Theorem mzpconstmpt

Description: A constant function expressed in maps-to notation is polynomial. This theorem and the several that follow ( mzpaddmpt , mzpmulmpt , mzpnegmpt , mzpsubmpt , mzpexpmpt ) can be used to build proofs that functions which are "manifestly polynomial", in the sense of being a maps-to containing constants, projections, and simple arithmetic operations, are actually polynomial functions. There is no mzpprojmpt because mzpproj is already expressed using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014)

	Ref	Expression
Assertion	mzpconstmpt	|- ( ( V e. _V /\ C e. ZZ ) -> ( x e. ( ZZ ^m V ) |-> C ) e. ( mzPoly ` V ) )

Proof

Step	Hyp	Ref	Expression
1		fconstmpt	|- ( ( ZZ ^m V ) X. { C } ) = ( x e. ( ZZ ^m V ) |-> C )
2		mzpconst	|- ( ( V e. _V /\ C e. ZZ ) -> ( ( ZZ ^m V ) X. { C } ) e. ( mzPoly ` V ) )
3	1 2	eqeltrrid	|- ( ( V e. _V /\ C e. ZZ ) -> ( x e. ( ZZ ^m V ) |-> C ) e. ( mzPoly ` V ) )