Working on Lang, Serge. "Basic Mathematics" (p. 100, example).
Let $S$ be the set of numbers x such that $1 \leq x \leq 2$. Let $T$ be the set of all numbers $5x$ with all x in $S$. We contend that $T$ is the set of numbers $y$ with $5 \leq y \leq 10$.
$ \def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \def\Ae#1{\qquad\mathbf{\forall E} \: #1 \\} \def\Ai#1{\qquad\mathbf{\forall I} \: #1 \\} \def\Ee#1{\qquad\mathbf{\exists E} \: #1 \\} \def\Ei#1{\qquad\mathbf{\exists I} \: #1 \\} \def\R#1{\qquad\mathbf{R} \: #1 \\} \def\ci#1{\qquad\mathbf{\land I} \: #1 \\} \def\ce#1{\qquad\mathbf{\land E} \: #1 \\} \def\ii#1{\qquad\mathbf{\to I} \: #1 \\} \def\ie#1{\qquad\mathbf{\to E} \: #1 \\} \def\be#1{\qquad\mathbf{\leftrightarrow E} \: #1 \\} \def\bi#1{\qquad\mathbf{\leftrightarrow I} \: #1 \\} \def\qi#1{\qquad\mathbf{=I}\\} \def\qe#1{\qquad\mathbf{=E} \: #1 \\} \def\ne#1{\qquad\mathbf{\neg E} \: #1 \\} \def\ni#1{\qquad\mathbf{\neg I} \: #1 \\} \def\IP#1{\qquad\mathbf{IP} \: #1 \\} \def\x#1{\qquad\mathbf{X} \: #1 \\} \def\DNE#1{\qquad\mathbf{DNE} \: #1 \\} $ $ \fitch{1.\, \forall x(x \in T \leftrightarrow \exists y(5 \leq 5y \leq 10 \land x=5y))\\ 2.\,\forall x(x \in T' \leftrightarrow (5 \leq x \leq 10))}{ \fitch{3.\, a \in T}{ 4.\,a \in T \leftrightarrow \exists y(5 \leq 5y \leq 10 \land a=5y) \Ae{1} 5.\,\exists y(5 \leq 5y \leq 10 \land a=5y) \be{4,3} \fitch{6.\, (5 \leq 5k \leq 10) \land a=5k}{ 7.\,(5 \leq 5k \leq 10) \ce{6} 8.\,a=5k \ce{6} 9.\,5 \leq a \leq 10 \qe{} }\\ 10.\,5 \leq a \leq 10 \Ee{5,6-9} 11.\,a \in T' \leftrightarrow 5 \leq a \leq 10 \Ae{2} 12.\,a \in T' \be{11,10} }\\ \fitch{13.\, a \in T'}{ 14.\,a \in T' \leftrightarrow 5 \leq a \leq 10 \Ae{2} 15.\,5 \leq a \leq 10 \be{14,13} 16.\,\exists y(a=y/5) \qquad?\\ \vdots\\ \exists y(5 \leq 5y \leq 10 \land a=5y)\\ a \in T \leftrightarrow \exists y(5 \leq 5y \leq 10 \land a=5y)\\ a \in T }\\ a \in T \leftrightarrow a \in T'\\ \forall x(x \in T \leftrightarrow x \in T')\\ T=T' } $
I have three questions:
- Are the premises correctly symbolised ?
- How can I justify line 16 ?
- How can I complete this proof ?