Theorem rspcl

Description: The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)

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

Proof

Step	Hyp	Ref	Expression
1		rspcl.k	⊢ 𝐾 = ( RSpan ‘ 𝑅 )
2		rspcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		rspcl.u	⊢ 𝑈 = ( LIdeal ‘ 𝑅 )
4		rlmlmod	⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod )
5		rlmbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ ( ringLMod ‘ 𝑅 ) )
6	2 5	eqtri	⊢ 𝐵 = ( Base ‘ ( ringLMod ‘ 𝑅 ) )
7		lidlval	⊢ ( LIdeal ‘ 𝑅 ) = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
8	3 7	eqtri	⊢ 𝑈 = ( LSubSp ‘ ( ringLMod ‘ 𝑅 ) )
9		rspval	⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
10	1 9	eqtri	⊢ 𝐾 = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
11	6 8 10	lspcl	⊢ ( ( ( ringLMod ‘ 𝑅 ) ∈ LMod ∧ 𝐺 ⊆ 𝐵 ) → ( 𝐾 ‘ 𝐺 ) ∈ 𝑈 )
12	4 11	sylan	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐺 ⊆ 𝐵 ) → ( 𝐾 ‘ 𝐺 ) ∈ 𝑈 )