Aug-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &    1 = (1r𝑅)    &   𝐼 = (1r𝑃)       (𝑅 ∈ CRing → ((𝑂𝐼)‘ 1 ) = 1 )

20.36.15.12  Univariate polynomials (examples)

Theoremlinply1 42699 A term of the form 𝑥𝐶 is a (univariate) polynomial, also called "linear polynomial". (Part of ply1remlem 24141). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝐶))    &   (𝜑𝐶𝐾)    &   (𝜑𝑅 ∈ Ring)       (𝜑𝐺𝐵)

Theoremlineval 42700 A term of the form 𝑥𝐶 evaluated for 𝑥 = 𝑉 results in 𝑉𝐶 (part of ply1remlem 24141). (Contributed by AV, 3-Jul-2019.)
𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝐴 = (algSc‘𝑃)    &   𝐺 = (𝑋 (𝐴𝐶))    &   (𝜑𝐶𝐾)    &   𝑂 = (eval1𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑉𝐾)       (𝜑 → ((𝑂𝐺)‘𝑉) = (𝑉(-g𝑅)𝐶))

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