Metamath Proof Explorer


Theorem ply1mpl0

Description: The univariate polynomial ring has the same zero as the corresponding multivariate polynomial ring. (Contributed by Stefan O'Rear, 23-Mar-2015) (Revised by Mario Carneiro, 3-Oct-2015)

Ref Expression
Hypotheses ply1mpl0.m
|- M = ( 1o mPoly R )
ply1mpl0.p
|- P = ( Poly1 ` R )
ply1mpl0.z
|- .0. = ( 0g ` P )
Assertion ply1mpl0
|- .0. = ( 0g ` M )

Proof

Step Hyp Ref Expression
1 ply1mpl0.m
 |-  M = ( 1o mPoly R )
2 ply1mpl0.p
 |-  P = ( Poly1 ` R )
3 ply1mpl0.z
 |-  .0. = ( 0g ` P )
4 eqidd
 |-  ( T. -> ( Base ` P ) = ( Base ` P ) )
5 eqid
 |-  ( Base ` P ) = ( Base ` P )
6 2 5 ply1bas
 |-  ( Base ` P ) = ( Base ` ( 1o mPoly R ) )
7 1 fveq2i
 |-  ( Base ` M ) = ( Base ` ( 1o mPoly R ) )
8 6 7 eqtr4i
 |-  ( Base ` P ) = ( Base ` M )
9 8 a1i
 |-  ( T. -> ( Base ` P ) = ( Base ` M ) )
10 eqid
 |-  ( +g ` P ) = ( +g ` P )
11 2 1 10 ply1plusg
 |-  ( +g ` P ) = ( +g ` M )
12 11 a1i
 |-  ( T. -> ( +g ` P ) = ( +g ` M ) )
13 12 oveqdr
 |-  ( ( T. /\ ( x e. ( Base ` P ) /\ y e. ( Base ` P ) ) ) -> ( x ( +g ` P ) y ) = ( x ( +g ` M ) y ) )
14 4 9 13 grpidpropd
 |-  ( T. -> ( 0g ` P ) = ( 0g ` M ) )
15 14 mptru
 |-  ( 0g ` P ) = ( 0g ` M )
16 3 15 eqtri
 |-  .0. = ( 0g ` M )