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 |
|
signsvf.e |
⊢ ( 𝜑 → 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) ) |
6 |
|
signsvf.0 |
⊢ ( 𝜑 → ( 𝐸 ‘ 0 ) ≠ 0 ) |
7 |
|
signsvf.f |
⊢ ( 𝜑 → 𝐹 = ( 𝐸 ++ 〈“ 𝐴 ”〉 ) ) |
8 |
|
signsvf.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
9 |
|
signsvf.n |
⊢ 𝑁 = ( ♯ ‘ 𝐸 ) |
10 |
|
signsvf.b |
⊢ 𝐵 = ( 𝐸 ‘ ( 𝑁 − 1 ) ) |
11 |
8
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
12 |
5
|
eldifad |
⊢ ( 𝜑 → 𝐸 ∈ Word ℝ ) |
13 |
|
wrdf |
⊢ ( 𝐸 ∈ Word ℝ → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ ℝ ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ ℝ ) |
15 |
9
|
oveq1i |
⊢ ( 𝑁 − 1 ) = ( ( ♯ ‘ 𝐸 ) − 1 ) |
16 |
|
eldifsn |
⊢ ( 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) ↔ ( 𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅ ) ) |
17 |
5 16
|
sylib |
⊢ ( 𝜑 → ( 𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅ ) ) |
18 |
|
lennncl |
⊢ ( ( 𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅ ) → ( ♯ ‘ 𝐸 ) ∈ ℕ ) |
19 |
|
fzo0end |
⊢ ( ( ♯ ‘ 𝐸 ) ∈ ℕ → ( ( ♯ ‘ 𝐸 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
20 |
17 18 19
|
3syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐸 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
21 |
15 20
|
eqeltrid |
⊢ ( 𝜑 → ( 𝑁 − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
22 |
14 21
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑁 − 1 ) ) ∈ ℝ ) |
23 |
22
|
recnd |
⊢ ( 𝜑 → ( 𝐸 ‘ ( 𝑁 − 1 ) ) ∈ ℂ ) |
24 |
10 23
|
eqeltrid |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
25 |
11 24
|
mulcomd |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) ) |
26 |
25
|
breq2d |
⊢ ( 𝜑 → ( 0 < ( 𝐴 · 𝐵 ) ↔ 0 < ( 𝐵 · 𝐴 ) ) ) |
27 |
10 22
|
eqeltrid |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
28 |
|
sgnmulsgp |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < ( 𝐴 · 𝐵 ) ↔ 0 < ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) ) ) |
29 |
8 27 28
|
syl2anc |
⊢ ( 𝜑 → ( 0 < ( 𝐴 · 𝐵 ) ↔ 0 < ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) ) ) |
30 |
26 29
|
bitr3d |
⊢ ( 𝜑 → ( 0 < ( 𝐵 · 𝐴 ) ↔ 0 < ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) ) ) |
31 |
30
|
biimpa |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 0 < ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) ) |
32 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) ) |
33 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 𝐵 ∈ ℂ ) |
34 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 𝐴 ∈ ℂ ) |
35 |
|
simpr |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 0 < ( 𝐵 · 𝐴 ) ) |
36 |
35
|
gt0ne0d |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( 𝐵 · 𝐴 ) ≠ 0 ) |
37 |
33 34 36
|
mulne0bad |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 𝐵 ≠ 0 ) |
38 |
10 37
|
eqnetrrid |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( 𝐸 ‘ ( 𝑁 − 1 ) ) ≠ 0 ) |
39 |
1 2 3 4 9
|
signsvtn0 |
⊢ ( ( 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐸 ‘ ( 𝑁 − 1 ) ) ≠ 0 ) → ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) = ( sgn ‘ ( 𝐸 ‘ ( 𝑁 − 1 ) ) ) ) |
40 |
10
|
fveq2i |
⊢ ( sgn ‘ 𝐵 ) = ( sgn ‘ ( 𝐸 ‘ ( 𝑁 − 1 ) ) ) |
41 |
39 40
|
eqtr4di |
⊢ ( ( 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐸 ‘ ( 𝑁 − 1 ) ) ≠ 0 ) → ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) = ( sgn ‘ 𝐵 ) ) |
42 |
32 38 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) = ( sgn ‘ 𝐵 ) ) |
43 |
42
|
fveq2d |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( sgn ‘ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) = ( sgn ‘ ( sgn ‘ 𝐵 ) ) ) |
44 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 𝐵 ∈ ℝ ) |
45 |
44
|
rexrd |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 𝐵 ∈ ℝ* ) |
46 |
|
sgnsgn |
⊢ ( 𝐵 ∈ ℝ* → ( sgn ‘ ( sgn ‘ 𝐵 ) ) = ( sgn ‘ 𝐵 ) ) |
47 |
45 46
|
syl |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( sgn ‘ ( sgn ‘ 𝐵 ) ) = ( sgn ‘ 𝐵 ) ) |
48 |
43 47
|
eqtrd |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( sgn ‘ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) = ( sgn ‘ 𝐵 ) ) |
49 |
48
|
oveq2d |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( ( sgn ‘ 𝐴 ) · ( sgn ‘ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ) = ( ( sgn ‘ 𝐴 ) · ( sgn ‘ 𝐵 ) ) ) |
50 |
31 49
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 0 < ( ( sgn ‘ 𝐴 ) · ( sgn ‘ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ) ) |
51 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 𝐴 ∈ ℝ ) |
52 |
|
sgnclre |
⊢ ( 𝐵 ∈ ℝ → ( sgn ‘ 𝐵 ) ∈ ℝ ) |
53 |
44 52
|
syl |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( sgn ‘ 𝐵 ) ∈ ℝ ) |
54 |
42 53
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ∈ ℝ ) |
55 |
|
sgnmulsgp |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ∈ ℝ ) → ( 0 < ( 𝐴 · ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ↔ 0 < ( ( sgn ‘ 𝐴 ) · ( sgn ‘ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ) ) ) |
56 |
51 54 55
|
syl2anc |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( 0 < ( 𝐴 · ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ↔ 0 < ( ( sgn ‘ 𝐴 ) · ( sgn ‘ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ) ) ) |
57 |
50 56
|
mpbird |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → 0 < ( 𝐴 · ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ) |
58 |
|
eqid |
⊢ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) = ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) |
59 |
1 2 3 4 5 6 7 8 9 58
|
signsvtp |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) ) ) → ( 𝑉 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐸 ) ) |
60 |
57 59
|
syldan |
⊢ ( ( 𝜑 ∧ 0 < ( 𝐵 · 𝐴 ) ) → ( 𝑉 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐸 ) ) |