Theorem rspssp

Description: The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015)

	Ref	Expression
Hypotheses	rspcl.k	⊢ 𝐾 = ( RSpan ‘ 𝑅 )
	rspssp.u	⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
Assertion	rspssp	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ⊆ 𝐼 ) → ( 𝐾 ‘ 𝐺 ) ⊆ 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		rspcl.k	⊢ 𝐾 = ( RSpan ‘ 𝑅 )
2		rspssp.u	⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
3		rlmlmod	⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod )
4		lidlval	⊢ ( LIdeal ‘ 𝑅 ) = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
5	2 4	eqtri	⊢ 𝑈 = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
6		rspval	⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
7	1 6	eqtri	⊢ 𝐾 = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
8	5 7	lspssp	⊢ ( ( ( ringLMod ‘ 𝑅 ) ∈ LMod ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ⊆ 𝐼 ) → ( 𝐾 ‘ 𝐺 ) ⊆ 𝐼 )
9	3 8	syl3an1	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐺 ⊆ 𝐼 ) → ( 𝐾 ‘ 𝐺 ) ⊆ 𝐼 )