Metamath Proof Explorer

Theorem ply1baspropd

Description: Property deduction for univariate polynomial base set. (Contributed by Stefan O'Rear, 27-Mar-2015)

Ref Expression
Hypotheses ply1baspropd.b1 
|- ( ph -> B = ( Base ` R ) )
ply1baspropd.b2 
|- ( ph -> B = ( Base ` S ) )
ply1baspropd.p 
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` S ) y ) )
Assertion ply1baspropd 
|- ( ph -> ( Base ` ( Poly1 ` R ) ) = ( Base ` ( Poly1 ` S ) ) )

Proof

Step Hyp Ref Expression
1 ply1baspropd.b1 
 |-  ( ph -> B = ( Base ` R ) )
2 ply1baspropd.b2 
 |-  ( ph -> B = ( Base ` S ) )
3 ply1baspropd.p 
 |-  ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` S ) y ) )
4 1 2 3 mplbaspropd 
 |-  ( ph -> ( Base ` ( 1o mPoly R ) ) = ( Base ` ( 1o mPoly S ) ) )
5 eqid 
 |-  ( Poly1 ` R ) = ( Poly1 ` R )
6 eqid 
 |-  ( PwSer1 ` R ) = ( PwSer1 ` R )
7 eqid 
 |-  ( Base ` ( Poly1 ` R ) ) = ( Base ` ( Poly1 ` R ) )
8 5 6 7 ply1bas 
 |-  ( Base ` ( Poly1 ` R ) ) = ( Base ` ( 1o mPoly R ) )
9 eqid 
 |-  ( Poly1 ` S ) = ( Poly1 ` S )
10 eqid 
 |-  ( PwSer1 ` S ) = ( PwSer1 ` S )
11 eqid 
 |-  ( Base ` ( Poly1 ` S ) ) = ( Base ` ( Poly1 ` S ) )
12 9 10 11 ply1bas 
 |-  ( Base ` ( Poly1 ` S ) ) = ( Base ` ( 1o mPoly S ) )
13 4 8 12 3eqtr4g 
 |-  ( ph -> ( Base ` ( Poly1 ` R ) ) = ( Base ` ( Poly1 ` S ) ) )