Theorem opprnzrb

Description: The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr . (Contributed by SN, 20-Jun-2025)

	Ref	Expression
Hypothesis	opprnzr.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
Assertion	opprnzrb	⊢ ( 𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing )

Proof

Step	Hyp	Ref	Expression
1		opprnzr.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
2	1	opprringb	⊢ ( 𝑅 ∈ Ring ↔ 𝑂 ∈ Ring )
3	2	anbi1i	⊢ ( ( 𝑅 ∈ Ring ∧ ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) ) ↔ ( 𝑂 ∈ Ring ∧ ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) ) )
4		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
5		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
6	4 5	isnzr	⊢ ( 𝑅 ∈ NzRing ↔ ( 𝑅 ∈ Ring ∧ ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) ) )
7	1 4	oppr1	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑂 )
8	1 5	oppr0	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑂 )
9	7 8	isnzr	⊢ ( 𝑂 ∈ NzRing ↔ ( 𝑂 ∈ Ring ∧ ( 1r ‘ 𝑅 ) ≠ ( 0g ‘ 𝑅 ) ) )
10	3 6 9	3bitr4i	⊢ ( 𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing )