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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 )