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Metamath Proof Explorer

Theorem ressmplbas

Description: A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015)

Ref Expression
Hypotheses ressmpl.s 
|- S = ( I mPoly R )
ressmpl.h 
|- H = ( R |`s T )
ressmpl.u 
|- U = ( I mPoly H )
ressmpl.b 
|- B = ( Base ` U )
ressmpl.1 
|- ( ph -> I e. V )
ressmpl.2 
|- ( ph -> T e. ( SubRing ` R ) )
ressmpl.p 
|- P = ( S |`s B )
Assertion ressmplbas 
|- ( ph -> B = ( Base ` P ) )

Proof

Step Hyp Ref Expression
1 ressmpl.s 
 |-  S = ( I mPoly R )
2 ressmpl.h 
 |-  H = ( R |`s T )
3 ressmpl.u 
 |-  U = ( I mPoly H )
4 ressmpl.b 
 |-  B = ( Base ` U )
5 ressmpl.1 
 |-  ( ph -> I e. V )
6 ressmpl.2 
 |-  ( ph -> T e. ( SubRing ` R ) )
7 ressmpl.p 
 |-  P = ( S |`s B )
8 eqid 
 |-  ( I mPwSer H ) = ( I mPwSer H )
9 eqid 
 |-  ( Base ` ( I mPwSer H ) ) = ( Base ` ( I mPwSer H ) )
10 eqid 
 |-  ( Base ` S ) = ( Base ` S )
11 1 2 3 4 5 6 8 9 10 ressmplbas2 
 |-  ( ph -> B = ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) )
12 inss2 
 |-  ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) C_ ( Base ` S )
13 11 12 eqsstrdi 
 |-  ( ph -> B C_ ( Base ` S ) )
14 7 10 ressbas2 
 |-  ( B C_ ( Base ` S ) -> B = ( Base ` P ) )
15 13 14 syl 
 |-  ( ph -> B = ( Base ` P ) )