Metamath Proof Explorer


Theorem axsegconlem5

Description: Lemma for axsegcon . Show that the distance between two points is nonnegative. (Contributed by Scott Fenton, 17-Sep-2013)

Ref Expression
Hypothesis axsegconlem2.1 𝑆 = Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴𝑝 ) − ( 𝐵𝑝 ) ) ↑ 2 )
Assertion axsegconlem5 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 0 ≤ ( √ ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 axsegconlem2.1 𝑆 = Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴𝑝 ) − ( 𝐵𝑝 ) ) ↑ 2 )
2 1 axsegconlem2 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑆 ∈ ℝ )
3 1 axsegconlem3 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 0 ≤ 𝑆 )
4 2 3 sqrtge0d ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 0 ≤ ( √ ‘ 𝑆 ) )