Step |
Hyp |
Ref |
Expression |
1 |
|
ply1term.1 |
⊢ 𝐹 = ( 𝑧 ∈ ℂ ↦ ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ) |
2 |
|
simplr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → 𝑁 ∈ ℕ0 ) |
3 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
4 |
2 3
|
eleqtrdi |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
5 |
|
fzss1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( 𝑁 ... 𝑁 ) ⊆ ( 0 ... 𝑁 ) ) |
6 |
4 5
|
syl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → ( 𝑁 ... 𝑁 ) ⊆ ( 0 ... 𝑁 ) ) |
7 |
|
elfz1eq |
⊢ ( 𝑘 ∈ ( 𝑁 ... 𝑁 ) → 𝑘 = 𝑁 ) |
8 |
7
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → 𝑘 = 𝑁 ) |
9 |
|
iftrue |
⊢ ( 𝑘 = 𝑁 → if ( 𝑘 = 𝑁 , 𝐴 , 0 ) = 𝐴 ) |
10 |
8 9
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → if ( 𝑘 = 𝑁 , 𝐴 , 0 ) = 𝐴 ) |
11 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
12 |
11
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) |
13 |
10 12
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → if ( 𝑘 = 𝑁 , 𝐴 , 0 ) ∈ ℂ ) |
14 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → 𝑧 ∈ ℂ ) |
15 |
2
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → 𝑁 ∈ ℕ0 ) |
16 |
8 15
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
17 |
14 16
|
expcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
18 |
13 17
|
mulcld |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) → ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
19 |
|
eldifn |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) → ¬ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) |
20 |
19
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → ¬ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ) |
21 |
2
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → 𝑁 ∈ ℕ0 ) |
22 |
21
|
nn0zd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → 𝑁 ∈ ℤ ) |
23 |
|
fzsn |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ... 𝑁 ) = { 𝑁 } ) |
24 |
23
|
eleq2d |
⊢ ( 𝑁 ∈ ℤ → ( 𝑘 ∈ ( 𝑁 ... 𝑁 ) ↔ 𝑘 ∈ { 𝑁 } ) ) |
25 |
|
elsn2g |
⊢ ( 𝑁 ∈ ℤ → ( 𝑘 ∈ { 𝑁 } ↔ 𝑘 = 𝑁 ) ) |
26 |
24 25
|
bitrd |
⊢ ( 𝑁 ∈ ℤ → ( 𝑘 ∈ ( 𝑁 ... 𝑁 ) ↔ 𝑘 = 𝑁 ) ) |
27 |
22 26
|
syl |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → ( 𝑘 ∈ ( 𝑁 ... 𝑁 ) ↔ 𝑘 = 𝑁 ) ) |
28 |
20 27
|
mtbid |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → ¬ 𝑘 = 𝑁 ) |
29 |
28
|
iffalsed |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → if ( 𝑘 = 𝑁 , 𝐴 , 0 ) = 0 ) |
30 |
29
|
oveq1d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 0 · ( 𝑧 ↑ 𝑘 ) ) ) |
31 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → 𝑧 ∈ ℂ ) |
32 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) → 𝑘 ∈ ( 0 ... 𝑁 ) ) |
33 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑁 ) → 𝑘 ∈ ℕ0 ) |
34 |
32 33
|
syl |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
35 |
|
expcl |
⊢ ( ( 𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
36 |
31 34 35
|
syl2an |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
37 |
36
|
mul02d |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → ( 0 · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
38 |
30 37
|
eqtrd |
⊢ ( ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... 𝑁 ) ∖ ( 𝑁 ... 𝑁 ) ) ) → ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
39 |
|
fzfid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → ( 0 ... 𝑁 ) ∈ Fin ) |
40 |
6 18 38 39
|
fsumss |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
41 |
2
|
nn0zd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → 𝑁 ∈ ℤ ) |
42 |
31 2
|
expcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → ( 𝑧 ↑ 𝑁 ) ∈ ℂ ) |
43 |
11 42
|
mulcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ∈ ℂ ) |
44 |
|
oveq2 |
⊢ ( 𝑘 = 𝑁 → ( 𝑧 ↑ 𝑘 ) = ( 𝑧 ↑ 𝑁 ) ) |
45 |
9 44
|
oveq12d |
⊢ ( 𝑘 = 𝑁 → ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ) |
46 |
45
|
fsum1 |
⊢ ( ( 𝑁 ∈ ℤ ∧ ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ∈ ℂ ) → Σ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ) |
47 |
41 43 46
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 𝑁 ... 𝑁 ) ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ) |
48 |
40 47
|
eqtr3d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ) |
49 |
48
|
mpteq2dva |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ ( 𝐴 · ( 𝑧 ↑ 𝑁 ) ) ) ) |
50 |
1 49
|
eqtr4id |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0 ) → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( if ( 𝑘 = 𝑁 , 𝐴 , 0 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |