Theorem rlm0

Description: Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	rlm0	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ ( ringLMod ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		rlmval	⊢ ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) )
2	1	a1i	⊢ ( ⊤ → ( ringLMod ‘ 𝑅 ) = ( ( subringAlg ‘ 𝑅 ) ‘ ( Base ‘ 𝑅 ) ) )
3		eqidd	⊢ ( ⊤ → ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) )
4		ssidd	⊢ ( ⊤ → ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) )
5	2 3 4	sralmod0	⊢ ( ⊤ → ( 0g ‘ 𝑅 ) = ( 0g ‘ ( ringLMod ‘ 𝑅 ) ) )
6	5	mptru	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ ( ringLMod ‘ 𝑅 ) )