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Metamath Proof Explorer

Theorem sn-rediv1d

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

Ref Expression
Hypothesis sn-rediv1d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion sn-rediv1d ( 𝜑 → ( 𝐴 /ℝ 1 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 sn-rediv1d.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 remullid ⊢ ( 𝐴 ∈ ℝ → ( 1 · 𝐴 ) = 𝐴 )
3 1 2 syl ⊢ ( 𝜑 → ( 1 · 𝐴 ) = 𝐴 )
4 1red ⊢ ( 𝜑 → 1 ∈ ℝ )
5 ax-1ne0 ⊢ 1 ≠ 0
6 5 a1i ⊢ ( 𝜑 → 1 ≠ 0 )
7 1 1 4 6 redivmuld ⊢ ( 𝜑 → ( ( 𝐴 /ℝ 1 ) = 𝐴 ↔ ( 1 · 𝐴 ) = 𝐴 ) )
8 3 7 mpbird ⊢ ( 𝜑 → ( 𝐴 /ℝ 1 ) = 𝐴 )