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 )