Theorem opprnzr

Description: The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015)

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

Proof

Step	Hyp	Ref	Expression
1		opprnzr.1	⊢ 𝑂 = ( oppr ‘ 𝑅 )
2	1	opprnzrb	⊢ ( 𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing )
3	2	biimpi	⊢ ( 𝑅 ∈ NzRing → 𝑂 ∈ NzRing )