P. 298 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isomnd 29701*	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)    &   ⊢ + = (+g‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    ⇒   ⊢ (𝑀 ∈ oMnd ↔ (𝑀 ∈ Mnd ∧ 𝑀 ∈ Toset ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 (𝑎 ≤ 𝑏 → (𝑎 + 𝑐) ≤ (𝑏 + 𝑐))))
Theorem	isogrp 29702	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))
Theorem	ogrpgrp 29703	An left ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.)
⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp)
Theorem	omndmnd 29704	A left ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
Theorem	omndtos 29705	A left ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.)
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
Theorem	omndadd 29706	In an ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 + 𝑍) ≤ (𝑌 + 𝑍))
Theorem	omndaddr 29707	In a right ordered monoid, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ (((oppg‘𝑀) ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑍 + 𝑋) ≤ (𝑍 + 𝑌))
Theorem	omndadd2d 29708	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)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑍)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑊)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))
Theorem	omndadd2rd 29709	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)    &   ⊢ + = (+g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑊 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑍)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑊)    &   ⊢ (𝜑 → (oppg‘𝑀) ∈ oMnd)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ≤ (𝑍 + 𝑊))
Theorem	submomnd 29710	A submonoid of an ordered monoid is also ordered. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ ((𝑀 ∈ oMnd ∧ (𝑀 ↾s 𝐴) ∈ Mnd) → (𝑀 ↾s 𝐴) ∈ oMnd)
Theorem	xrge0omnd 29711	The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.)
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd
Theorem	omndmul2 29712	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)    &   ⊢ · = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))
Theorem	omndmul3 29713	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)    &   ⊢ · = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑃 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ≤ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 0 ≤ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑃 · 𝑋))
Theorem	omndmul 29714	In a commutative ordered monoid, the ordering is compatible with group power. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ≤ = (le‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    ⇒   ⊢ (𝜑 → (𝑁 · 𝑋) ≤ (𝑁 · 𝑌))
Theorem	ogrpinvOLD 29715	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 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 )
Theorem	ogrpinv0le 29716	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ≤ = (le‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 ≤ 𝑋 ↔ (𝐼‘𝑋) ≤ 0 ))
Theorem	ogrpsub 29717	In an ordered group, the ordering is compatible with group subtraction. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ≤ = (le‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 − 𝑍) ≤ (𝑌 − 𝑍))
Theorem	ogrpaddlt 29718	In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 + 𝑍) < (𝑌 + 𝑍))
Theorem	ogrpaddltbi 29719	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 + 𝑍) < (𝑌 + 𝑍)))
Theorem	ogrpaddltrd 29720	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 < 𝑌)    ⇒   ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌))
Theorem	ogrpaddltrbid 29721	In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 4-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 < 𝑌 ↔ (𝑍 + 𝑋) < (𝑍 + 𝑌)))
Theorem	ogrpsublt 29722	In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (𝑋 − 𝑍) < (𝑌 − 𝑍))
Theorem	ogrpinv0lt 29723	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵) → ( 0 < 𝑋 ↔ (𝐼‘𝑋) < 0 ))
Theorem	ogrpinvlt 29724	In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ < = (lt‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ oGrp ∧ (oppg‘𝐺) ∈ oGrp) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝐼‘𝑌) < (𝐼‘𝑋)))
20.3.9.3  Signum in an ordered monoid
Syntax	csgns 29725	Extend class notation to include the Signum function.
class sgns
Definition	df-sgns 29726*	Signum function for a structure. See also df-sgn 13827 for the version for extended reals. (Contributed by Thierry Arnoux, 10-Sep-2018.)
⊢ sgns = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟) ↦ if(𝑥 = (0g‘𝑟), 0, if((0g‘𝑟)(lt‘𝑟)𝑥, 1, -1))))
Theorem	sgnsv 29727*	The sign mapping. (Contributed by Thierry Arnoux, 9-Sep-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ < = (lt‘𝑅)    &   ⊢ 𝑆 = (sgns‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, if( 0 < 𝑥, 1, -1))))
Theorem	sgnsval 29728	The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ < = (lt‘𝑅)    &   ⊢ 𝑆 = (sgns‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) → (𝑆‘𝑋) = if(𝑋 = 0 , 0, if( 0 < 𝑋, 1, -1)))
Theorem	sgnsf 29729	The sign function. (Contributed by Thierry Arnoux, 9-Sep-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ < = (lt‘𝑅)    &   ⊢ 𝑆 = (sgns‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝑆:𝐵⟶{-1, 0, 1})
20.3.9.4  The Archimedean property for generic ordered algebraic structures
Syntax	cinftm 29730	Class notation for the infinitesimal relation.
