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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisinftm 30301* Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    · = (.g𝑊)    &    < = (lt‘𝑊)       ((𝑊𝑉𝑋𝐵𝑌𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))

Theoremisarchi 30302* Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (⋘‘𝑊)       (𝑊𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ¬ 𝑥 < 𝑦))

Theorempnfinf 30303 Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018.)
(𝐴 ∈ ℝ+𝐴(⋘‘ℝ*𝑠)+∞)

Theoremxrnarchi 30304 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
¬ ℝ*𝑠 ∈ Archi

Theoremisarchi2 30305* Alternative way to express the predicate "𝑊 is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    · = (.g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)       ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 (𝑛 · 𝑥))))

Theoremsubmarchi 30306 A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)

Theoremisarchi3 30307* This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)       (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑦𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))

Theoremarchirng 30308* Property of Archimedean ordered groups, framing positive 𝑌 between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    = (le‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑0 < 𝑌)       (𝜑 → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))

Theoremarchirngz 30309* Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    = (le‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑 → (oppg𝑊) ∈ oGrp)       (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌𝑌 ((𝑛 + 1) · 𝑋)))

Theoremarchiexdiv 30310* In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)       (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋𝐵𝑌𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋))

Theoremarchiabllem1a 30311* Lemma for archiabl 30318: In case an archimedean group 𝑊 admits a smallest positive element 𝑈, then any positive element 𝑋 of 𝑊 can be written as (𝑛 · 𝑈) with 𝑛 ∈ ℕ. Since the reciprocal holds for negative elements, 𝑊 is then isomorphic to . (Contributed by Thierry Arnoux, 12-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)    &   (𝜑𝑋𝐵)    &   (𝜑0 < 𝑋)       (𝜑 → ∃𝑛 ∈ ℕ 𝑋 = (𝑛 · 𝑈))

Theoremarchiabllem1b 30312* Lemma for archiabl 30318. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)       ((𝜑𝑦𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))

Theoremarchiabllem1 30313* Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &   (𝜑𝑈𝐵)    &   (𝜑0 < 𝑈)    &   ((𝜑𝑥𝐵0 < 𝑥) → 𝑈 𝑥)       (𝜑𝑊 ∈ Abel)

Theoremarchiabllem2a 30314* Lemma for archiabl 30318, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))    &   (𝜑𝑋𝐵)    &   (𝜑0 < 𝑋)       (𝜑 → ∃𝑐𝐵 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) 𝑋))

Theoremarchiabllem2c 30315* Lemma for archiabl 30318. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ¬ (𝑋 + 𝑌) < (𝑌 + 𝑋))

Theoremarchiabllem2b 30316* Lemma for archiabl 30318. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremarchiabllem2 30317* Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (le‘𝑊)    &    < = (lt‘𝑊)    &    · = (.g𝑊)    &   (𝜑𝑊 ∈ oGrp)    &   (𝜑𝑊 ∈ Archi)    &    + = (+g𝑊)    &   (𝜑 → (oppg𝑊) ∈ oGrp)    &   ((𝜑𝑎𝐵0 < 𝑎) → ∃𝑏𝐵 ( 0 < 𝑏𝑏 < 𝑎))       (𝜑𝑊 ∈ Abel)

Theoremarchiabl 30318 Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

20.3.9.5  Semiring left modules

Syntaxcslmd 30319 Extend class notation with class of all semimodules.
class SLMod

Definitiondf-slmd 30320* Define the class of all (left) modules over semirings, i.e. semimodules, which are generalizations of left modules. A semimodule is a commutative monoid (=vectors) together with a semiring (=scalars) and a left scalar product connecting them. (0[,]+∞) for example is not a full fledged left module, but is a semimodule. Definition of [Golan] p. 149. (Contributed by Thierry Arnoux, 21-Mar-2018.)
SLMod = {𝑔 ∈ CMnd ∣ [(Base‘𝑔) / 𝑣][(+g𝑔) / 𝑎][( ·𝑠𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞𝑘𝑟𝑘𝑥𝑣𝑤𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r𝑓)𝑠𝑤) = 𝑤 ∧ ((0g𝑓)𝑠𝑤) = (0g𝑔))))}

Theoremisslmd 30321* The predicate "is a semimodule". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   𝑂 = (0g𝐹)       (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤 ∧ (𝑂 · 𝑤) = 0 ))))

