Theorem mplring

Description: The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015)

	Ref	Expression
Hypothesis	mplgrp.p	|- P = ( I mPoly R )
Assertion	mplring	|- ( ( I e. V /\ R e. Ring ) -> P e. Ring )

Proof

Step	Hyp	Ref	Expression
1		mplgrp.p	|- P = ( I mPoly R )
2		eqid	|- ( I mPwSer R ) = ( I mPwSer R )
3		eqid	|- ( Base ` P ) = ( Base ` P )
4		simpl	|- ( ( I e. V /\ R e. Ring ) -> I e. V )
5		simpr	|- ( ( I e. V /\ R e. Ring ) -> R e. Ring )
6	2 1 3 4 5	mplsubrg	|- ( ( I e. V /\ R e. Ring ) -> ( Base ` P ) e. ( SubRing ` ( I mPwSer R ) ) )
7	1 2 3	mplval2	|- P = ( ( I mPwSer R ) |`s ( Base ` P ) )
8	7	subrgring	|- ( ( Base ` P ) e. ( SubRing ` ( I mPwSer R ) ) -> P e. Ring )
9	6 8	syl	|- ( ( I e. V /\ R e. Ring ) -> P e. Ring )