Theorem rsp0

Description: The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015)

	Ref	Expression
Hypotheses	rspcl.k	⊢ 𝐾 = ( RSpan ‘ 𝑅 )
	rsp0.z	⊢ 0 = ( 0g ‘ 𝑅 )
Assertion	rsp0	⊢ ( 𝑅 ∈ Ring → ( 𝐾 ‘ { 0 } ) = { 0 } )

Proof

Step	Hyp	Ref	Expression
1		rspcl.k	⊢ 𝐾 = ( RSpan ‘ 𝑅 )
2		rsp0.z	⊢ 0 = ( 0g ‘ 𝑅 )
3		rlmlmod	⊢ ( 𝑅 ∈ Ring → ( ringLMod ‘ 𝑅 ) ∈ LMod )
4		rlm0	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ ( ringLMod ‘ 𝑅 ) )
5	2 4	eqtri	⊢ 0 = ( 0g ‘ ( ringLMod ‘ 𝑅 ) )
6		rspval	⊢ ( RSpan ‘ 𝑅 ) = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
7	1 6	eqtri	⊢ 𝐾 = ( LSpan ‘ ( ringLMod ‘ 𝑅 ) )
8	5 7	lspsn0	⊢ ( ( ringLMod ‘ 𝑅 ) ∈ LMod → ( 𝐾 ‘ { 0 } ) = { 0 } )
9	3 8	syl	⊢ ( 𝑅 ∈ Ring → ( 𝐾 ‘ { 0 } ) = { 0 } )