Metamath Proof Explorer


Theorem fzrev3i

Description: The "complement" of a member of a finite set of sequential integers. (Contributed by NM, 20-Nov-2005)

Ref Expression
Assertion fzrev3i ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → ( ( 𝑀 + 𝑁 ) − 𝐾 ) ∈ ( 𝑀 ... 𝑁 ) )

Proof

Step Hyp Ref Expression
1 elfzelz ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝐾 ∈ ℤ )
2 fzrev3 ( 𝐾 ∈ ℤ → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( ( 𝑀 + 𝑁 ) − 𝐾 ) ∈ ( 𝑀 ... 𝑁 ) ) )
3 1 2 syl ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( ( 𝑀 + 𝑁 ) − 𝐾 ) ∈ ( 𝑀 ... 𝑁 ) ) )
4 3 ibi ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → ( ( 𝑀 + 𝑁 ) − 𝐾 ) ∈ ( 𝑀 ... 𝑁 ) )