Step |
Hyp |
Ref |
Expression |
1 |
|
signsv.p |
⊢ ⨣ = ( 𝑎 ∈ { - 1 , 0 , 1 } , 𝑏 ∈ { - 1 , 0 , 1 } ↦ if ( 𝑏 = 0 , 𝑎 , 𝑏 ) ) |
2 |
|
signsv.w |
⊢ 𝑊 = { 〈 ( Base ‘ ndx ) , { - 1 , 0 , 1 } 〉 , 〈 ( +g ‘ ndx ) , ⨣ 〉 } |
3 |
|
signsv.t |
⊢ 𝑇 = ( 𝑓 ∈ Word ℝ ↦ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ↦ ( 𝑊 Σg ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝑓 ‘ 𝑖 ) ) ) ) ) ) |
4 |
|
signsv.v |
⊢ 𝑉 = ( 𝑓 ∈ Word ℝ ↦ Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) ) |
5 |
|
wrd0 |
⊢ ∅ ∈ Word ℝ |
6 |
1 2 3 4
|
signsvvfval |
⊢ ( ∅ ∈ Word ℝ → ( 𝑉 ‘ ∅ ) = Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ ∅ ) ) if ( ( ( 𝑇 ‘ ∅ ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ ∅ ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) ) |
7 |
5 6
|
ax-mp |
⊢ ( 𝑉 ‘ ∅ ) = Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ ∅ ) ) if ( ( ( 𝑇 ‘ ∅ ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ ∅ ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) |
8 |
|
hash0 |
⊢ ( ♯ ‘ ∅ ) = 0 |
9 |
8
|
oveq2i |
⊢ ( 1 ..^ ( ♯ ‘ ∅ ) ) = ( 1 ..^ 0 ) |
10 |
|
0le1 |
⊢ 0 ≤ 1 |
11 |
|
1z |
⊢ 1 ∈ ℤ |
12 |
|
0z |
⊢ 0 ∈ ℤ |
13 |
|
fzon |
⊢ ( ( 1 ∈ ℤ ∧ 0 ∈ ℤ ) → ( 0 ≤ 1 ↔ ( 1 ..^ 0 ) = ∅ ) ) |
14 |
11 12 13
|
mp2an |
⊢ ( 0 ≤ 1 ↔ ( 1 ..^ 0 ) = ∅ ) |
15 |
10 14
|
mpbi |
⊢ ( 1 ..^ 0 ) = ∅ |
16 |
9 15
|
eqtri |
⊢ ( 1 ..^ ( ♯ ‘ ∅ ) ) = ∅ |
17 |
16
|
sumeq1i |
⊢ Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ ∅ ) ) if ( ( ( 𝑇 ‘ ∅ ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ ∅ ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) = Σ 𝑗 ∈ ∅ if ( ( ( 𝑇 ‘ ∅ ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ ∅ ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) |
18 |
|
sum0 |
⊢ Σ 𝑗 ∈ ∅ if ( ( ( 𝑇 ‘ ∅ ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ ∅ ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) = 0 |
19 |
7 17 18
|
3eqtri |
⊢ ( 𝑉 ‘ ∅ ) = 0 |