Theoremslmdlema 30322 Lemma for properties of a semimodule. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   𝑂 = (0g𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑅 · 𝑌) ∈ 𝑉 ∧ (𝑅 · (𝑌 + 𝑋)) = ((𝑅 · 𝑌) + (𝑅 · 𝑋)) ∧ ((𝑄 𝑅) · 𝑌) = ((𝑄 · 𝑌) + (𝑅 · 𝑌))) ∧ (((𝑄 × 𝑅) · 𝑌) = (𝑄 · (𝑅 · 𝑌)) ∧ ( 1 · 𝑌) = 𝑌 ∧ (𝑂 · 𝑌) = 0 )))

Theoremlmodslmd 30323 Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ LMod → 𝑊 ∈ SLMod)

Theoremslmdcmn 30324 A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ CMnd)

Theoremslmdmnd 30325 A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ Mnd)

Theoremslmdsrg 30326 The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ SLMod → 𝐹 ∈ SRing)

Theoremslmdbn0 30327 The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑊)       (𝑊 ∈ SLMod → 𝐵 ≠ ∅)

Theoremslmdacl 30328 Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)

Theoremslmdmcl 30329 Closure of ring multiplication for a semimodule. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = (.r𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)

Theoremslmdsn0 30330 The set of scalars in a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       (𝑊 ∈ SLMod → 𝐵 ≠ ∅)

Theoremslmdvacl 30331 Closure of vector addition for a semiring left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Theoremslmdass 30332 Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ SLMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theoremslmdvscl 30333 Closure of scalar product for a semiring left module. (hvmulcl 28446 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ SLMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)

Theoremslmdvsdi 30334 Distributive law for scalar product. (ax-hvdistr1 28441 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ SLMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))

Theoremslmdvsdir 30335 Distributive law for scalar product. (ax-hvdistr1 28441 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))

Theoremslmdvsass 30336 Associative law for scalar product. (ax-hvmulass 28440 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝐹)       ((𝑊 ∈ SLMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))

Theoremslmd0cl 30337 The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)       (𝑊 ∈ SLMod → 0𝐾)

Theoremslmd1cl 30338 The ring unit in a semiring left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    1 = (1r𝐹)       (𝑊 ∈ SLMod → 1𝐾)

Theoremslmdvs1 30339 Scalar product with ring unit. (ax-hvmulid 28439 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 1 · 𝑋) = 𝑋)

Theoremslmd0vcl 30340 The zero vector is a vector. (ax-hv0cl 28436 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ SLMod → 0𝑉)

Theoremslmd0vlid 30341 Left identity law for the zero vector. (hvaddid2 28456 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)

Theoremslmd0vrid 30342 Right identity law for the zero vector. (ax-hvaddid 28437 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)

Theoremslmd0vs 30343 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28443 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )

Theoremslmdvs0 30344 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28457 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ SLMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )

20.3.9.6  Finitely supported group sums - misc additions

Theoremgsumle 30345 A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐹𝑟 𝐺)       (𝜑 → (𝑀 Σg 𝐹) (𝑀 Σg 𝐺))

Theoremgsummpt2co 30346* Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐸𝑉)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥𝐴) → 𝐷𝐸)    &   𝐹 = (𝑥𝐴𝐷)       (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦𝐸 ↦ (𝑊 Σg (𝑥 ∈ (𝐹 “ {𝑦}) ↦ 𝐶)))))

Theoremgsummpt2d 30347* Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 18761. (Contributed by Thierry Arnoux, 27-Apr-2020.)
𝑧𝐶    &   𝑦𝜑    &   𝐵 = (Base‘𝑊)    &   (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)    &   (𝜑 → Rel 𝐴)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ CMnd)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)       (𝜑 → (𝑊 Σg (𝑥𝐴𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))

Theoremgsumvsca1 30348* Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.)
𝐵 = (Base‘𝑊)    &   𝐺 = (Scalar‘𝑊)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑃𝐾)    &   ((𝜑𝑘𝐴) → 𝑄𝐵)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘𝐴𝑄))))

Theoremgsumvsca2 30349* Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) (Proof shortened by AV, 12-Dec-2019.)
𝐵 = (Base‘𝑊)    &   𝐺 = (Scalar‘𝑊)    &    0 = (0g𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   (𝜑𝐾 ⊆ (Base‘𝐺))    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝑊 ∈ SLMod)    &   (𝜑𝑄𝐵)    &   ((𝜑𝑘𝐴) → 𝑃𝐾)       (𝜑 → (𝑊 Σg (𝑘𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘𝐴𝑃)) · 𝑄))