class ⋘
Syntax	carchi 29731	Class notation for the Archimedean property.
class Archi
Definition	df-inftm 29732*	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)) ∧ ((0g‘𝑤)(lt‘𝑤)𝑥 ∧ ∀𝑛 ∈ ℕ (𝑛(.g‘𝑤)𝑥)(lt‘𝑤)𝑦))})
Definition	df-archi 29733	A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅}
Theorem	inftmrel 29734	The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵))
Theorem	isinftm 29735*	Express 𝑥 is infinitesimal with respect to 𝑦 for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ < = (lt‘𝑊)    ⇒   ⊢ ((𝑊 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(⋘‘𝑊)𝑌 ↔ ( 0 < 𝑋 ∧ ∀𝑛 ∈ ℕ (𝑛 · 𝑋) < 𝑌)))
Theorem	isarchi 29736*	Express the predicate "𝑊 is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (⋘‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦))
Theorem	pnfinf 29737	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)+∞)
Theorem	xrnarchi 29738	The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
⊢ ¬ ℝ*𝑠 ∈ Archi
Theorem	isarchi2 29739*	Alternative way to express the predicate "𝑊 is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    ⇒   ⊢ ((𝑊 ∈ Toset ∧ 𝑊 ∈ Mnd) → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 ≤ (𝑛 · 𝑥))))
Theorem	submarchi 29740	A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.)
⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi)
Theorem	isarchi3 29741*	This is the usual definition of the Archimedean property for an ordered group. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    ⇒   ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))))
Theorem	archirng 29742*	Property of Archimedean ordered groups, framing positive 𝑌 between multiples of 𝑋. (Contributed by Thierry Arnoux, 12-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    &   ⊢ (𝜑 → 0 < 𝑌)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
Theorem	archirngz 29743*	Property of Archimedean left and right ordered groups. (Contributed by Thierry Arnoux, 6-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℤ ((𝑛 · 𝑋) < 𝑌 ∧ 𝑌 ≤ ((𝑛 + 1) · 𝑋)))
Theorem	archiexdiv 29744*	In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    ⇒   ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋))
Theorem	archiabllem1a 29745*	Lemma for archiabl 29752: 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 ℤ. (Contributed by Thierry Arnoux, 12-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝑋 = (𝑛 · 𝑈))
Theorem	archiabllem1b 29746*	Lemma for archiabl 29752. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥)    ⇒   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ∃𝑛 ∈ ℤ 𝑦 = (𝑛 · 𝑈))
Theorem	archiabllem1 29747*	Archimedean ordered groups with a minimal positive value are abelian. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑈)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥) → 𝑈 ≤ 𝑥)    ⇒   ⊢ (𝜑 → 𝑊 ∈ Abel)
Theorem	archiabllem2a 29748*	Lemma for archiabl 29752, which requires the group to be both left- and right-ordered. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 0 < 𝑋)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝐵 ( 0 < 𝑐 ∧ (𝑐 + 𝑐) ≤ 𝑋))
Theorem	archiabllem2c 29749*	Lemma for archiabl 29752. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ¬ (𝑋 + 𝑌) < (𝑌 + 𝑋))
Theorem	archiabllem2b 29750*	Lemma for archiabl 29752. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	archiabllem2 29751*	Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ ≤ = (le‘𝑊)    &   ⊢ < = (lt‘𝑊)    &   ⊢ · = (.g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ oGrp)    &   ⊢ (𝜑 → 𝑊 ∈ Archi)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎))    ⇒   ⊢ (𝜑 → 𝑊 ∈ Abel)
Theorem	archiabl 29752	Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
20.3.9.5  Semiring left modules
Syntax	cslmd 29753	Extend class notation with class of all semimodules.
