Theorem div1

Description: A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	div1	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 / 1 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mulid2	⊢ ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = 𝐴 )
2		ax-1cn	⊢ 1 ∈ ℂ
3		ax-1ne0	⊢ 1 ≠ 0
4	2 3	pm3.2i	⊢ ( 1 ∈ ℂ ∧ 1 ≠ 0 )
5		divmul	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ( 1 ∈ ℂ ∧ 1 ≠ 0 ) ) → ( ( 𝐴 / 1 ) = 𝐴 ↔ ( 1 · 𝐴 ) = 𝐴 ) )
6	4 5	mp3an3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐴 / 1 ) = 𝐴 ↔ ( 1 · 𝐴 ) = 𝐴 ) )
7	6	anidms	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 / 1 ) = 𝐴 ↔ ( 1 · 𝐴 ) = 𝐴 ) )
8	1 7	mpbird	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 / 1 ) = 𝐴 )