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Metamath Proof Explorer

Theorem sn-redividd

Description: A number divided by itself is 1. (Contributed by SN, 2-Apr-2026)

Ref Expression
Hypotheses sn-rediv0d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
sn-rediv0d.z ⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion sn-redividd ( 𝜑 → ( 𝐴 /ℝ 𝐴 ) = 1 )

Proof

Step Hyp Ref Expression
1 sn-rediv0d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 sn-rediv0d.z ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 eqidd ⊢ ( 𝜑 → 𝐴 = 𝐴 )
4 1 1 2 rediveq1d ⊢ ( 𝜑 → ( ( 𝐴 /ℝ 𝐴 ) = 1 ↔ 𝐴 = 𝐴 ) )
5 3 4 mpbird ⊢ ( 𝜑 → ( 𝐴 /ℝ 𝐴 ) = 1 )