Theoremgsummptres 30350* Extend a finite group sum by padding outside with zeroes. Proof generated using OpenAI's proof assistant. (Contributed by Thierry Arnoux, 11-Jul-2020.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑥 ∈ (𝐴𝐷)) → 𝐶 = 0 )       (𝜑 → (𝐺 Σg (𝑥𝐴𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴𝐷) ↦ 𝐶)))

Theoremxrge0tsmsd 30351* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   (𝜑𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑠))), ℝ*, < ))       (𝜑 → (𝐺 tsums 𝐹) = {𝑆})

Theoremxrge0tsmsbi 30352 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))       (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = (𝐺 tsums 𝐹)))

Theoremxrge0tsmseq 30353 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
𝐺 = (ℝ*𝑠s (0[,]+∞))    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴⟶(0[,]+∞))    &   (𝜑𝐶 ∈ (𝐺 tsums 𝐹))       (𝜑𝐶 = (𝐺 tsums 𝐹))

Theoremrngurd 30354* Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑1𝐵)    &   ((𝜑𝑥𝐵) → ( 1 · 𝑥) = 𝑥)    &   ((𝜑𝑥𝐵) → (𝑥 · 1 ) = 𝑥)       (𝜑1 = (1r𝑅))

Theoremress1r 30355 1r is unaffected by restriction. This is a bit more generic than subrg1 19186. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑆 = (𝑅s 𝐴)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 1𝐴𝐴𝐵) → 1 = (1r𝑆))

Theoremdvrdir 30356 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝑈)) → ((𝑋 + 𝑌) / 𝑍) = ((𝑋 / 𝑍) + (𝑌 / 𝑍)))

Theoremrdivmuldivd 30357 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    + = (+g𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝐵)    &   (𝜑𝑊𝑈)       (𝜑 → ((𝑋 / 𝑌) · (𝑍 / 𝑊)) = ((𝑋 · 𝑍) / (𝑌 · 𝑊)))

Theoremringinvval 30358* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
𝐵 = (Base‘𝑅)    &    = (.r𝑅)    &    1 = (1r𝑅)    &   𝑁 = (invr𝑅)    &   𝑈 = (Unit‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) = (𝑦𝑈 (𝑦 𝑋) = 1 ))

Theoremdvrcan5 30359 Cancellation law for common factor in ratio. (divcan5 11079 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝑈𝑍𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌))

Theoremsubrgchr 30360 If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
(𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅s 𝐴)) = (chr‘𝑅))

20.3.9.8  Ordered rings and fields

Syntaxcorng 30361 Extend class notation with the class of all ordered rings.
class oRing

Syntaxcofld 30362 Extend class notation with the class of all ordered fields.
class oField

Definitiondf-orng 30363* Define class of all ordered rings. An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g𝑟) / 𝑧][(.r𝑟) / 𝑡][(le‘𝑟) / 𝑙]𝑎𝑣𝑏𝑣 ((𝑧𝑙𝑎𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}

Definitiondf-ofld 30364 Define class of all ordered fields. An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
oField = (Field ∩ oRing)

Theoremisorng 30365* An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    = (le‘𝑅)       (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎𝐵𝑏𝐵 (( 0 𝑎0 𝑏) → 0 (𝑎 · 𝑏))))

Theoremorngring 30366 An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ Ring)

Theoremorngogrp 30367 An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝑅 ∈ oRing → 𝑅 ∈ oGrp)

Theoremisofld 30368 An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing))

Theoremorngmul 30369 In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝑅)    &    = (le‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ oRing ∧ (𝑋𝐵0 𝑋) ∧ (𝑌𝐵0 𝑌)) → 0 (𝑋 · 𝑌))

Theoremorngsqr 30370 In an ordered ring, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝑅)    &    = (le‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ oRing ∧ 𝑋𝐵) → 0 (𝑋 · 𝑋))

Theoremornglmulle 30371 In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (le‘𝑅)    &   (𝜑𝑋 𝑌)    &   (𝜑0 𝑍)       (𝜑 → (𝑍 · 𝑋) (𝑍 · 𝑌))

Theoremorngrmulle 30372 In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    = (le‘𝑅)    &   (𝜑𝑋 𝑌)    &   (𝜑0 𝑍)       (𝜑 → (𝑋 · 𝑍) (𝑌 · 𝑍))

Theoremornglmullt 30373 In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋 < 𝑌)    &   (𝜑0 < 𝑍)       (𝜑 → (𝑍 · 𝑋) < (𝑍 · 𝑌))

