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 = ( Poly1𝑅 )
ply1mpl0.z 0 = ( 0g𝑃 )
Assertion ply1mpl0 0 = ( 0g𝑀 )

Proof

Step Hyp Ref Expression
1 ply1mpl0.m 𝑀 = ( 1o mPoly 𝑅 )
2 ply1mpl0.p 𝑃 = ( Poly1𝑅 )
3 ply1mpl0.z 0 = ( 0g𝑃 )
4 eqidd ( ⊤ → ( 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 ( ⊤ → ( Base ‘ 𝑃 ) = ( Base ‘ 𝑀 ) )
10 eqid ( +g𝑃 ) = ( +g𝑃 )
11 2 1 10 ply1plusg ( +g𝑃 ) = ( +g𝑀 )
12 11 a1i ( ⊤ → ( +g𝑃 ) = ( +g𝑀 ) )
13 12 oveqdr ( ( ⊤ ∧ ( 𝑥 ∈ ( Base ‘ 𝑃 ) ∧ 𝑦 ∈ ( Base ‘ 𝑃 ) ) ) → ( 𝑥 ( +g𝑃 ) 𝑦 ) = ( 𝑥 ( +g𝑀 ) 𝑦 ) )
14 4 9 13 grpidpropd ( ⊤ → ( 0g𝑃 ) = ( 0g𝑀 ) )
15 14 mptru ( 0g𝑃 ) = ( 0g𝑀 )
16 3 15 eqtri 0 = ( 0g𝑀 )