Metamath Proof Explorer


Theorem elfzole1

Description: A member in a half-open integer interval is greater than or equal to the lower bound. (Contributed by Stefan O'Rear, 15-Aug-2015)

Ref Expression
Assertion elfzole1 ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀𝐾 )

Proof

Step Hyp Ref Expression
1 elfzoelz ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝐾 ∈ ℤ )
2 elfzoel1 ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ℤ )
3 elfzoel2 ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
4 elfzo ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀𝐾𝐾 < 𝑁 ) ) )
5 1 2 3 4 syl3anc ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀𝐾𝐾 < 𝑁 ) ) )
6 5 ibi ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → ( 𝑀𝐾𝐾 < 𝑁 ) )
7 6 simpld ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀𝐾 )