Theorem drnginvrl

Description: Property of the multiplicative inverse in a division ring. ( recid2 analog.) (Contributed by NM, 19-Apr-2014)

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

Proof

Step	Hyp	Ref	Expression
1		drnginvrl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		drnginvrl.z	⊢ 0 = ( 0g ‘ 𝑅 )
3		drnginvrl.t	⊢ · = ( .r ‘ 𝑅 )
4		drnginvrl.u	⊢ 1 = ( 1r ‘ 𝑅 )
5		drnginvrl.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
6		eqid	⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
7	1 6 2	drngunit	⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ) )
8		drngring	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )
9	6 5 3 4	unitlinv	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ ( Unit ‘ 𝑅 ) ) → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 )
10	9	ex	⊢ ( 𝑅 ∈ Ring → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 ) )
11	8 10	syl	⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 ) )
12	7 11	sylbird	⊢ ( 𝑅 ∈ DivRing → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 ) )
13	12	3impib	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( 𝐼 ‘ 𝑋 ) · 𝑋 ) = 1 )