Theoremorngrmullt 30374 In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋 < 𝑌)    &   (𝜑0 < 𝑍)       (𝜑 → (𝑋 · 𝑍) < (𝑌 · 𝑍))

Theoremorngmullt 30375 In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   (𝜑𝑅 ∈ oRing)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑0 < 𝑋)    &   (𝜑0 < 𝑌)       (𝜑0 < (𝑋 · 𝑌))

Theoremofldfld 30376 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Field)

Theoremofldtos 30377 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
(𝐹 ∈ oField → 𝐹 ∈ Toset)

Theoremorng0le1 30378 In an ordered ring, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    = (le‘𝐹)       (𝐹 ∈ oRing → 0 1 )

Theoremofldlt1 30379 In an ordered field, the ring unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
0 = (0g𝐹)    &    1 = (1r𝐹)    &    < = (lt‘𝐹)       (𝐹 ∈ oField → 0 < 1 )

Theoremofldchr 30380 The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.) (Proof shortened by AV, 6-Oct-2020.)
(𝐹 ∈ oField → (chr‘𝐹) = 0)

Theoremsuborng 30381 Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝑅 ∈ oRing ∧ (𝑅s 𝐴) ∈ Ring) → (𝑅s 𝐴) ∈ oRing)

Theoremsubofld 30382 Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
((𝐹 ∈ oField ∧ (𝐹s 𝐴) ∈ Field) → (𝐹s 𝐴) ∈ oField)

Theoremisarchiofld 30383* Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.)
𝐵 = (Base‘𝑊)    &   𝐻 = (ℤRHom‘𝑊)    &    < = (lt‘𝑊)       (𝑊 ∈ oField → (𝑊 ∈ Archi ↔ ∀𝑥𝐵𝑛 ∈ ℕ 𝑥 < (𝐻𝑛)))

20.3.9.9  Ring homomorphisms - misc additions

Theoremrhmdvdsr 30384 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝑋 = (Base‘𝑅)    &    = (∥r𝑅)    &    / = (∥r𝑆)       (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴𝑋𝐵𝑋) ∧ 𝐴 𝐵) → (𝐹𝐴) / (𝐹𝐵))

Theoremrhmopp 30385 A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr𝑅) RingHom (oppr𝑆)))

Theoremelrhmunit 30386 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹𝐴) ∈ (Unit‘𝑆))

Theoremrhmdvd 30387 A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
𝑈 = (Unit‘𝑆)    &   𝑋 = (Base‘𝑅)    &    / = (/r𝑆)    &    · = (.r𝑅)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋) ∧ ((𝐹𝐵) ∈ 𝑈 ∧ (𝐹𝐶) ∈ 𝑈)) → ((𝐹𝐴) / (𝐹𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶))))

Theoremrhmunitinv 30388 Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr𝑅)‘𝐴)) = ((invr𝑆)‘(𝐹𝐴)))

Theoremkerunit 30389 If a unit element lies in the kernel of a ring homomorphism, then 0 = 1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑆)    &    1 = (1r𝑆)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑈 ∩ (𝐹 “ { 0 })) ≠ ∅) → 1 = 0 )

20.3.9.10  Scalar restriction operation

Syntaxcresv 30390 Extend class notation with the scalar restriction operation.
class v

Definitiondf-resv 30391* Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.)
v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)⟩)))

Theoremreldmresv 30392 The scalar restriction is a proper operator, so it can be used with ovprc1 6962. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Rel dom ↾v

Theoremresvval 30393 Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝑊𝑋𝐴𝑌) → 𝑅 = if(𝐵𝐴, 𝑊, (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩)))

Theoremresvid2 30394 General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = 𝑊)

Theoremresvval2 30395 Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       ((¬ 𝐵𝐴𝑊𝑋𝐴𝑌) → 𝑅 = (𝑊 sSet ⟨(Scalar‘ndx), (𝐹s 𝐴)⟩))

Theoremresvsca 30396 Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)       (𝐴𝑉 → (𝐹s 𝐴) = (Scalar‘𝑅))

Theoremresvlem 30397 Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝑅 = (𝑊v 𝐴)    &   𝐶 = (𝐸𝑊)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 5       (𝐴𝑉𝐶 = (𝐸𝑅))

Theoremresvbas 30398 Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 = (Base‘𝐻))

Theoremresvplusg 30399 +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    + = (+g𝐺)       (𝐴𝑉+ = (+g𝐻))

Theoremresvvsca 30400 ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.)
𝐻 = (𝐺v 𝐴)    &    · = ( ·𝑠𝐺)       (𝐴𝑉· = ( ·𝑠𝐻))

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