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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tosglb 29001 | Same theorem as toslub 28999, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by AV, 28-Sep-2020.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ < = (lt‘𝐾) & ⊢ (𝜑 → 𝐾 ∈ Toset) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ((glb‘𝐾)‘𝐴) = inf(𝐴, 𝐵, < )) | ||
Theorem | clatp0cl 29002 | 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 → 0 ∈ 𝐵) | ||
Theorem | clatp1cl 29003 | 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 → 1 ∈ 𝐵) | ||
Axiom | ax-xrssca 29004 | 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.) |
⊢ ℝfld = (Scalar‘ℝ*𝑠) | ||
Axiom | ax-xrsvsca 29005 | 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) | ||
Theorem | xrs0 29006 | The zero of the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 11951 and df-xrs 15985), however it has a zero. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
⊢ 0 = (0g‘ℝ*𝑠) | ||
Theorem | xrslt 29007 | The "strictly less than" relation for the extended real structure. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ < = (lt‘ℝ*𝑠) | ||
Theorem | xrsinvgval 29008 | The inversion operation in the extended real numbers. The extended real is not a group, as its addition is not associative. (cf. xaddass 11951 and df-xrs 15985), however it has an inversion operation. (Contributed by Thierry Arnoux, 13-Jun-2017.) |
⊢ (𝐵 ∈ ℝ* → ((invg‘ℝ*𝑠)‘𝐵) = -𝑒𝐵) | ||
Theorem | xrsmulgzz 29009 | 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)) | ||
Theorem | xrstos 29010 | The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.) |
⊢ ℝ*𝑠 ∈ Toset | ||
Theorem | xrsclat 29011 | The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.) |
⊢ ℝ*𝑠 ∈ CLat | ||
Theorem | xrsp0 29012 | 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) | ||
Theorem | xrsp1 29013 | The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) |
⊢ +∞ = (1.‘ℝ*𝑠) | ||
Theorem | ressmulgnn 29014 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝐴 ⊆ (Base‘𝐺) & ⊢ ∗ = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) | ||
Theorem | ressmulgnn0 29015 | Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝐴 ⊆ (Base‘𝐺) & ⊢ ∗ = (.g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ (0g‘𝐺) = (0g‘𝐻) ⇒ ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) | ||
Theorem | xrge0base 29016 | The base of the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | ||
Theorem | xrge00 29017 | The zero of the extended nonnegative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | ||
Theorem | xrge0plusg 29018 | 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*𝑠 ↾s (0[,]+∞))) | ||
Theorem | xrge0le 29019 | The lower-or-equal relation in the extended real numbers. (Contributed by Thierry Arnoux, 14-Mar-2018.) |
⊢ ≤ = (le‘(ℝ*𝑠 ↾s (0[,]+∞))) | ||
Theorem | xrge0mulgnn0 29020 | The group multiple function in the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ (0[,]+∞)) → (𝐴(.g‘(ℝ*𝑠 ↾s (0[,]+∞)))𝐵) = (𝐴 ·e 𝐵)) | ||
Theorem | xrge0addass 29021 | 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))) | ||
Theorem | xrge0addgt0 29022 | The sum of nonnegative and positive numbers is positive. See addgtge0 10395. (Contributed by Thierry Arnoux, 6-Jul-2017.) |
⊢ (((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞)) ∧ 0 < 𝐴) → 0 < (𝐴 +𝑒 𝐵)) | ||
Theorem | xrge0adddir 29023 | 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))) | ||
Theorem | xrge0adddi 29024 | 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))) | ||
Theorem | xrge0npcan 29025 | Extended nonnegative real version of npcan 10169. (Contributed by Thierry Arnoux, 9-Jun-2017.) |
⊢ ((𝐴 ∈ (0[,]+∞) ∧ 𝐵 ∈ (0[,]+∞) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +𝑒 -𝑒𝐵) +𝑒 𝐵) = 𝐴) | ||
Theorem | fsumrp0cl 29026* | Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.) |
⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) | ||
Theorem | abliso 29027 | 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) | ||
Syntax | comnd 29028 | Extend class notation with the class of all right ordered monoids. |
class oMnd | ||
Syntax | cogrp 29029 | Extend class notation with the class of all right ordered groups. |
class oGrp | ||
Definition | df-omnd 29030* | 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 (oppg‘𝑀). (Contributed by Thierry Arnoux, 13-Mar-2018.) |
⊢ oMnd = {𝑔 ∈ Mnd ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑝][(le‘𝑔) / 𝑙](𝑔 ∈ Toset ∧ ∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ∀𝑐 ∈ 𝑣 (𝑎𝑙𝑏 → (𝑎𝑝𝑐)𝑙(𝑏𝑝𝑐)))} | ||
Definition | df-ogrp 29031 | 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) | ||
Theorem | isomnd 29032* | 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 29033 | 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 29034 | An left ordered group is a group. (Contributed by Thierry Arnoux, 9-Jul-2018.) |
⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp) | ||
Theorem | omndmnd 29035 | A left ordered monoid is a monoid. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd) | ||
Theorem | omndtos 29036 | A left ordered monoid is a totally ordered set. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset) | ||
Theorem | omndadd 29037 | 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 29038 | 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 29039 | 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 29040 | 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 29041 | 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 29042 | The nonnegative extended real numbers form an ordered monoid. (Contributed by Thierry Arnoux, 22-Mar-2018.) |
⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ oMnd | ||
Theorem | omndmul2 29043 | 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 29044 | 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 29045 | 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 29046 | 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 29047 | 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 29048 | 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 29049 | 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 29050 | 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 29051 | 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 29052 | 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 29053 | 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 29054 | 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 29055 | 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))) | ||
Syntax | csgns 29056 | Extend class notation to include the Signum function. |
class sgns | ||
Definition | df-sgns 29057* | Signum function for a structure. See also df-sgn 13675 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 29058* | 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 29059 | 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 29060 | 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}) | ||
Syntax | cinftm 29061 | Class notation for the infinitesimal relation. |
class ⋘ | ||
Syntax | carchi 29062 | Class notation for the Archimedean property. |
class Archi | ||
Definition | df-inftm 29063* | 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 29064 | A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ Archi = {𝑤 ∣ (⋘‘𝑤) = ∅} | ||
Theorem | inftmrel 29065 | The infinitesimal relation for a structure 𝑊. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (⋘‘𝑊) ⊆ (𝐵 × 𝐵)) | ||
Theorem | isinftm 29066* | 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 29067* | 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 29068 | 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 29069 | The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.) |
⊢ ¬ ℝ*𝑠 ∈ Archi | ||
Theorem | isarchi2 29070* | 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 29071 | A submonoid is archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
⊢ (((𝑊 ∈ Toset ∧ 𝑊 ∈ Archi) ∧ 𝐴 ∈ (SubMnd‘𝑊)) → (𝑊 ↾s 𝐴) ∈ Archi) | ||
Theorem | isarchi3 29072* | 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 29073* | 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 29074* | 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 29075* | 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 29076* | Lemma for archiabl 29083: 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 29077* | Lemma for archiabl 29083. (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 29078* | 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 29079* | Lemma for archiabl 29083, 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 29080* | Lemma for archiabl 29083. (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 29081* | Lemma for archiabl 29083. (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 29082* | 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 29083 | Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.) |
⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel) | ||
Syntax | cslmd 29084 | Extend class notation with class of all semimodules. |
class SLMod | ||
Definition | df-slmd 29085* | 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 29086* | 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 29087 | 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 29088 | Left semimodules generalize the notion of left modules. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑊 ∈ LMod → 𝑊 ∈ SLMod) | ||
Theorem | slmdcmn 29089 | A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) | ||
Theorem | slmdmnd 29090 | A semimodule is a monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ (𝑊 ∈ SLMod → 𝑊 ∈ Mnd) | ||
Theorem | slmdsrg 29091 | 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 29092 | The base set of a semimodule is nonempty. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 𝐵 ≠ ∅) | ||
Theorem | slmdacl 29093 | 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 29094 | 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 29095 | 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 29096 | 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 29097 | 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 29098 | Closure of scalar product for a semiring left module. (hvmulcl 27254 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 29099 | Distributive law for scalar product. (ax-hvdistr1 27249 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 29100 | Distributive law for scalar product. (ax-hvdistr1 27249 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))) |
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