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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremtosglbOLD 28801 Same theorem as toslub 28798, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.) Obsolete version of tosglb 28800 as of 28-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝐾)    &    < = (lt‘𝐾)    &   (𝜑𝐾 ∈ Toset)    &   (𝜑𝐴𝐵)       (𝜑 → ((glb‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
 
20.3.7.4  Complete lattices
 
Theoremclatp0cl 28802 The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
𝐵 = (Base‘𝑊)    &    0 = (0.‘𝑊)       (𝑊 ∈ CLat → 0𝐵)
 
Theoremclatp1cl 28803 The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.)
𝐵 = (Base‘𝑊)    &    1 = (1.‘𝑊)       (𝑊 ∈ CLat → 1𝐵)
 
20.3.7.5  Extended reals Structure - misc additions
 
Axiomax-xrssca 28804 Assume the scalar component of the extended real structure is the field of the real numbers (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
fld = (Scalar‘ℝ*𝑠)
 
Axiomax-xrsvsca 28805 Assume the scalar product of the extended real structure is the extended real number multiplication operation (this has to be defined in the main body of set.mm). (Contributed by Thierry Arnoux, 22-Oct-2017.)
·e = ( ·𝑠 ‘ℝ*𝑠)
 
Theoremxrs0 28806 The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 11822 and df-xrs 15873), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.)
0 = (0g‘ℝ*𝑠)
 
Theoremxrslt 28807 The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.)
< = (lt‘ℝ*𝑠)
 
Theoremxrsinvgval 28808 The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 11822 and df-xrs 15873), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.)
(𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵)
 
Theoremxrsmulgzz 28809 The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℝ*) → (𝐴(.g‘ℝ*𝑠)𝐵) = (𝐴 ·e 𝐵))
 
Theoremxrstos 28810 The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
*𝑠 ∈ Toset
 
Theoremxrsclat 28811 The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)
*𝑠 ∈ CLat
 
Theoremxrsp0 28812 The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Proof shortened by AV, 28-Sep-2020.)
-∞ = (0.‘ℝ*𝑠)
 
Theoremxrsp1 28813 The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
+∞ = (1.‘ℝ*𝑠)
 
Theoremressmulgnn 28814 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐴 ⊆ (Base‘𝐺)    &    = (.g𝐺)    &   𝐼 = (invg𝐺)       ((𝑁 ∈ ℕ ∧ 𝑋𝐴) → (𝑁(.g𝐻)𝑋) = (𝑁 𝑋))
 
Theoremressmulgnn0 28815 Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
𝐻 = (𝐺s 𝐴)    &   𝐴 ⊆ (Base‘𝐺)    &    = (.g𝐺)    &   𝐼 = (invg𝐺)    &   (0g𝐺) = (0g𝐻)       ((𝑁 ∈ ℕ0𝑋𝐴) → (𝑁(.g𝐻)𝑋) = (𝑁 𝑋))
 
20.3.7.6  The extended nonnegative real numbers commutative monoid
 
Theoremxrge0base 28816 The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
(0[,]+∞) = (Base‘(ℝ*𝑠s (0[,]+∞)))
 
Theoremxrge00 28817 The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
0 = (0g‘(ℝ*𝑠s (0[,]+∞)))
 
Theoremxrge0plusg 28818 The additive law of the extended nonnegative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
+𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))
 
Theoremxrge0le 28819 The lower-or-equal relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.)
≤ = (le‘(ℝ*𝑠s (0[,]+∞)))
 
Theoremxrge0mulgnn0 28820 The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
((𝐴 ∈ ℕ0𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵))
 
Theoremxrge0addass 28821 Associativity of extended nonnegative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) +𝑒 𝐶) = (𝐴 +𝑒 (𝐵 +𝑒 𝐶)))
 
Theoremxrge0addgt0 28822 The sum of nonnegative and positive numbers is positive. See addgtge0 10269. (Contributed by Thierry Arnoux, 6-Jul-2017.)
(((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵))
 
Theoremxrge0adddir 28823 Right-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → ((𝐴 +𝑒 𝐵) ·e 𝐶) = ((𝐴 ·e 𝐶) +𝑒 (𝐵 ·e 𝐶)))
 
Theoremxrge0adddi 28824 Left-distributivity of extended nonnegative real multiplication over addition. (Contributed by Thierry Arnoux, 6-Sep-2018.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝐶 ·e (𝐴 +𝑒 𝐵)) = ((𝐶 ·e 𝐴) +𝑒 (𝐶 ·e 𝐵)))
 
Theoremxrge0npcan 28825 Extended nonnegative real version of npcan 10045. (Contributed by Thierry Arnoux, 9-Jun-2017.)
((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴)
 
Theoremfsumrp0cl 28826* Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
(𝜑𝐴 ∈ Fin)    &   ((𝜑𝑘𝐴) → 𝐵 ∈ (0[,)+∞))       (𝜑 → Σ𝑘𝐴 𝐵 ∈ (0[,)+∞))
 
20.3.8  Algebra
 
20.3.8.1  Monoids Homomorphisms
 
Theoremabliso 28827 The image of an Abelian group by a group isomorphism is also Abelian. (Contributed by Thierry Arnoux, 8-Mar-2018.)
((𝑀 ∈ Abel ∧ 𝐹 ∈ (𝑀 GrpIso 𝑁)) → 𝑁 ∈ Abel)
 
20.3.8.2  Ordered monoids and groups
 
Syntaxcomnd 28828 Extend class notation with the class of all right ordered monoids.
class oMnd
 
Syntaxcogrp 28829 Extend class notation with the class of all right ordered groups.
class oGrp
 
Definitiondf-omnd 28830* Define class of all right ordered monoids. An ordered monoid is a monoid with a total ordering compatible with its operation. It is possible to use this definition also for left ordered monoids, by applying it to (oppg𝑀). (Contributed by Thierry Arnoux, 13-Mar-2018.)
oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+g𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎𝑣𝑏𝑣𝑐𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))}
 
Definitiondf-ogrp 28831 Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 13-Mar-2018.)
oGrp = (Grp ∩ oMnd)
 
Theoremisomnd 28832* A (left) ordered monoid is a monoid with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (le‘𝑀)       (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 (𝑎 𝑏 → (𝑎 + 𝑐) (𝑏 + 𝑐))))
 
Theoremisogrp 28833 A (left) ordered group is a group with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.)
(𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd))
 
Theoremogrpgrp 28834 An left ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
(𝐺 ∈ oGrp → 𝐺 ∈ Grp)
 
Theoremomndmnd 28835 A left ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
 
Theoremomndtos 28836 A left ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
(𝑀 ∈ oMnd → 𝑀 ∈ Toset)
 
Theoremomndadd 28837 In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 + 𝑍) (𝑌 + 𝑍))
 
Theoremomndaddr 28838 In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)       (((oppg𝑀) ∈ oMnd ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑍 + 𝑋) (𝑍 + 𝑌))
 
Theoremomndadd2d 28839 In a commutative left ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑𝑀 ∈ CMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))
 
Theoremomndadd2rd 28840 In a left- and right- ordered monoid, the ordering is compatible with monoid addition. Double addition version. (Contributed by Thierry Arnoux, 2-May-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑊𝐵)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 𝑍)    &   (𝜑𝑌 𝑊)    &   (𝜑 → (oppg𝑀) ∈ oMnd)       (𝜑 → (𝑋 + 𝑌) (𝑍 + 𝑊))
 
Theoremsubmomnd 28841 A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
((𝑀 ∈ oMnd ∧ (𝑀s 𝐴) ∈ Mnd) → (𝑀s 𝐴) ∈ oMnd)
 
Theoremxrge0omnd 28842 The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.)
(ℝ*𝑠s (0[,]+∞)) ∈ oMnd
 
Theoremomndmul2 28843 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
 
Theoremomndmul3 28844 In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &    0 = (0g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑃 ∈ ℕ0)    &   (𝜑𝑁𝑃)    &   (𝜑𝑋𝐵)    &   (𝜑0 𝑋)       (𝜑 → (𝑁 · 𝑋) (𝑃 · 𝑋))
 
Theoremomndmul 28845 In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑀)    &    = (le‘𝑀)    &    · = (.g𝑀)    &   (𝜑𝑀 ∈ oMnd)    &   (𝜑𝑀 ∈ CMnd)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 𝑌)       (𝜑 → (𝑁 · 𝑋) (𝑁 · 𝑌))
 
