Theorem rspssid

Description: The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015)

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

Proof

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