Theorem drgext0g

Description: The additive neutral element of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023)

	Ref	Expression
Hypotheses	drgext.b	⊢ 𝐵 = ( ( subringAlg ‘ 𝐸 ) ‘ 𝑈 )
	drgext.1	⊢ ( 𝜑 → 𝐸 ∈ DivRing )
	drgext.2	⊢ ( 𝜑 → 𝑈 ∈ ( SubRing ‘ 𝐸 ) )
Assertion	drgext0g	⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		drgext.b	⊢ 𝐵 = ( ( subringAlg ‘ 𝐸 ) ‘ 𝑈 )
2		drgext.1	⊢ ( 𝜑 → 𝐸 ∈ DivRing )
3		drgext.2	⊢ ( 𝜑 → 𝑈 ∈ ( SubRing ‘ 𝐸 ) )
4	1	a1i	⊢ ( 𝜑 → 𝐵 = ( ( subringAlg ‘ 𝐸 ) ‘ 𝑈 ) )
5		eqidd	⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐸 ) )
6		eqid	⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 )
7	6	subrgss	⊢ ( 𝑈 ∈ ( SubRing ‘ 𝐸 ) → 𝑈 ⊆ ( Base ‘ 𝐸 ) )
8	3 7	syl	⊢ ( 𝜑 → 𝑈 ⊆ ( Base ‘ 𝐸 ) )
9	4 5 8	sralmod0	⊢ ( 𝜑 → ( 0g ‘ 𝐸 ) = ( 0g ‘ 𝐵 ) )