Theorem drnginvrrd

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

	Ref	Expression
Hypotheses	drnginvrld.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	drnginvrld.0	⊢ 0 = ( 0g ‘ 𝑅 )
	drnginvrld.t	⊢ · = ( .r ‘ 𝑅 )
	drnginvrld.u	⊢ 1 = ( 1r ‘ 𝑅 )
	drnginvrld.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
	drnginvrld.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
	drnginvrld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	drnginvrld.1	⊢ ( 𝜑 → 𝑋 ≠ 0 )
Assertion	drnginvrrd	⊢ ( 𝜑 → ( 𝑋 · ( 𝐼 ‘ 𝑋 ) ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		drnginvrld.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		drnginvrld.0	⊢ 0 = ( 0g ‘ 𝑅 )
3		drnginvrld.t	⊢ · = ( .r ‘ 𝑅 )
4		drnginvrld.u	⊢ 1 = ( 1r ‘ 𝑅 )
5		drnginvrld.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
6		drnginvrld.r	⊢ ( 𝜑 → 𝑅 ∈ DivRing )
7		drnginvrld.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8		drnginvrld.1	⊢ ( 𝜑 → 𝑋 ≠ 0 )
9	1 2 3 4 5	drnginvrr	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝑋 · ( 𝐼 ‘ 𝑋 ) ) = 1 )
10	6 7 8 9	syl3anc	⊢ ( 𝜑 → ( 𝑋 · ( 𝐼 ‘ 𝑋 ) ) = 1 )