Theorem mpllmod

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

	Ref	Expression
Hypothesis	mplgrp.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
Assertion	mpllmod	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ LMod )

Proof

Step	Hyp	Ref	Expression
1		mplgrp.p	⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
2		eqid	⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
3		simpl	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝐼 ∈ 𝑉 )
4		simpr	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑅 ∈ Ring )
5	2 3 4	psrlmod	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 𝐼 mPwSer 𝑅 ) ∈ LMod )
6		eqid	⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
7	2 1 6 3 4	mpllss	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( Base ‘ 𝑃 ) ∈ ( LSubSp ‘ ( 𝐼 mPwSer 𝑅 ) ) )
8	1 2 6	mplval2	⊢ 𝑃 = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ 𝑃 ) )
9		eqid	⊢ ( LSubSp ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( LSubSp ‘ ( 𝐼 mPwSer 𝑅 ) )
10	8 9	lsslmod	⊢ ( ( ( 𝐼 mPwSer 𝑅 ) ∈ LMod ∧ ( Base ‘ 𝑃 ) ∈ ( LSubSp ‘ ( 𝐼 mPwSer 𝑅 ) ) ) → 𝑃 ∈ LMod )
11	5 7 10	syl2anc	⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ LMod )