Theorem fz0add1fz1

Description: Translate membership in a 0-based half-open integer range into membership in a 1-based finite sequence of integers. (Contributed by Alexander van der Vekens, 23-Nov-2017)

	Ref	Expression
Assertion	fz0add1fz1	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑋 + 1 ) ∈ ( 1 ... 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		1z	⊢ 1 ∈ ℤ
2		fzoaddel	⊢ ( ( 𝑋 ∈ ( 0 ..^ 𝑁 ) ∧ 1 ∈ ℤ ) → ( 𝑋 + 1 ) ∈ ( ( 0 + 1 ) ..^ ( 𝑁 + 1 ) ) )
3	1 2	mpan2	⊢ ( 𝑋 ∈ ( 0 ..^ 𝑁 ) → ( 𝑋 + 1 ) ∈ ( ( 0 + 1 ) ..^ ( 𝑁 + 1 ) ) )
4	3	adantl	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑋 + 1 ) ∈ ( ( 0 + 1 ) ..^ ( 𝑁 + 1 ) ) )
5		0p1e1	⊢ ( 0 + 1 ) = 1
6	5	oveq1i	⊢ ( ( 0 + 1 ) ..^ ( 𝑁 + 1 ) ) = ( 1 ..^ ( 𝑁 + 1 ) )
7		nn0z	⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ )
8		fzval3	⊢ ( 𝑁 ∈ ℤ → ( 1 ... 𝑁 ) = ( 1 ..^ ( 𝑁 + 1 ) ) )
9	8	eqcomd	⊢ ( 𝑁 ∈ ℤ → ( 1 ..^ ( 𝑁 + 1 ) ) = ( 1 ... 𝑁 ) )
10	7 9	syl	⊢ ( 𝑁 ∈ ℕ0 → ( 1 ..^ ( 𝑁 + 1 ) ) = ( 1 ... 𝑁 ) )
11	6 10	eqtrid	⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 + 1 ) ..^ ( 𝑁 + 1 ) ) = ( 1 ... 𝑁 ) )
12	11	eleq2d	⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑋 + 1 ) ∈ ( ( 0 + 1 ) ..^ ( 𝑁 + 1 ) ) ↔ ( 𝑋 + 1 ) ∈ ( 1 ... 𝑁 ) ) )
13	12	adantr	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → ( ( 𝑋 + 1 ) ∈ ( ( 0 + 1 ) ..^ ( 𝑁 + 1 ) ) ↔ ( 𝑋 + 1 ) ∈ ( 1 ... 𝑁 ) ) )
14	4 13	mpbid	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ ( 0 ..^ 𝑁 ) ) → ( 𝑋 + 1 ) ∈ ( 1 ... 𝑁 ) )