Quantifier-Free Formula of Arithmetic is Provable

Theorem
Let $\phi$ be a sentence in the language of arithmetic.

Suppose $\phi$ contains no quantifiers.

Suppose also that $\N \models \phi$.
 * That is, the natural numbers model $\phi$.

Then $\phi$ is a theorem of minimal arithmetic.

Proof
By Existence of Negation Normal Form of Statement, $\phi$ is logically equivalent to a WFF $\psi$ such that:
 * The only logical connectives are $\set {\neg, \land, \lor}$.
 * The connective $\neg$ only appears in front of a predicate symbol.

Proceed by induction on the structure of WFFs in the language of arithmetic.

Predicate Symbol
Let $\alpha, \beta$ be terms.

Let $a, b \in \N$ be:
 * $a = \map {\operatorname{val}_\N} \alpha$
 * $b = \map {\operatorname{val}_\N} \beta$

where $\map {\operatorname{val}_\AA} \tau$ is the value of $\tau$.

Suppose $\psi$ is $\alpha = \beta$.

The result follows from Equality of Terms of Natural Numbers is Provable.

Suppose $\psi$ is $\alpha < \beta$.

Then $a < b$.

By Equality of Terms of Natural Numbers is Provable:\
 * $\alpha = \sqbrk a$
 * $\beta = \sqbrk b$

are theorems, where $\sqbrk x$ is the unary representation of $x$.

The result follows from Substitution Property of Equality and Ordering of Natural Numbers is Provable.

By definition of language of arithmetic, these are the only predicate symbols.

Negation
By definition of negation normal form, $\neg$ is only applied to predicate symbols.

That is, only in the form:
 * $\neg \map P {\tau_1, \dotsc, \tau_n}$

Let $\alpha, \beta, a, b$ be defined as above.

Suppose $\psi$ is $\neg {\alpha = \beta}$.

Then $a \ne b$.

The result follows from Substitution Property of Equality and Inequality of Natural Numbers is Provable.

Suppose $\psi$ is $\neg {\alpha < \beta}$.

Then $a \ge b$.

The result follows from Substitution Property of Equality and Negation of Ordering of Natural Numbers is Provable.

Connectives
Let $A, B$ be WFFs.

By definition of negation normal form, the only other connectives are $\land$ and $\lor$.

Suppose $\psi$ is:
 * $A \land B$

Then $\N \models A$ and $\N \models B$ by definition of value of formula.

By the inductive hypothesis, $A$ and $B$ are theorems.

The result follows from Rule of Conjunction.

Suppose $\psi$ is:
 * $A \lor B$

Then $\N \models A$ or $\N \models B$.

Suppose that $\N \models A$.

Then, by the inductive hypothesis, $A$ is a theorem.

The result follows from Rule of Addition.

Likewise, if $\N \models B$, the result follows from Rule of Addition.

By the Principle of Structural Induction, every $\psi$ in this form has this property.

But by the definition of logical equivalence, $\phi$ can be derived from $\psi$.

Hence the result.