Theorem sralmod0

Description: The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014)

	Ref	Expression
Hypotheses	sralmod0.a	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
	sralmod0.z	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝑊 ) )
	sralmod0.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
Assertion	sralmod0	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		sralmod0.a	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
2		sralmod0.z	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝑊 ) )
3		sralmod0.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
4		eqidd	⊢ ( 𝜑 → ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) )
5	1 3	srabase	⊢ ( 𝜑 → ( Base ‘ 𝑊 ) = ( Base ‘ 𝐴 ) )
6	1 3	sraaddg	⊢ ( 𝜑 → ( +g ‘ 𝑊 ) = ( +g ‘ 𝐴 ) )
7	6	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Base ‘ 𝑊 ) ∧ 𝑏 ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑎 ( +g ‘ 𝑊 ) 𝑏 ) = ( 𝑎 ( +g ‘ 𝐴 ) 𝑏 ) )
8	4 5 7	grpidpropd	⊢ ( 𝜑 → ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝐴 ) )
9	2 8	eqtrd	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝐴 ) )