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 |
|
simpll |
⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ) |
6 |
5
|
eldifad |
⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → 𝐹 ∈ Word ℝ ) |
7 |
1 2 3 4
|
signstcl |
⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 1 } ) |
8 |
6 7
|
sylancom |
⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 1 } ) |
9 |
1 2 3 4
|
signstfvneq0 |
⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ≠ 0 ) |
10 |
|
eldifsn |
⊢ ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ ( { - 1 , 0 , 1 } ∖ { 0 } ) ↔ ( ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 0 , 1 } ∧ ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ≠ 0 ) ) |
11 |
8 9 10
|
sylanbrc |
⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ ( { - 1 , 0 , 1 } ∖ { 0 } ) ) |
12 |
|
tpcomb |
⊢ { - 1 , 0 , 1 } = { - 1 , 1 , 0 } |
13 |
12
|
difeq1i |
⊢ ( { - 1 , 0 , 1 } ∖ { 0 } ) = ( { - 1 , 1 , 0 } ∖ { 0 } ) |
14 |
|
neg1ne0 |
⊢ - 1 ≠ 0 |
15 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
16 |
|
diftpsn3 |
⊢ ( ( - 1 ≠ 0 ∧ 1 ≠ 0 ) → ( { - 1 , 1 , 0 } ∖ { 0 } ) = { - 1 , 1 } ) |
17 |
14 15 16
|
mp2an |
⊢ ( { - 1 , 1 , 0 } ∖ { 0 } ) = { - 1 , 1 } |
18 |
13 17
|
eqtri |
⊢ ( { - 1 , 0 , 1 } ∖ { 0 } ) = { - 1 , 1 } |
19 |
11 18
|
eleqtrdi |
⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 𝑁 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ 𝑁 ) ∈ { - 1 , 1 } ) |