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

Metamath Proof Explorer

Theorem rerecne0d

Description: The reciprocal of a nonzero number is nonzero. (Contributed by SN, 4-Apr-2026)

Ref Expression
Hypotheses sn-rereccld.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
sn-rereccld.z ⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion rerecne0d ( 𝜑 → ( 1 /ℝ 𝐴 ) ≠ 0 )

Proof

Step Hyp Ref Expression
1 sn-rereccld.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 sn-rereccld.z ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 ax-1ne0 ⊢ 1 ≠ 0
4 1red ⊢ ( 𝜑 → 1 ∈ ℝ )
5 4 1 2 redivne0bd ⊢ ( 𝜑 → ( 1 ≠ 0 ↔ ( 1 /ℝ 𝐴 ) ≠ 0 ) )
6 3 5 mpbii ⊢ ( 𝜑 → ( 1 /ℝ 𝐴 ) ≠ 0 )