Theorem opsr0

Description: Zero in the ordered power series ring. (Contributed by Stefan O'Rear, 23-Mar-2015) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	opsr0.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
	opsr0.o	⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
	opsr0.t	⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
Assertion	opsr0	⊢ ( 𝜑 → ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑂 ) )

Proof

Step	Hyp	Ref	Expression
1		opsr0.s	⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2		opsr0.o	⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
3		opsr0.t	⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
4		eqidd	⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) )
5	1 2 3	opsrbas	⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑂 ) )
6	1 2 3	opsrplusg	⊢ ( 𝜑 → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑂 ) )
7	6	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑂 ) 𝑦 ) )
8	4 5 7	grpidpropd	⊢ ( 𝜑 → ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑂 ) )