Theorem 2halves

Description: Two halves make a whole. (Contributed by NM, 11-Apr-2005)

	Ref	Expression
Assertion	2halves	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 / 2 ) + ( 𝐴 / 2 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		2times	⊢ ( 𝐴 ∈ ℂ → ( 2 · 𝐴 ) = ( 𝐴 + 𝐴 ) )
2	1	oveq1d	⊢ ( 𝐴 ∈ ℂ → ( ( 2 · 𝐴 ) / 2 ) = ( ( 𝐴 + 𝐴 ) / 2 ) )
3		2cn	⊢ 2 ∈ ℂ
4		2ne0	⊢ 2 ≠ 0
5		divcan3	⊢ ( ( 𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( ( 2 · 𝐴 ) / 2 ) = 𝐴 )
6	3 4 5	mp3an23	⊢ ( 𝐴 ∈ ℂ → ( ( 2 · 𝐴 ) / 2 ) = 𝐴 )
7		2cnne0	⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 )
8		divdir	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( 𝐴 + 𝐴 ) / 2 ) = ( ( 𝐴 / 2 ) + ( 𝐴 / 2 ) ) )
9	7 8	mp3an3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐴 + 𝐴 ) / 2 ) = ( ( 𝐴 / 2 ) + ( 𝐴 / 2 ) ) )
10	9	anidms	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 + 𝐴 ) / 2 ) = ( ( 𝐴 / 2 ) + ( 𝐴 / 2 ) ) )
11	2 6 10	3eqtr3rd	⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 / 2 ) + ( 𝐴 / 2 ) ) = 𝐴 )