TheoremogrpinvOLD 28846 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 30-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵0 𝑋) → (𝐼𝑋) 0 )
 
Theoremogrpinv0le 28847 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 𝑋 ↔ (𝐼𝑋) 0 ))
 
Theoremogrpsub 28848 In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝐺)    &    = (le‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 𝑌) → (𝑋 𝑍) (𝑌 𝑍))
 
Theoremogrpaddlt 28849 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
 
Theoremogrpaddltbi 28850 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
 
Theoremogrpaddltrd 28851 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑 → (oppg𝐺) ∈ oGrp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑋 < 𝑌)       (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌))
 
Theoremogrpaddltrbid 28852 In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑 → (oppg𝐺) ∈ oGrp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌)))
 
Theoremogrpsublt 28853 In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ oGrp ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ 𝑋 < 𝑌) → (𝑋 𝑍) < (𝑌 𝑍))
 
Theoremogrpinv0lt 28854 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &   𝐼 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ oGrp ∧ 𝑋𝐵) → ( 0 < 𝑋 ↔ (𝐼𝑋) < 0 ))
 
Theoremogrpinvlt 28855 In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
𝐵 = (Base‘𝐺)    &    < = (lt‘𝐺)    &   𝐼 = (invg𝐺)       (((𝐺 ∈ oGrp ∧ (oppg𝐺) ∈ oGrp) ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝐼𝑌) < (𝐼𝑋)))
 
20.3.8.3  Signum in an ordered monoid
 
Syntaxcsgns 28856 Extend class notation to include the Signum function.
class sgns
 
Definitiondf-sgns 28857* Signum function for a structure. See also df-sgn 13538 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g𝑟), 0, if((0g𝑟)(lt‘𝑟)𝑥, 1, -1))))
 
Theoremsgnsv 28858* The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
 
Theoremsgnsval 28859 The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       ((𝑅𝑉𝑋𝐵) → (𝑆𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
 
Theoremsgnsf 28860 The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    < = (lt‘𝑅)    &   𝑆 = (sgns𝑅)       (𝑅𝑉𝑆:𝐵⟶{-1, 0, 1})
 
20.3.8.4  The Archimedean property for generic ordered algebraic structures
 
Syntaxcinftm 28861 Class notation for the infinitesimal relation.
class
 
Syntaxcarchi 28862 Class notation for the Archimedean property.
class Archi
 
Definitiondf-inftm 28863* Define the relation "𝑥 is infinitesimal with respect to 𝑦 " for a structure 𝑤. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⋘ = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ ((0g𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g𝑤)𝑥)(lt‘𝑤)𝑦))})
 
Definitiondf-archi 28864 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
 
Theoreminftmrel 28865 The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
𝐵 = (Base‘𝑊)       (𝑊𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
 
Theoremisinftm 28866* 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 28867* 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 28868 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 28869 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
¬ ℝ*𝑠 ∈ Archi
 
Theoremisarchi2 28870* 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 28871 A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
(((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊s 𝐴) ∈ Archi)
 
Theoremisarchi3 28872* 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 28873* 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 28874* 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 28875* 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 28876* Lemma for archiabl 28883: 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 28877* Lemma for archiabl 28883. (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 28878* 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 28879* Lemma for archiabl 28883, 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 28880* Lemma for archiabl 28883. (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 28881* Lemma for archiabl 28883. (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 28882* 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 28883 Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
 
20.3.8.5  Semiring left modules
 
Syntaxcslmd 28884 Extend class notation with class of all semimodules.
class SLMod
 
Definitiondf-slmd 28885* 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 28886* 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 28887 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 28888 Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ LMod → 𝑊 ∈ SLMod)
 
Theoremslmdcmn 28889 A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
 
Theoremslmdmnd 28890 A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
(𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
 
Theoremslmdsrg 28891 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 28892 The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.)
𝐵 = (Base‘𝑊)       (𝑊 ∈ SLMod → 𝐵 ≠ ∅)
 
Theoremslmdacl 28893 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 28894 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 28895 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 28896 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 28897 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 28898 Closure of scalar product for a semiring left module. (hvmulcl 27046 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 28899 Distributive law for scalar product. (ax-hvdistr1 27041 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 28900 Distributive law for scalar product. (ax-hvdistr1 27041 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)))
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