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		nzrring	⊢ ( 𝑅 ∈ NzRing → 𝑅 ∈ Ring )
3	1	opprring	⊢ ( 𝑅 ∈ Ring → 𝑂 ∈ Ring )
4	2 3	syl	⊢ ( 𝑅 ∈ NzRing → 𝑂 ∈ Ring )
5		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
6	5	isnzr2	⊢ ( 𝑅 ∈ NzRing ↔ ( 𝑅 ∈ Ring ∧ 2o ≼ ( Base ‘ 𝑅 ) ) )
7	6	simprbi	⊢ ( 𝑅 ∈ NzRing → 2o ≼ ( Base ‘ 𝑅 ) )
8	1 5	opprbas	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑂 )
9	8	isnzr2	⊢ ( 𝑂 ∈ NzRing ↔ ( 𝑂 ∈ Ring ∧ 2o ≼ ( Base ‘ 𝑅 ) ) )
10	4 7 9	sylanbrc	⊢ ( 𝑅 ∈ NzRing → 𝑂 ∈ NzRing )