Theorem drnginvrcld

Description: Closure of the multiplicative inverse in a division ring. ( reccld analog). (Contributed by SN, 14-Aug-2024)

	Ref	Expression
Hypotheses	invrcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	invrcl.z	⊢ 0 = ( 0g ‘ 𝑅 )
	invrcl.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
	drnginvrcld.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
	drnginvrcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	drnginvrcld.1	⊢ ( 𝜑 → 𝑋 ≠ 0 )
Assertion	drnginvrcld	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		invrcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		invrcl.z	⊢ 0 = ( 0g ‘ 𝑅 )
3		invrcl.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
4		drnginvrcld.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
5		drnginvrcld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		drnginvrcld.1	⊢ ( 𝜑 → 𝑋 ≠ 0 )
7	1 2 3	drnginvrcl	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )
8	4 5 6 7	syl3anc	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )