Theorem opprdrng

Description: The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014)

	Ref	Expression
Hypothesis	opprdrng.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
Assertion	opprdrng	⊢ ( 𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing )

Proof

Step	Hyp	Ref	Expression
1		opprdrng.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
2	1	opprringb	⊢ ( 𝑅 ∈ Ring ↔ 𝑂 ∈ Ring )
3	2	anbi1i	⊢ ( ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ) ↔ ( 𝑂 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ) )
4		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
5		eqid	⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑅 )
6		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
7	4 5 6	isdrng	⊢ ( 𝑅 ∈ DivRing ↔ ( 𝑅 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ) )
8	1 4	opprbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 )
9	5 1	opprunit	⊢ ( Unit ‘ 𝑅 ) = ( Unit ‘ 𝑂 )
10	1 6	oppr0	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑂 )
11	8 9 10	isdrng	⊢ ( 𝑂 ∈ DivRing ↔ ( 𝑂 ∈ Ring ∧ ( Unit ‘ 𝑅 ) = ( ( Base ‘ 𝑅 ) ∖ { ( 0g ‘ 𝑅 ) } ) ) )
12	3 7 11	3bitr4i	⊢ ( 𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing )