Metamath Proof Explorer


Theorem axsegconlem6

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

Ref Expression
Hypothesis axsegconlem2.1 𝑆 = Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴𝑝 ) − ( 𝐵𝑝 ) ) ↑ 2 )
Assertion axsegconlem6 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴𝐵 ) → 0 < ( √ ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 axsegconlem2.1 𝑆 = Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴𝑝 ) − ( 𝐵𝑝 ) ) ↑ 2 )
2 1 axsegconlem4 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( √ ‘ 𝑆 ) ∈ ℝ )
3 2 3adant3 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴𝐵 ) → ( √ ‘ 𝑆 ) ∈ ℝ )
4 1 axsegconlem5 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 0 ≤ ( √ ‘ 𝑆 ) )
5 4 3adant3 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴𝐵 ) → 0 ≤ ( √ ‘ 𝑆 ) )
6 eqeelen ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 = 𝐵 ↔ Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴𝑝 ) − ( 𝐵𝑝 ) ) ↑ 2 ) = 0 ) )
7 1 eqeq1i ( 𝑆 = 0 ↔ Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴𝑝 ) − ( 𝐵𝑝 ) ) ↑ 2 ) = 0 )
8 6 7 bitr4di ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 = 𝐵𝑆 = 0 ) )
9 1 axsegconlem2 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑆 ∈ ℝ )
10 1 axsegconlem3 ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 0 ≤ 𝑆 )
11 sqrt00 ( ( 𝑆 ∈ ℝ ∧ 0 ≤ 𝑆 ) → ( ( √ ‘ 𝑆 ) = 0 ↔ 𝑆 = 0 ) )
12 9 10 11 syl2anc ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( √ ‘ 𝑆 ) = 0 ↔ 𝑆 = 0 ) )
13 8 12 bitr4d ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 = 𝐵 ↔ ( √ ‘ 𝑆 ) = 0 ) )
14 13 necon3bid ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴𝐵 ↔ ( √ ‘ 𝑆 ) ≠ 0 ) )
15 14 biimp3a ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴𝐵 ) → ( √ ‘ 𝑆 ) ≠ 0 )
16 3 5 15 ne0gt0d ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴𝐵 ) → 0 < ( √ ‘ 𝑆 ) )