hcju/proofwikips huggingface dataset:
Newton's Laws of Motion/Third Law
== Physical Law == '''Newton's third law of motion''' is one of three physical laws that forms the basis for classical mechanics. === Statement of Law ===  :To every force there is always an equal and opposite force. That is, the forces of two bodies on each other are always equal and are directed in opposite directions.  As {{AuthorRef|Isaac Newton}} himself put it: :''Whenever one body exerts a force on a second body, the second body exerts a force on the first body. These forces are equal in magnitude and opposite in direction.'' == Also known as == This law is also referred to as the '''law of action and reaction'''. It is also often referred to as just '''Newton's third law'''. == Also see == * Newton's First Law of Motion * Newton's Second Law of Motion {{Namedfor|Isaac Newton}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Newton's Second Law of Motion|next = Definition:Rate of Change}}: Chapter $\text{VI}$: On the Seashore * {{BookReference|Classical Mechanics|1965|J.W. Leech|ed = 2nd|edpage = Second Edition|prev = Newton's Second Law of Motion|next = Definition:Force}}: Chapter $\text {I}$: Introduction: $(3)$ * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Force/Unit|next = Newton's Law of Universal Gravitation/Historical Note}}: $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Action and Reaction * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's Second Law of Motion|next = Newton's Method|entry = Newton's laws of motion}}
Newton's Laws of Motion/First Law
== Physical Law == '''Newton's first law of motion''' is one of three physical laws that forms the basis for classical mechanics. === Statement of Law ===  :Every body remains in a state of constant velocity unless it is acted upon by an external unbalanced force.   This of course includes it being stationary, that is, with a constant velocity of zero.  This tendency is sometimes called inertia. As {{AuthorRef|Isaac Newton}} himself put it: :''A body remains at rest or, if already in motion, remains in uniform motion with constant speed in a straight line, unless it is acted on by an unbalanced external force.'' == Also known as == This law is also sometimes called the '''principle of inertia'''. It is also often referred to as just '''Newton's first law'''. == Also see == * Newton's Second Law of Motion * Newton's Third Law of Motion It is worth pointing out that '''Newton's first law of motion''' is in fact a special case of Newton's Second Law of Motion, putting $\mathbf F = 0$. {{Namedfor|Isaac Newton}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Newton's Laws of Motion|next = Newton's Second Law of Motion}}: Chapter $\text{VI}$: On the Seashore * {{BookReference|Classical Mechanics|1965|J.W. Leech|ed = 2nd|edpage = Second Edition|prev = Definition:Mass|next = Newton's Second Law of Motion}}: Chapter $\text {I}$: Introduction: $(1)$ * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Newton's Laws of Motion|next = Definition:Stationary}}: $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Inertia * {{BookReference|Tensor Calculus and Relativity|1975|Derek F. Lawden|ed = 3rd|edpage = Third Edition|next = Definition:Frame of Reference}}: Chapter $1$ Special Principle of Relativity. Lorentz Transformations: $1$. Newton's laws of motion * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's Laws of Motion|next = Newton's Second Law of Motion|entry = Newton's laws of motion}}
Newton's Laws of Motion/Second Law
== Physical Law == '''Newton's second law of motion''' is one of three physical laws that forms the basis for classical mechanics. === Statement of Law ===  :The total force applied on a body is equal to the derivative with respect to time of the linear momentum of the body: :$\mathbf F = \map {\dfrac \d {\d t} } {m \bsv}$  As {{AuthorRef|Isaac Newton}} himself put it: :''The acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.'' == Also known as == This law is also often referred to as just '''Newton's second law'''. == Also see == * At velocities near the speed of light, see Einstein's Law of Motion. * Newton's First Law of Motion * Newton's Third Law of Motion {{Namedfor|Isaac Newton}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Newton's First Law of Motion|next = Newton's Third Law of Motion}}: Chapter $\text{VI}$: On the Seashore * {{BookReference|Classical Mechanics|1965|J.W. Leech|ed = 2nd|edpage = Second Edition|prev = Newton's First Law of Motion|next = Newton's Third Law of Motion}}: Chapter $\text {I}$: Introduction: $(2)$ * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Mass|next = Definition:Gram}}: $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Mass * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Derivative|next = Definition:Free Fall}}: $1$: The Nature of Differential Equations: $\S 1$: Introduction: $(1)$ * {{BookReference|Engineering Mathematics|1977|A.J.M. Spencer|volume = I|prev = Definition:Differential Equation|next = Definition:Ordinary Differential Equation}}: Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Volume of Sphere/Proof by Archimedes|next = Definition:Linear Momentum}}: Chapter $\text {B}.7$: A Simple Approach to $E = M c^2$ * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Einstein's Mass-Energy Equation|next = Motion of Rocket in Outer Space/Proof 2}}: Chapter $\text {B}.8$: Rocket Propulsion in Outer Space * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Volume of Solid of Revolution/Historical Note|next = Definition:Velocity}}: Chapter $8$: The System of the World: Newton * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Speed of Light|next = Definition:Newton (Unit)|entry = constant|subentry = in physical laws}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's First Law of Motion|next = Newton's Third Law of Motion|entry = Newton's laws of motion}}
Newton's Law of Cooling
== Physical Law == The rate at which a hot body loses heat is proportional to the difference in temperature between it and its surroundings. {{Namedfor|Isaac Newton|cat = Newton}} == Historical Note == {{:Newton's Law of Cooling/Historical Note}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Law of Mass Action|next = Newton's Law of Cooling/Historical Note}}: $1$: The Nature of Differential Equations: $\S 4$: Growth, Decay and Chemical Reactions: Problem $5$
Acceleration Due to Gravity
== Physical Law == A body $B$ in a uniform gravitational field experiences a force which gives rise to a constant acceleration independent of the mass of the body. If the force due to the gravitational field is the '''only''' force on the body, it is said to be in free fall. == Derivation == This law can be derived from Newton's Law of Universal Gravitation. Let the mass of $B$ be $m$. Let the mass of the body $P$ which gives rise to the gravitational field be $M$. Then the force on $B$ is given by: :$F = G \dfrac {M m} {r^2}$ where: :$G$ is the gravitational constant :$r$ is the distance between the centers of gravity of $B$ and $P$. The assumption is that $M$ is orders of magnitude greater than $m$, and $r$ is also several orders of magnitude greater than the displacements observed on $B$ in the local frame. Then: :$F = m \dfrac {G M} {r^2}$ But from Newton's Second Law of Motion: :$F = m a$ where $a$ is the magnitude of the acceleration which would be imparted to the body if no other force were acting on it. Hence: :$a = \dfrac {G M} {r^2}$ which is, as we said, independent of the mass of the object.  == Gravity on Earth == {{:Acceleration Due to Gravity/Earth's Surface|Gravity on Earth}} == Historical Note == {{:Acceleration Due to Gravity/Historical Note}} == Sources == * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Constant of Proportion|next = Definition:Gravity}}: $\text {I}$: Motion, Sound and Heat: Chapter $2$: Falling Bodies: Free Fall
Newton's Law of Universal Gravitation
== Physical Law == Let $a$ and $b$ be particles with mass $m_a$ and $m_b$ respectively. Then $a$ and $b$ exert a force upon each other whose magnitude and direction are given by '''Newton's law of universal gravitation''': :$\mathbf F_{a b} \propto \dfrac {m_a m_b {\mathbf r_{b a} } } {r^3}$ where: :$\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$ :$\mathbf r_{b a}$ is the displacement vector from $b$ to $a$ :$r$ is the distance between $a$ and $b$. Thus it is seen that the direction of $\mathbf F_{a b}$ is specifically '''towards''' $a$. By exchanging $a$ and $b$ in the above, it is seen that $b$ exerts the same force on $a$ as $a$ does on $b$, but in the opposite direction, that is, towards $b$. === Gravitational Constant === {{:Definition:Gravitational Constant}} Thus the equation becomes: :$\mathbf F_{a b} = \dfrac {G m_a m_b \mathbf r_{b a} } {r^3}$ == Also presented as == :$\mathbf F_{a b} \propto \dfrac {m_a m_b \hat {\mathbf r}_{b a} } {r^2}$ where $\hat {\mathbf r}_{a b}$ is the unit vector in the direction from $b$ to $a$. == Also known as == '''Newton's Law of Universal Gravitation''' is also known as just '''Newton's Law of Gravitation'''. Some sources refer to it as the '''inverse square law of gravitation'''. {{Namedfor|Isaac Newton|cat = Newton}} == Historical Note == {{:Newton's Law of Universal Gravitation/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Kepler's Third Law of Planetary Motion|next = Newton's Laws of Motion}}: Chapter $\text{VI}$: On the Seashore * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Mathematician:Frederick William Herschel|next = Definition:Gravitational Constant}}: $\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: The Gravitational Constant * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Kepler's Laws of Planetary Motion/Historical Note|next = Definition:Gravitational Constant}}: $\S 3.21$: Newton's Law of Gravitation: $(11)$ * {{BookReference|Electromagnetism|1990|I.S. Grant|author2 = W.R. Phillips|ed = 2nd|edpage = Second Edition|next = Coulomb's Law of Electrostatics}}: Chapter $1$: Force and energy in electrostatics * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Kepler's Laws of Planetary Motion|next = Book:Isaac Newton/Philosophiae Naturalis Principia Mathematica}}: Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Acceleration is Second Derivative of Displacement with respect to Time|next = Newton's Law of Universal Gravitation/Historical Note}}: Chapter $8$: The System of the World: Newton * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Inverse for Complex Multiplication|next = Definition:Inverse Trigonometric Function|entry = inverse square law of gravitation}}
Fermat's Principle of Least Time
== Physical Law == === Original formulation === The path taken by a ray of light from one point to another is the one that can be traversed in the least time. === Modern version === The optical path length must be stationary, that is, be either a minimum, maximum or a point of inflection. == Proof == This follows from the Huygens-Fresnel Principle.  == Also see == * Snell-Descartes Law {{Namedfor|Pierre de Fermat|cat = Fermat}} == Historical Note == {{:Fermat's Principle of Least Time/Historical Note}} == Sources == * {{BookReference|Men of Mathematics|1937|Eric Temple Bell|prev = Definition:Euler-Lagrange Equation/Historical Note|next = Fermat's Principle of Least Time/Historical Note}}: Chapter $\text{IV}$: The Prince of Amateurs * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Mathematician:Willebrord van Royen Snell|next = Brachistochrone is Cycloid/Proof 1}}: $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Heron's Principle of Reflection|next = Fermat's Principle of Least Time/Historical Note}}: Chapter $\text {A}.7$: Heron (first century A.D.) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Snell-Descartes Law/Historical Note|next = Definition:Folium of Descartes/Historical Note}}: Chapter $\text {A}.13$: Fermat ({{DateRange|1601|1665}})
Snell-Descartes Law
== Physical Law == Consider a ray of light crossing the threshold between two media. Let its speed: :in medium $1$ be $v_1$ :in medium $2$ be $v_2$. Let it meet the threshold at: :an angle $\alpha_1$ from the vertical in medium 1 :an angle $\alpha_2$ from the vertical in medium 2. Then the '''Snell-Descartes law''' states that: :$\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$ == Proof == The '''Snell-Descartes law''' can be derived from Fermat's Principle of Least Time as follows: Let the ray of light travel from $A$ to $P$ in the medium $1$. Then let it travel from $P$ to $B$ in medium $2$. :350px The total time $T$ required for that journey is: :$T = \dfrac {\sqrt {a^2 + x^2} } {v_1} + \dfrac {\sqrt {b^2 + \paren {c - x}^2} } {v_2}$ from the geometry of the above diagram. From Fermat's Principle of Least Time, this time will be a minimum. From Derivative at Maximum or Minimum, we need: :$\dfrac {\d T} {\d x} = 0$. So: :$\dfrac x {v_1 \sqrt {a^2 + x^2} } = \dfrac {c - x} {v_2 \sqrt {b^2 + \paren {c - x}^2} }$ which leads directly to: :$\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$ by definition of sine.  == Also presented as == This law can also be seen expressed as: :$\dfrac {\sin \alpha_1} {\sin \alpha_2} = \dfrac {v_1} {v_2}$ {{Namedfor|Willebrord van Royen Snell|name2 = René Descartes|cat = Snell|cat2 = Descartes}} == Historical Note == {{:Snell-Descartes Law/Historical Note}} == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Definition:Brachistochrone|next = Mathematician:Willebrord van Royen Snell}}: $1$: The Nature of Differential Equations: $\S 6$: The Brachistochrone. Fermat and the Bernoullis * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Largest Rectangle with Given Perimeter is Square/Historical Note|next = Snell-Descartes Law/Historical Note}}: Chapter $\text {A}.13$: Fermat ({{DateRange|1601|1665}}) * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Length of Arc of Nephroid|next = Brachistochrone is Cycloid/Proof 1}}: Chapter $\text {B}.22$: Bernoulli's Solution of the Brachistochrone Problem
Einstein's Law of Motion
== Physical Law == The force and acceleration on a body of constant rest mass are related by the equation: : $\mathbf F = \dfrac {m_0 \mathbf a} {\left({1 - \dfrac{v^2}{c^2}}\right)^{\tfrac 3 2}}$ where: : $\mathbf F$ is the force on the body : $\mathbf a$ is the acceleration induced on the body : $v$ is the magnitude of the velocity of the body : $c$ is the speed of light : $m_0$ is the rest mass of the body. == Proof == Into Newton's Second Law of Motion: : $\mathbf F = \dfrac {\mathrm d}{\mathrm d t} \left({m \mathbf v}\right)$ we substitute Einstein's Mass-Velocity Equation: : $m = \dfrac {m_0} {\sqrt {1 - \dfrac {v^2} {c^2}}}$ to obtain: : $\mathbf F = \dfrac {\mathrm d} {\mathrm d t} \left({\dfrac {m_0 \mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2}}}}\right)$ Then we perform the differentiation {{WRT|Differentiation}} time: {{begin-eqn}} {{eqn | l = \frac{\mathrm d}{\mathrm d t} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right)       | r = \frac{\mathrm d}{\mathrm d v} \left({\frac {\mathbf v}{\sqrt{1 - \dfrac {v^2}{c^2} } } }\right) \frac{\mathrm d v}{\mathrm d t}       | c = Chain Rule for Derivatives }} {{eqn | r = \mathbf a \left({\frac {\sqrt{1 - \dfrac {v^2}{c^2} } - \dfrac v 2 \dfrac 1 {\sqrt{1 - \dfrac {v^2}{c^2} } } \dfrac{-2 v}{c^2} } {1 - \dfrac {v^2}{c^2} } }\right)       | c = Chain Rule for Derivatives, Quotient Rule, etc. }} {{eqn | r = \mathbf a \left({\frac {c^2 \left({1 - \dfrac {v^2}{c^2} }\right) + v^2} {c^2 \left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right)       | c =  }} {{eqn | r = \mathbf a \left({\frac 1 {\left({1 - \dfrac {v^2}{c^2} }\right)^{3/2} } }\right)       | c =  }} {{end-eqn}} Thus we arrive at the form: : $\mathbf F = \dfrac {m_0 \mathbf a} {\left({1 - \dfrac{v^2}{c^2}}\right)^{\tfrac 3 2}}$  == Comment == Thus we see that at low velocities (i.e. much less than that of light), the well-known equation $\mathbf F = m \mathbf a$ holds to a high degree of accuracy. {{namedfor|Albert Einstein|cat = Einstein}} == Sources == * {{BookReference|Calculus Gems|1992|George F. Simmons|prev = Einstein's Mass-Velocity Equation|next = Einstein's Mass-Energy Equation}}: Chapter $\text {B}.7$: A Simple Approach to $E = M c^2$
Hooke's Law
== Physical Law == '''Hooke's Law''' applies to an ideal spring: :$\mathbf F = -k \mathbf x$ where: :$\mathbf F$ is the force caused by a displacement $\mathbf x$ :$k$ is the spring force constant. The negative sign indicates that the force pulls in the opposite direction to that of the displacement imposed. :''The strain is proportional to the stress.'' == Also see == * Dimension of Spring Force Constant: the dimension of $k$ is $\mathsf {M T}^{-2}$. == Sources == * {{BookReference|Understanding Physics|1966|Isaac Asimov|prev = Definition:Elasticity|next = Definition:Elastic Limit}}: $\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: The Gravitational Constant * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Robert Hooke|next = Definition:Hopf Bifurcation|entry = Hooke's law}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Robert Hooke|next = Definition:Hopf Bifurcation|entry = Hooke's law}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Homotopy|next = Mathematician:Guillaume de l'Hôpital|entry = Hooke's law}}
Leibniz's Formula for Pi/Lemma
== Lemma == {{begin-eqn}} {{eqn | l = \frac 1 {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \cdots + t^{4 n} - \frac {t^{4 n + 2} } {1 + t^2}       | c =  }} {{eqn | r = \paren {\sum_{k \mathop = 0}^{2 n} \paren {-1}^k t^{2 k} } - \frac {t^{4 n + 2} } {1 + t^2}       | c =  }} {{end-eqn}} This holds for all real $t \in \R$. == Proof == {{begin-eqn}} {{eqn | l = \frac {1 - \paren {-t^2}^{2 n + 1} } {1 - \paren {-t^2} }       | r = \sum_{k \mathop = 0}^{2 n} \paren {-t^2}^k       | c = Sum of Geometric Sequence }} {{eqn | ll= \leadsto       | l = \frac {1 + \paren {t^2}^{2 n + 1} } {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \cdots + t^{4 n}       | c =  }} {{eqn | ll= \leadsto       | l = \frac 1 {1 + t^2} + \frac {t^{4 n + 2} } {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \cdots + t^{4 n}       | c =  }} {{eqn | ll= \leadsto       | l = \frac 1 {1 + t^2}       | r = 1 - t^2 + t^4 - t^6 + \cdots + t^{4 n} - \frac {t^{4 n + 2} } {1 + t^2}       | c =  }} {{end-eqn}} From Square of Real Number is Non-Negative, we have that: :$t^2 \ge 0$ for all real $t$. So $- t^2 \le 0$ and so $- t^2 \ne 1$. So the conditions of Sum of Geometric Sequence are satisfied, and so the above argument holds for all real $t$.
Stabilizer is Subgroup/Corollary
Stabilizer is Subgroup/Corollary 0 8606 479172 378345 2020-07-22T06:48:35Z Prime.mover 59 Prime.mover moved page Stabilizer is Subgroup/Corollary 2 to Stabilizer is Subgroup/Corollary wikitext text/x-wiki == Corollary to Stabilizer is Subgroup ==  Let $G$ be a group whose identity is $e$. Let $G$ act on a set $X$. Let $x \in X$. Then: :$\forall g, h \in G: g * x = h * x \iff g^{-1} h \in \Stab x$  == Proof == {{begin-eqn}} {{eqn | l = h * x       | r = g * x       | c =  }} {{eqn | ll= \leadstoandfrom       | l = g^{-1} * \paren {h * x}       | r = g^{-1} * \paren {g * x}       | c =  }} {{eqn | ll= \leadstoandfrom       | l = \paren {g^{-1} h} * x       | r = \paren {g^{-1} g} * x       | c =  }} {{eqn | ll= \leadstoandfrom       | l = \paren {g^{-1} h} * x       | r = e * x       | c =  }} {{eqn | ll= \leadstoandfrom       | l = \paren {g^{-1} h} * x       | r = x       | c =  }} {{eqn | ll= \leadstoandfrom       | l = g^{-1} h       | o = \in       | r = \Stab x       | c =  }} {{end-eqn}}  == Sources == * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Definition:Transitive Group Action|next = Orbit-Stabilizer Theorem/Proof 2}}: $\S 6.5$. Orbits: Lemma $\text {(ii)}$
Upper Bound is Lower Bound for Inverse Ordering
{{refactor|Not trivial. Result may need rephrasing using iff instead}} == Definition == Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$. Let $M$ be an upper bound for $\left({T, \preceq}\right)$. Let $\succeq$ be the dual ordering of $\preceq$. Then $M$ is a lower bound for $\left({T, \succeq}\right)$. == Proof == Let $M$ be an upper bound for $\left({T, \preceq}\right)$. That is: :$\forall a \in T: a \preceq M$ By definition of dual ordering, it follows that: :$\forall a \in T: M \succeq a$ That is, $M$ is a lower bound for $\left({T, \succeq}\right)$.   {{expand|Proof via duality}} == Also see == * Lower Bound is Upper Bound for Inverse Ordering
Euclid's Lemma for Irreducible Elements
== Lemma == Let $\struct {D, +, \times}$ be a Euclidean domain whose unity is $1$. Let $p$ be an irreducible element of $D$. Let $a, b \in D$ such that: :$p \divides a \times b$ where $\divides$ means '''is a divisor of'''. Then $p \divides a$ or $p \divides b$. === General Result === {{:Euclid's Lemma for Irreducible Elements/General Result}} == Proof == Let $p \divides a \times b$. Suppose $p \nmid a$. Then from the definition of irreducible: : $p \perp a$ Thus from Euclid's Lemma for Euclidean Domains: :$p \divides b$ Similarly, if $p \nmid b$: :$p \divides a$ So: :$p \divides a b \implies p \divides a$ or: :$p \divides b$ as we needed to show.  {{Namedfor|Euclid}} == Also see == * Euclid's Lemma for Prime Divisors, for the usual statement of this result, which is this lemma as applied specifically to the integers. == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Euclid's Lemma for Euclidean Domains|next = Euclid's Lemma for Irreducible Elements/General Result}}: Chapter $6$: Polynomials and Euclidean Rings: $\S 29$. Irreducible elements: Theorem $56 \ \text{(i)}$