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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 
 |-  ( Base ` ( Poly1 ` R ) ) = ( Base ` ( Poly1 ` R ) )
7 5 6 ply1bas 
 |-  ( Base ` ( Poly1 ` R ) ) = ( Base ` ( 1o mPoly R ) )
8 eqid 
 |-  ( Poly1 ` S ) = ( Poly1 ` S )
9 eqid 
 |-  ( Base ` ( Poly1 ` S ) ) = ( Base ` ( Poly1 ` S ) )
10 8 9 ply1bas 
 |-  ( Base ` ( Poly1 ` S ) ) = ( Base ` ( 1o mPoly S ) )
11 4 7 10 3eqtr4g 
 |-  ( ph -> ( Base ` ( Poly1 ` R ) ) = ( Base ` ( Poly1 ` S ) ) )