Step |
Hyp |
Ref |
Expression |
1 |
|
axsegconlem2.1 |
⊢ 𝑆 = Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑝 ) − ( 𝐵 ‘ 𝑝 ) ) ↑ 2 ) |
2 |
1
|
axsegconlem4 |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( √ ‘ 𝑆 ) ∈ ℝ ) |
3 |
2
|
3adant3 |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ≠ 𝐵 ) → ( √ ‘ 𝑆 ) ∈ ℝ ) |
4 |
1
|
axsegconlem5 |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → 0 ≤ ( √ ‘ 𝑆 ) ) |
5 |
4
|
3adant3 |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ≠ 𝐵 ) → 0 ≤ ( √ ‘ 𝑆 ) ) |
6 |
|
eqeelen |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 = 𝐵 ↔ Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑝 ) − ( 𝐵 ‘ 𝑝 ) ) ↑ 2 ) = 0 ) ) |
7 |
1
|
eqeq1i |
⊢ ( 𝑆 = 0 ↔ Σ 𝑝 ∈ ( 1 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑝 ) − ( 𝐵 ‘ 𝑝 ) ) ↑ 2 ) = 0 ) |
8 |
6 7
|
bitr4di |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 = 𝐵 ↔ 𝑆 = 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 ) ) → ( 𝐴 ≠ 𝐵 ↔ ( √ ‘ 𝑆 ) ≠ 0 ) ) |
15 |
14
|
biimp3a |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ≠ 𝐵 ) → ( √ ‘ 𝑆 ) ≠ 0 ) |
16 |
3 5 15
|
ne0gt0d |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ≠ 𝐵 ) → 0 < ( √ ‘ 𝑆 ) ) |