Theorem drnginvrcl

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

	Ref	Expression
Hypotheses	drnginvrcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	drnginvrcl.z	⊢ 0 = ( 0g ‘ 𝑅 )
	drnginvrcl.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
Assertion	drnginvrcl	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 )

Proof

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