Let $\eqclass x {16}, \eqclass y {16} \in \struct {\Z_{16}, +}$. Then: {{begin-eqn}} {{eqn | l = \map \phi {\eqclass x {16} } \times \map \phi {\eqclass y {16} }       | r = \map \phi {x + 16 m_1} \times \map \phi {y + 16 m_2}       | c = {{Defof|Residue Class}}: for some representative $m_1, m_2 \in \Z$ }} {{eqn | r = 3 \uparrow \paren {x + 16 m_1} \times 3 \uparrow \paren {y + 16 m_2}       | c = using Knuth uparrow notation $3 \uparrow k := 3^k$ }} {{eqn | r = 3 \uparrow \paren {x + 16 m_1 + y + 16 m_2}       | c = Product of Powers }} {{eqn | r = 3 \uparrow \paren {\paren {x + y} + 16 \paren {m_1 + m_2} }       | c =  }} {{eqn | r = 3 \uparrow \paren {\eqclass {x + y} {16} }       | c = {{Defof|Residue Class}} and {{Defof|Modulo Addition}} }} {{eqn | r = \map \phi {\eqclass x {16} + \eqclass y {16} }       | c = Definition of $\phi$ }} {{end-eqn}} Thus it is seen that $\phi$ is a group homomorphism. {{qed|lemma}} It remains to be seen that $\phi$ is a bijection. Because $17$ is prime: $\forall x \in \Z, 1 \le x < 17: x \perp 17$ where $\perp$ denotes coprimality. Thus by definition of multiplicative group of reduced residues modulo $17$: :$\order {\struct {\Z'_{17}, \times} } = 16$ where $\order {\, \cdot \,}$ denotes the order of a group. Similarly, by definition of additive group of integers modulo $16$: :$\order {\struct {\Z_{16}, +} } = 16$ So: :$\order {\struct {\Z'_{17}, \times} } = \order {\struct {\Z_{16}, +} }$ which is a necessary condition for group isomorphism. {{qed|lemma}} Now we have: {{begin-eqn}} {{eqn | l = 16       | o = \equiv       | r = 0       | rr= \pmod {16}       | c =  }} {{eqn | ll= \leadsto       | l = \map \phi {\eqclass {16} {16} }       | r = \map \phi {\eqclass 0 {16} }       | c =  }} {{eqn | ll= \leadsto       | l = \map \phi {\eqclass {16} {16} }       | r = \eqclass 1 {17}       | c = Group Homomorphism Preserves Identity }} {{eqn | n = 1       | ll= \leadsto       | l = 3^{16}       | o = \equiv       | r = 1       | rr= \pmod {17}       | c = Definition of $\phi$ }} {{end-eqn}} Now let $\eqclass x {16}, \eqclass y {16} \in \Z_{16}$ such that $\map \phi {\eqclass x {16} } = \map \phi {\eqclass y {16} }$. We have: {{begin-eqn}} {{eqn | l = \map \phi {\eqclass x {16} }       | r = \map \phi {\eqclass y {16} }       | c =  }} {{eqn | ll= \leadsto       | lo= \forall m_1, m_2 \in \Z       | l = \map \phi {x + 16 m_1}       | r = \map \phi {y + 16 m_2}       | c = {{Defof|Residue Class}} }} {{eqn | ll= \leadsto       | l = 3 \uparrow \paren {x + 16 m_1}       | r = 3 \uparrow \paren {y + 16 m_2}       | c = Definition of $\phi$ }} {{eqn | ll= \leadsto       | l = 3^x \paren {3^{16} }^{m_1}       | r = 3^y \paren {3^{16} }^{m_2}       | c = Product of Powers, Power of Power }} {{eqn | ll= \leadsto       | l = 3^x \times 1^{m_1}       | r = 3^y \times 1^{m_2}       | c = as $3^{16} = 1 \pmod {17}$ from $(1)$ }} {{eqn | ll= \leadsto       | l = 3^x       | r = 3^y       | c =  }} {{eqn | ll= \leadsto       | l = x       | r = y       | c =  }} {{end-eqn}} Thus $\phi$ is an injection. From Equivalence of Mappings between Sets of Same Cardinality it follows that $\phi$ is a bijection. {{qed|lemma}} Thus $\phi$ is a bijective group homomorphism. Hence the result by definition of group isomorphism. {{Qed}}