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Metamath Proof Explorer

Theorem ply1basOLD

Description: Obsolete version of ply1bas as of 20-May-2025. (Contributed by Mario Carneiro, 9-Feb-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ply1val.1 
|- P = ( Poly1 ` R )
ply1lss.2 
|- S = ( PwSer1 ` R )
ply1lss.u 
|- U = ( Base ` P )
Assertion ply1basOLD 
|- U = ( Base ` ( 1o mPoly R ) )

Proof

Step Hyp Ref Expression
1 ply1val.1 
 |-  P = ( Poly1 ` R )
2 ply1lss.2 
 |-  S = ( PwSer1 ` R )
3 ply1lss.u 
 |-  U = ( Base ` P )
4 eqid 
 |-  ( 1o mPoly R ) = ( 1o mPoly R )
5 eqid 
 |-  ( 1o mPwSer R ) = ( 1o mPwSer R )
6 eqid 
 |-  ( Base ` ( 1o mPoly R ) ) = ( Base ` ( 1o mPoly R ) )
7 eqid 
 |-  ( Base ` S ) = ( Base ` S )
8 2 7 5 psr1bas2 
 |-  ( Base ` S ) = ( Base ` ( 1o mPwSer R ) )
9 4 5 6 8 mplbasss 
 |-  ( Base ` ( 1o mPoly R ) ) C_ ( Base ` S )
10 1 2 ply1val 
 |-  P = ( S |`s ( Base ` ( 1o mPoly R ) ) )
11 10 7 ressbas2 
 |-  ( ( Base ` ( 1o mPoly R ) ) C_ ( Base ` S ) -> ( Base ` ( 1o mPoly R ) ) = ( Base ` P ) )
12 9 11 ax-mp 
 |-  ( Base ` ( 1o mPoly R ) ) = ( Base ` P )
13 3 12 eqtr4i 
 |-  U = ( Base ` ( 1o mPoly R ) )