Theorem xp1d2m1eqxm1d2

Description: A complex number increased by 1, then divided by 2, then decreased by 1 equals the complex number decreased by 1 and then divided by 2. (Contributed by AV, 24-May-2020)

	Ref	Expression
Assertion	xp1d2m1eqxm1d2	⊢ ( 𝑋 ∈ ℂ → ( ( ( 𝑋 + 1 ) / 2 ) − 1 ) = ( ( 𝑋 − 1 ) / 2 ) )

Proof

Step	Hyp	Ref	Expression
1		peano2cn	⊢ ( 𝑋 ∈ ℂ → ( 𝑋 + 1 ) ∈ ℂ )
2	1	halfcld	⊢ ( 𝑋 ∈ ℂ → ( ( 𝑋 + 1 ) / 2 ) ∈ ℂ )
3		peano2cnm	⊢ ( ( ( 𝑋 + 1 ) / 2 ) ∈ ℂ → ( ( ( 𝑋 + 1 ) / 2 ) − 1 ) ∈ ℂ )
4	2 3	syl	⊢ ( 𝑋 ∈ ℂ → ( ( ( 𝑋 + 1 ) / 2 ) − 1 ) ∈ ℂ )
5		peano2cnm	⊢ ( 𝑋 ∈ ℂ → ( 𝑋 − 1 ) ∈ ℂ )
6	5	halfcld	⊢ ( 𝑋 ∈ ℂ → ( ( 𝑋 − 1 ) / 2 ) ∈ ℂ )
7		2cnd	⊢ ( 𝑋 ∈ ℂ → 2 ∈ ℂ )
8		2ne0	⊢ 2 ≠ 0
9	8	a1i	⊢ ( 𝑋 ∈ ℂ → 2 ≠ 0 )
10		1cnd	⊢ ( 𝑋 ∈ ℂ → 1 ∈ ℂ )
11	2 10 7	subdird	⊢ ( 𝑋 ∈ ℂ → ( ( ( ( 𝑋 + 1 ) / 2 ) − 1 ) · 2 ) = ( ( ( ( 𝑋 + 1 ) / 2 ) · 2 ) − ( 1 · 2 ) ) )
12	1 7 9	divcan1d	⊢ ( 𝑋 ∈ ℂ → ( ( ( 𝑋 + 1 ) / 2 ) · 2 ) = ( 𝑋 + 1 ) )
13	7	mullidd	⊢ ( 𝑋 ∈ ℂ → ( 1 · 2 ) = 2 )
14	12 13	oveq12d	⊢ ( 𝑋 ∈ ℂ → ( ( ( ( 𝑋 + 1 ) / 2 ) · 2 ) − ( 1 · 2 ) ) = ( ( 𝑋 + 1 ) − 2 ) )
15	5 7 9	divcan1d	⊢ ( 𝑋 ∈ ℂ → ( ( ( 𝑋 − 1 ) / 2 ) · 2 ) = ( 𝑋 − 1 ) )
16		2m1e1	⊢ ( 2 − 1 ) = 1
17	16	a1i	⊢ ( 𝑋 ∈ ℂ → ( 2 − 1 ) = 1 )
18	17	oveq2d	⊢ ( 𝑋 ∈ ℂ → ( 𝑋 − ( 2 − 1 ) ) = ( 𝑋 − 1 ) )
19		id	⊢ ( 𝑋 ∈ ℂ → 𝑋 ∈ ℂ )
20	19 7 10	subsub3d	⊢ ( 𝑋 ∈ ℂ → ( 𝑋 − ( 2 − 1 ) ) = ( ( 𝑋 + 1 ) − 2 ) )
21	15 18 20	3eqtr2rd	⊢ ( 𝑋 ∈ ℂ → ( ( 𝑋 + 1 ) − 2 ) = ( ( ( 𝑋 − 1 ) / 2 ) · 2 ) )
22	11 14 21	3eqtrd	⊢ ( 𝑋 ∈ ℂ → ( ( ( ( 𝑋 + 1 ) / 2 ) − 1 ) · 2 ) = ( ( ( 𝑋 − 1 ) / 2 ) · 2 ) )
23	4 6 7 9 22	mulcan2ad	⊢ ( 𝑋 ∈ ℂ → ( ( ( 𝑋 + 1 ) / 2 ) − 1 ) = ( ( 𝑋 − 1 ) / 2 ) )