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 ∈ ℤ V ⟼ C ∈ mzPoly ⁡ V

Proof

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