This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )