Metamath Proof Explorer


Theorem subrgmpl

Description: A subring of the base ring induces a subring of polynomials. (Contributed by Mario Carneiro, 3-Jul-2015)

Ref Expression
Hypotheses subrgmpl.s
|- S = ( I mPoly R )
subrgmpl.h
|- H = ( R |`s T )
subrgmpl.u
|- U = ( I mPoly H )
subrgmpl.b
|- B = ( Base ` U )
Assertion subrgmpl
|- ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B e. ( SubRing ` S ) )

Proof

Step Hyp Ref Expression
1 subrgmpl.s
 |-  S = ( I mPoly R )
2 subrgmpl.h
 |-  H = ( R |`s T )
3 subrgmpl.u
 |-  U = ( I mPoly H )
4 subrgmpl.b
 |-  B = ( Base ` U )
5 simpl
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> I e. V )
6 simpr
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> T e. ( SubRing ` R ) )
7 eqid
 |-  ( I mPwSer H ) = ( I mPwSer H )
8 eqid
 |-  ( Base ` ( I mPwSer H ) ) = ( Base ` ( I mPwSer H ) )
9 eqid
 |-  ( Base ` S ) = ( Base ` S )
10 1 2 3 4 5 6 7 8 9 ressmplbas2
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B = ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) )
11 eqid
 |-  ( I mPwSer R ) = ( I mPwSer R )
12 11 2 7 8 subrgpsr
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( Base ` ( I mPwSer H ) ) e. ( SubRing ` ( I mPwSer R ) ) )
13 subrgrcl
 |-  ( T e. ( SubRing ` R ) -> R e. Ring )
14 13 adantl
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> R e. Ring )
15 11 1 9 5 14 mplsubrg
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( Base ` S ) e. ( SubRing ` ( I mPwSer R ) ) )
16 subrgin
 |-  ( ( ( Base ` ( I mPwSer H ) ) e. ( SubRing ` ( I mPwSer R ) ) /\ ( Base ` S ) e. ( SubRing ` ( I mPwSer R ) ) ) -> ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) e. ( SubRing ` ( I mPwSer R ) ) )
17 12 15 16 syl2anc
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) e. ( SubRing ` ( I mPwSer R ) ) )
18 10 17 eqeltrd
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B e. ( SubRing ` ( I mPwSer R ) ) )
19 inss2
 |-  ( ( Base ` ( I mPwSer H ) ) i^i ( Base ` S ) ) C_ ( Base ` S )
20 10 19 eqsstrdi
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B C_ ( Base ` S ) )
21 1 11 9 mplval2
 |-  S = ( ( I mPwSer R ) |`s ( Base ` S ) )
22 21 subsubrg
 |-  ( ( Base ` S ) e. ( SubRing ` ( I mPwSer R ) ) -> ( B e. ( SubRing ` S ) <-> ( B e. ( SubRing ` ( I mPwSer R ) ) /\ B C_ ( Base ` S ) ) ) )
23 15 22 syl
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> ( B e. ( SubRing ` S ) <-> ( B e. ( SubRing ` ( I mPwSer R ) ) /\ B C_ ( Base ` S ) ) ) )
24 18 20 23 mpbir2and
 |-  ( ( I e. V /\ T e. ( SubRing ` R ) ) -> B e. ( SubRing ` S ) )