Metamath Proof Explorer


Theorem mhp0cl

Description: The zero polynomial is homogeneous. Under df-mhp , it has any (nonnegative integer) degree which loosely corresponds to the value "undefined". The values -oo and 0 are also used in Metamath (by df-mdeg and df-dgr respectively) and the literature: https://math.stackexchange.com/a/1796314/593843 . (Contributed by SN, 12-Sep-2023)

Ref Expression
Hypotheses mhp0cl.h H=ImHomPR
mhp0cl.0 0˙=0R
mhp0cl.d D=h0I|h-1Fin
mhp0cl.i φIV
mhp0cl.r φRGrp
mhp0cl.n φN0
Assertion mhp0cl φD×0˙HN

Proof

Step Hyp Ref Expression
1 mhp0cl.h H=ImHomPR
2 mhp0cl.0 0˙=0R
3 mhp0cl.d D=h0I|h-1Fin
4 mhp0cl.i φIV
5 mhp0cl.r φRGrp
6 mhp0cl.n φN0
7 eqid ImPolyR=ImPolyR
8 eqid BaseImPolyR=BaseImPolyR
9 eqid 0ImPolyR=0ImPolyR
10 7 3 2 9 4 5 mpl0 φ0ImPolyR=D×0˙
11 7 mplgrp IVRGrpImPolyRGrp
12 4 5 11 syl2anc φImPolyRGrp
13 8 9 grpidcl ImPolyRGrp0ImPolyRBaseImPolyR
14 12 13 syl φ0ImPolyRBaseImPolyR
15 10 14 eqeltrrd φD×0˙BaseImPolyR
16 fczsupp0 D×0˙supp0˙=
17 0ss gD|fld𝑠0g=N
18 16 17 eqsstri D×0˙supp0˙gD|fld𝑠0g=N
19 18 a1i φD×0˙supp0˙gD|fld𝑠0g=N
20 1 7 8 2 3 4 5 6 15 19 ismhp2 φD×0˙HN