Theorem drnginvrn0

Description: The multiplicative inverse in a division ring is nonzero. ( recne0 analog). (Contributed by NM, 19-Apr-2014)

	Ref	Expression
Hypotheses	invrcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	invrcl.z	⊢ 0 = ( 0g ‘ 𝑅 )
	invrcl.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
Assertion	drnginvrn0	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		invrcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		invrcl.z	⊢ 0 = ( 0g ‘ 𝑅 )
3		invrcl.i	⊢ 𝐼 = ( invr ‘ 𝑅 )
4		drngring	⊢ ( 𝑅 ∈ DivRing → 𝑅 ∈ Ring )
5		eqid	⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
6	5 3	unitinvcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ ( Unit ‘ 𝑅 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) )
7	6	ex	⊢ ( 𝑅 ∈ Ring → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) ) )
8	4 7	syl	⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) → ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) ) )
9	1 5 2	drngunit	⊢ ( 𝑅 ∈ DivRing → ( 𝑋 ∈ ( Unit ‘ 𝑅 ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ) )
10	1 5 2	drngunit	⊢ ( 𝑅 ∈ DivRing → ( ( 𝐼 ‘ 𝑋 ) ∈ ( Unit ‘ 𝑅 ) ↔ ( ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑋 ) ≠ 0 ) ) )
11	8 9 10	3imtr3d	⊢ ( 𝑅 ∈ DivRing → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑋 ) ≠ 0 ) ) )
12	11	3impib	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( ( 𝐼 ‘ 𝑋 ) ∈ 𝐵 ∧ ( 𝐼 ‘ 𝑋 ) ≠ 0 ) )
13	12	simprd	⊢ ( ( 𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 𝐼 ‘ 𝑋 ) ≠ 0 )