Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) = ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) |
2 |
1
|
dfitg |
⊢ ∫ ∅ 𝐴 d 𝑥 = Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) ) ) |
3 |
|
ifan |
⊢ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) = if ( 𝑥 ∈ ∅ , if ( 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) , 0 ) |
4 |
|
noel |
⊢ ¬ 𝑥 ∈ ∅ |
5 |
4
|
iffalsei |
⊢ if ( 𝑥 ∈ ∅ , if ( 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) , 0 ) = 0 |
6 |
3 5
|
eqtri |
⊢ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) = 0 |
7 |
6
|
mpteq2i |
⊢ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) = ( 𝑥 ∈ ℝ ↦ 0 ) |
8 |
|
fconstmpt |
⊢ ( ℝ × { 0 } ) = ( 𝑥 ∈ ℝ ↦ 0 ) |
9 |
7 8
|
eqtr4i |
⊢ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) = ( ℝ × { 0 } ) |
10 |
9
|
fveq2i |
⊢ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) ) = ( ∫2 ‘ ( ℝ × { 0 } ) ) |
11 |
|
itg20 |
⊢ ( ∫2 ‘ ( ℝ × { 0 } ) ) = 0 |
12 |
10 11
|
eqtri |
⊢ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) ) = 0 |
13 |
12
|
oveq2i |
⊢ ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) ) ) = ( ( i ↑ 𝑘 ) · 0 ) |
14 |
|
ax-icn |
⊢ i ∈ ℂ |
15 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 3 ) → 𝑘 ∈ ℕ0 ) |
16 |
|
expcl |
⊢ ( ( i ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( i ↑ 𝑘 ) ∈ ℂ ) |
17 |
14 15 16
|
sylancr |
⊢ ( 𝑘 ∈ ( 0 ... 3 ) → ( i ↑ 𝑘 ) ∈ ℂ ) |
18 |
17
|
mul01d |
⊢ ( 𝑘 ∈ ( 0 ... 3 ) → ( ( i ↑ 𝑘 ) · 0 ) = 0 ) |
19 |
13 18
|
eqtrid |
⊢ ( 𝑘 ∈ ( 0 ... 3 ) → ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) ) ) = 0 ) |
20 |
19
|
sumeq2i |
⊢ Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) ) ) = Σ 𝑘 ∈ ( 0 ... 3 ) 0 |
21 |
|
fzfi |
⊢ ( 0 ... 3 ) ∈ Fin |
22 |
21
|
olci |
⊢ ( ( 0 ... 3 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... 3 ) ∈ Fin ) |
23 |
|
sumz |
⊢ ( ( ( 0 ... 3 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... 3 ) ∈ Fin ) → Σ 𝑘 ∈ ( 0 ... 3 ) 0 = 0 ) |
24 |
22 23
|
ax-mp |
⊢ Σ 𝑘 ∈ ( 0 ... 3 ) 0 = 0 |
25 |
20 24
|
eqtri |
⊢ Σ 𝑘 ∈ ( 0 ... 3 ) ( ( i ↑ 𝑘 ) · ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( ( 𝑥 ∈ ∅ ∧ 0 ≤ ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) ) , ( ℜ ‘ ( 𝐴 / ( i ↑ 𝑘 ) ) ) , 0 ) ) ) ) = 0 |
26 |
2 25
|
eqtri |
⊢ ∫ ∅ 𝐴 d 𝑥 = 0 |