To prove that $\nu_p$ is a valuation it is necessary to demonstrate: {{begin-axiom}} {{axiom | n = V1         | q = \forall q, r \in \Q         | ml= \nu_p \left({q r}\right)         | mo= =         | mr= \nu_p \left({q}\right) + \nu_p \left({r}\right) }} {{axiom | n = V2         | q = \forall q \in \Q         | ml= \nu_p \left({q}\right) = +\infty         | mo= \iff         | mr= q = 0 }} {{axiom | n = V3         | q = \forall q, r \in \Q         | ml= \nu_p \left({q + r}\right)         | mo= \ge         | mr= \min \left\{ {\nu_p \left({q}\right), \nu_p \left({r}\right) }\right\} }} {{end-axiom}} Let $q := \dfrac a b, r := \dfrac c d \in \Q$. === Axiom $(V1)$ === {{begin-eqn}} {{eqn | l = \nu_p \left({q r}\right)       | r = \nu_p \left({\frac a b \cdot \frac c d}\right) }} {{eqn | r = \nu_p \left({\frac {a c} {b d} }\right)       | c = {{Defof|Rational Multiplication}} }} {{eqn | r = \nu_p^\Z \left({a c}\right) - \nu_p^\Z \left({b d}\right)       | c = {{Defof|P-adic Valuation/Rational Numbers|$p$-adic Valuation}} }} {{eqn | r = \left({\nu_p^\Z \left({a}\right) + \nu_p^\Z \left({c}\right)}\right) - \left({\nu_p^\Z \left({b}\right) + \nu_p^\Z \left({d}\right)}\right)       | c = $p$-adic Valuation on Integers is Valuation: Axiom $V1$ }} {{eqn | r = \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right) + \nu_p^\Z \left({c}\right) - \nu_p^\Z \left({d}\right)       | c = Integer Addition is Commutative }} {{eqn | r = \nu_p \left({\frac a b}\right) + \nu_p \left({\frac c d}\right)       | c = {{Defof|P-adic Valuation|$p$-adic Valuation}} }} {{eqn | r = \nu_p \left({q}\right) + \nu_p \left({r}\right)       | c =  }} {{end-eqn}} {{qed|lemma}} === Axiom $(V2)$ === {{begin-eqn}} {{eqn | l = \dfrac a b       | r = 0 }} {{eqn | ll= \iff       | l = a       | r = 0       | c = {{Defof|Rational Number}} }} {{eqn | ll= \iff       | l = \nu_p^\Z \left({a}\right)       | r = +\infty       | c = {{Defof|P-adic Valuation|$p$-adic Valuation}} }} {{eqn | ll= \iff       | l = \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)       | r = +\infty       | c = as $b \ne 0$ }} {{eqn | ll= \iff       | l = \nu_p \left({\frac a b}\right)       | r = +\infty       | c = {{Defof|P-adic Valuation|$p$-adic Valuation}} }} {{end-eqn}} {{qed|lemma}} === Axiom $(V3)$ === From P-adic Valuation on Integers follows that: {{begin-eqn}} {{eqn | l = \nu_p \left({\frac a b + \dfrac c d}\right)       | r = \nu_p \left({\frac {a d + b c} {b d} }\right)       | c = {{Defof|Rational Addition}} }} {{eqn | r = \nu_p^\Z \left({a d + c b}\right) - \nu_p^\Z \left({b d}\right)       | c = {{Defof|P-adic Valuation|$p$-adic Valuation}} }} {{eqn | o = \ge       | r = \min \left\{ {\nu_p^\Z \left({a d}\right), \nu_p^\Z \left({c b}\right)}\right\} - \nu_p^\Z \left({b d}\right)       | c = $p$-adic Valuation on Integers is Valuation: Axiom $V3$ }} {{eqn | r = \min \left\{ {\nu_p^\Z \left({a}\right) + \nu_p^\Z \left({d}\right), \nu_p^\Z \left({c}\right) + \nu_p^\Z \left({b}\right)}\right\} - \nu_p^\Z \left({b}\right) - \nu_p^\Z \left({d}\right)       | c = $p$-adic Valuation on Integers is Valuation: Axiom $V1$ }} {{eqn | r = \min \left\{ {\nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right), \nu_p^\Z \left({c}\right) - \nu_p^\Z \left({d}\right)}\right\} }} {{eqn | r = \min \left\{ {\nu_p \left({\frac a b}\right), \nu_p \left({\frac c d}\right)}\right\}       | c = {{Defof|P-adic Valuation|$p$-adic Valuation}} }} {{end-eqn}} Hence: :$\nu_p \left({\dfrac a b + \dfrac c d}\right) \ge \min \left\{ {\nu_p \left({\dfrac a b}\right), \nu_p \left({\dfrac c d}\right)}\right\}$ Thus $\nu_p: \Q \to \Z \cup \left\{ {+\infty}\right\}$ is a valuation on $\Q$ by definition. {{qed}}