Metamath Proof Explorer

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∈V∧C∈ℤ→x∈ℤV⟼C∈mzPoly⁡V

Proof

Step Hyp Ref Expression
1 fconstmpt ⊢ℤV×C=x∈ℤV⟼C
2 mzpconst ⊢V∈V∧C∈ℤ→ℤV×C∈mzPoly⁡V
3 1 2 eqeltrrid ⊢V∈V∧C∈ℤ→x∈ℤV⟼C∈mzPoly⁡V