class SLMod
Definition	df-slmd 29754*	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][(+g‘𝑔) / 𝑎][( ·𝑠 ‘𝑔) / 𝑠][(Scalar‘𝑔) / 𝑓][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ SRing ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤 ∧ ((0g‘𝑓)𝑠𝑤) = (0g‘𝑔))))}
Theorem	isslmd 29755*	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)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 𝑂 = (0g‘𝐹)    ⇒   ⊢ (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((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 ))))
Theorem	slmdlema 29756	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)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 𝑂 = (0g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((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 )))
Theorem	lmodslmd 29757	Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ (𝑊 ∈ LMod → 𝑊 ∈ SLMod)
Theorem	slmdcmn 29758	A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
Theorem	slmdmnd 29759	A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
Theorem	slmdsrg 29760	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)
Theorem	slmdbn0 29761	The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅)
Theorem	slmdacl 29762	Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Theorem	slmdmcl 29763	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)    &   ⊢ · = (.r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
Theorem	slmdsn0 29764	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 ≠ ∅)
Theorem	slmdvacl 29765	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)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
Theorem	slmdass 29766	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)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	slmdvscl 29767	Closure of scalar product for a semiring left module. (hvmulcl 27870 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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
Theorem	slmdvsdi 29768	Distributive law for scalar product. (ax-hvdistr1 27865 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
Theorem	slmdvsdir 29769	Distributive law for scalar product. (ax-hvdistr1 27865 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
Theorem	slmdvsass 29770	Associative law for scalar product. (ax-hvmulass 27864 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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ × = (.r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Theorem	slmd0cl 29771	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 = (0g‘𝐹)    ⇒   ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾)
Theorem	slmd1cl 29772	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 = (1r‘𝐹)    ⇒   ⊢ (𝑊 ∈ SLMod → 1 ∈ 𝐾)
Theorem	slmdvs1 29773	Scalar product with ring unit. (ax-hvmulid 27863 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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋)
Theorem	slmd0vcl 29774	The zero vector is a vector. (ax-hv0cl 27860 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 = (0g‘𝑊)    ⇒   ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝑉)
Theorem	slmd0vlid 29775	Left identity law for the zero vector. (hvaddid2 27880 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋)
Theorem	slmd0vrid 29776	Right identity law for the zero vector. (ax-hvaddid 27861 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋)
Theorem	slmd0vs 29777	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27867 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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑂 = (0g‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 )
Theorem	slmdvs0 29778	Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 27881 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )
20.3.9.6  Finitely supported group sums - misc additions
Theorem	gsumle 29779	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)    &   ⊢ (𝜑 → 𝑀 ∈ oMnd)    &   ⊢ (𝜑 → 𝑀 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 ∘𝑟 ≤ 𝐺)    ⇒   ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺))
Theorem	gsummpt2co 29780*	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 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐷)    ⇒   ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶)))))
Theorem	gsummpt2d 29781*	Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 18371. (Contributed by Thierry Arnoux, 27-Apr-2020.)
⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)    &   ⊢ (𝜑 → Rel 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑊 ∈ CMnd)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷)))))
Theorem	gsumvsca1 29782*	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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑊 ∈ SLMod)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
Theorem	gsumvsca2 29783*	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 = (0g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝑊 ∈ SLMod)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄))
Theorem	gsummptres 29784*	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 = (0g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐷)) → 𝐶 = 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶)))
Theorem	xrge0tsmsd 29785*	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*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))), ℝ*, < ))    ⇒   ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝑆})
Theorem	xrge0tsmsbi 29786	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*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    ⇒   ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = ∪ (𝐺 tsums 𝐹)))
Theorem	xrge0tsmseq 29787	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*𝑠 ↾s (0[,]+∞))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞))    &   ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹))    ⇒   ⊢ (𝜑 → 𝐶 = ∪ (𝐺 tsums 𝐹))
20.3.9.7  Rings - misc additions
Theorem	rngurd 29788*	Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 1 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥)    ⇒   ⊢ (𝜑 → 1 = (1r‘𝑅))
Theorem	ress1r 29789	1r is unaffected by restriction. This is a bit more generic than subrg1 18790. (Contributed by Thierry Arnoux, 6-Sep-2018.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 = (1r‘𝑆))
Theorem	dvrdir 29790	Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 + 𝑌) / 𝑍) = ((𝑋 / 𝑍) + (𝑌 / 𝑍)))
Theorem	rdivmuldivd 29791	Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑊 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑋 / 𝑌) · (𝑍 / 𝑊)) = ((𝑋 · 𝑍) / (𝑌 · 𝑊)))
Theorem	ringinvval 29792*	The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∗ = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 ))
Theorem	dvrcan5 29793	Cancellation law for common factor in ratio. (divcan5 10727 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ / = (/r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌))
Theorem	subrgchr 29794	If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅 ↾s 𝐴)) = (chr‘𝑅))
20.3.9.8  Ordered rings and fields
Syntax	corng 29795	Extend class notation with the class of all ordered rings.
class oRing
Syntax	cofld 29796	Extend class notation with the class of all ordered fields.
class oField
Definition	df-orng 29797*	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][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))}
Definition	df-ofld 29798	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)
Theorem	isorng 29799*	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 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏))))
Theorem	orngring 29800	An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.)
⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring)