| Step |
Hyp |
Ref |
Expression |
| 1 |
|
etransclem35.p |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
| 2 |
|
etransclem35.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
| 3 |
|
etransclem35.f |
⊢ 𝐹 = ( 𝑥 ∈ ℝ ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) ) |
| 4 |
|
etransclem35.c |
⊢ 𝐶 = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = 𝑛 } ) |
| 5 |
|
etransclem35.d |
⊢ 𝐷 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ) |
| 6 |
|
reelprrecn |
⊢ ℝ ∈ { ℝ , ℂ } |
| 7 |
6
|
a1i |
⊢ ( 𝜑 → ℝ ∈ { ℝ , ℂ } ) |
| 8 |
|
reopn |
⊢ ℝ ∈ ( topGen ‘ ran (,) ) |
| 9 |
|
tgioo4 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
| 10 |
8 9
|
eleqtri |
⊢ ℝ ∈ ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
| 11 |
10
|
a1i |
⊢ ( 𝜑 → ℝ ∈ ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) |
| 12 |
|
nnm1nn0 |
⊢ ( 𝑃 ∈ ℕ → ( 𝑃 − 1 ) ∈ ℕ0 ) |
| 13 |
1 12
|
syl |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℕ0 ) |
| 14 |
|
etransclem5 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ ℝ ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥 ∈ ℝ ↦ ( ( 𝑥 − 𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) |
| 15 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
| 16 |
7 11 1 2 3 13 14 4 15
|
etransclem31 |
⊢ ( 𝜑 → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) = Σ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) ) |
| 17 |
|
nfv |
⊢ Ⅎ 𝑐 𝜑 |
| 18 |
|
nfcv |
⊢ Ⅎ 𝑐 ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
| 19 |
4 13
|
etransclem16 |
⊢ ( 𝜑 → ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∈ Fin ) |
| 20 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) |
| 21 |
4 13
|
etransclem12 |
⊢ ( 𝜑 → ( 𝐶 ‘ ( 𝑃 − 1 ) ) = { 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) } ) |
| 22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( 𝐶 ‘ ( 𝑃 − 1 ) ) = { 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) } ) |
| 23 |
20 22
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑐 ∈ { 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) } ) |
| 24 |
|
rabid |
⊢ ( 𝑐 ∈ { 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) } ↔ ( 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∧ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) ) |
| 25 |
23 24
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∧ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) ) |
| 26 |
25
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) |
| 27 |
26
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( 𝑃 − 1 ) = Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) ) |
| 28 |
27
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ! ‘ ( 𝑃 − 1 ) ) = ( ! ‘ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) ) ) |
| 29 |
28
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) = ( ( ! ‘ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) ) |
| 30 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑐 |
| 31 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( 0 ... 𝑀 ) ∈ Fin ) |
| 32 |
|
nn0ex |
⊢ ℕ0 ∈ V |
| 33 |
|
fzssnn0 |
⊢ ( 0 ... ( 𝑃 − 1 ) ) ⊆ ℕ0 |
| 34 |
|
mapss |
⊢ ( ( ℕ0 ∈ V ∧ ( 0 ... ( 𝑃 − 1 ) ) ⊆ ℕ0 ) → ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ⊆ ( ℕ0 ↑m ( 0 ... 𝑀 ) ) ) |
| 35 |
32 33 34
|
mp2an |
⊢ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ⊆ ( ℕ0 ↑m ( 0 ... 𝑀 ) ) |
| 36 |
25
|
simpld |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ) |
| 37 |
35 36
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑐 ∈ ( ℕ0 ↑m ( 0 ... 𝑀 ) ) ) |
| 38 |
30 31 37
|
mccl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ( ! ‘ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) ∈ ℕ ) |
| 39 |
29 38
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) ∈ ℕ ) |
| 40 |
39
|
nnzd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) ∈ ℤ ) |
| 41 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑃 ∈ ℕ ) |
| 42 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑀 ∈ ℕ0 ) |
| 43 |
|
elmapi |
⊢ ( 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) → 𝑐 : ( 0 ... 𝑀 ) ⟶ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 44 |
36 43
|
syl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑐 : ( 0 ... 𝑀 ) ⟶ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 45 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 0 ∈ ℤ ) |
| 46 |
41 42 44 45
|
etransclem10 |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) ∈ ℤ ) |
| 47 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( 1 ... 𝑀 ) ∈ Fin ) |
| 48 |
1
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑃 ∈ ℕ ) |
| 49 |
44
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑐 : ( 0 ... 𝑀 ) ⟶ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 50 |
|
fz1ssfz0 |
⊢ ( 1 ... 𝑀 ) ⊆ ( 0 ... 𝑀 ) |
| 51 |
50
|
sseli |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ∈ ( 0 ... 𝑀 ) ) |
| 52 |
51
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑗 ∈ ( 0 ... 𝑀 ) ) |
| 53 |
|
0zd |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 0 ∈ ℤ ) |
| 54 |
48 49 52 53
|
etransclem3 |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ∈ ℤ ) |
| 55 |
47 54
|
fprodzcl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ∈ ℤ ) |
| 56 |
46 55
|
zmulcld |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ∈ ℤ ) |
| 57 |
40 56
|
zmulcld |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) ∈ ℤ ) |
| 58 |
57
|
zcnd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) ∈ ℂ ) |
| 59 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
| 60 |
13 59
|
eleqtrdi |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
| 61 |
|
eluzfz2 |
⊢ ( ( 𝑃 − 1 ) ∈ ( ℤ≥ ‘ 0 ) → ( 𝑃 − 1 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 62 |
60 61
|
syl |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 63 |
|
eluzfz1 |
⊢ ( ( 𝑃 − 1 ) ∈ ( ℤ≥ ‘ 0 ) → 0 ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 64 |
60 63
|
syl |
⊢ ( 𝜑 → 0 ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 65 |
62 64
|
ifcld |
⊢ ( 𝜑 → if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 66 |
65
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 67 |
66 5
|
fmptd |
⊢ ( 𝜑 → 𝐷 : ( 0 ... 𝑀 ) ⟶ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 68 |
|
ovex |
⊢ ( 0 ... ( 𝑃 − 1 ) ) ∈ V |
| 69 |
|
ovex |
⊢ ( 0 ... 𝑀 ) ∈ V |
| 70 |
68 69
|
elmap |
⊢ ( 𝐷 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ↔ 𝐷 : ( 0 ... 𝑀 ) ⟶ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 71 |
67 70
|
sylibr |
⊢ ( 𝜑 → 𝐷 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ) |
| 72 |
2 59
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
| 73 |
|
fzsscn |
⊢ ( 0 ... ( 𝑃 − 1 ) ) ⊆ ℂ |
| 74 |
67
|
ffvelcdmda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 75 |
73 74
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) ∈ ℂ ) |
| 76 |
|
fveq2 |
⊢ ( 𝑗 = 0 → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 0 ) ) |
| 77 |
72 75 76
|
fsum1p |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = ( ( 𝐷 ‘ 0 ) + Σ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) ) ) |
| 78 |
5
|
a1i |
⊢ ( 𝜑 → 𝐷 = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ) ) |
| 79 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 = 0 ) → 𝑗 = 0 ) |
| 80 |
79
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑗 = 0 ) → if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) = ( 𝑃 − 1 ) ) |
| 81 |
|
eluzfz1 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → 0 ∈ ( 0 ... 𝑀 ) ) |
| 82 |
72 81
|
syl |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
| 83 |
78 80 82 13
|
fvmptd |
⊢ ( 𝜑 → ( 𝐷 ‘ 0 ) = ( 𝑃 − 1 ) ) |
| 84 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
| 85 |
84
|
oveq1i |
⊢ ( ( 0 + 1 ) ... 𝑀 ) = ( 1 ... 𝑀 ) |
| 86 |
85
|
sumeq1i |
⊢ Σ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) |
| 87 |
86
|
a1i |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) ) |
| 88 |
5
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) → ( 𝐷 ‘ 𝑗 ) = if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ) |
| 89 |
51 65 88
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) = if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ) |
| 90 |
|
0red |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 0 ∈ ℝ ) |
| 91 |
|
1red |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 1 ∈ ℝ ) |
| 92 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ∈ ℤ ) |
| 93 |
92
|
zred |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ∈ ℝ ) |
| 94 |
|
0lt1 |
⊢ 0 < 1 |
| 95 |
94
|
a1i |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 0 < 1 ) |
| 96 |
|
elfzle1 |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 1 ≤ 𝑗 ) |
| 97 |
90 91 93 95 96
|
ltletrd |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 0 < 𝑗 ) |
| 98 |
90 97
|
gtned |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → 𝑗 ≠ 0 ) |
| 99 |
98
|
neneqd |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → ¬ 𝑗 = 0 ) |
| 100 |
99
|
iffalsed |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) = 0 ) |
| 101 |
100
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) = 0 ) |
| 102 |
89 101
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) = 0 ) |
| 103 |
102
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = Σ 𝑗 ∈ ( 1 ... 𝑀 ) 0 ) |
| 104 |
|
fzfi |
⊢ ( 1 ... 𝑀 ) ∈ Fin |
| 105 |
104
|
olci |
⊢ ( ( 1 ... 𝑀 ) ⊆ ( ℤ≥ ‘ 𝐴 ) ∨ ( 1 ... 𝑀 ) ∈ Fin ) |
| 106 |
|
sumz |
⊢ ( ( ( 1 ... 𝑀 ) ⊆ ( ℤ≥ ‘ 𝐴 ) ∨ ( 1 ... 𝑀 ) ∈ Fin ) → Σ 𝑗 ∈ ( 1 ... 𝑀 ) 0 = 0 ) |
| 107 |
105 106
|
mp1i |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑀 ) 0 = 0 ) |
| 108 |
87 103 107
|
3eqtrd |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = 0 ) |
| 109 |
83 108
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝐷 ‘ 0 ) + Σ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) ) = ( ( 𝑃 − 1 ) + 0 ) ) |
| 110 |
1
|
nncnd |
⊢ ( 𝜑 → 𝑃 ∈ ℂ ) |
| 111 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
| 112 |
110 111
|
subcld |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℂ ) |
| 113 |
112
|
addridd |
⊢ ( 𝜑 → ( ( 𝑃 − 1 ) + 0 ) = ( 𝑃 − 1 ) ) |
| 114 |
77 109 113
|
3eqtrd |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) |
| 115 |
|
fveq1 |
⊢ ( 𝑐 = 𝐷 → ( 𝑐 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 116 |
115
|
sumeq2sdv |
⊢ ( 𝑐 = 𝐷 → Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) ) |
| 117 |
116
|
eqeq1d |
⊢ ( 𝑐 = 𝐷 → ( Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) ↔ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) ) |
| 118 |
117
|
elrab |
⊢ ( 𝐷 ∈ { 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) } ↔ ( 𝐷 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∧ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝐷 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) ) |
| 119 |
71 114 118
|
sylanbrc |
⊢ ( 𝜑 → 𝐷 ∈ { 𝑐 ∈ ( ( 0 ... ( 𝑃 − 1 ) ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) } ) |
| 120 |
119 21
|
eleqtrrd |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) |
| 121 |
115
|
fveq2d |
⊢ ( 𝑐 = 𝐷 → ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) = ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) |
| 122 |
121
|
prodeq2ad |
⊢ ( 𝑐 = 𝐷 → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) = ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) |
| 123 |
122
|
oveq2d |
⊢ ( 𝑐 = 𝐷 → ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
| 124 |
|
fveq1 |
⊢ ( 𝑐 = 𝐷 → ( 𝑐 ‘ 0 ) = ( 𝐷 ‘ 0 ) ) |
| 125 |
124
|
breq2d |
⊢ ( 𝑐 = 𝐷 → ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) ↔ ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) ) ) |
| 126 |
124
|
oveq2d |
⊢ ( 𝑐 = 𝐷 → ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) = ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) |
| 127 |
126
|
fveq2d |
⊢ ( 𝑐 = 𝐷 → ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) = ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) |
| 128 |
127
|
oveq2d |
⊢ ( 𝑐 = 𝐷 → ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) |
| 129 |
126
|
oveq2d |
⊢ ( 𝑐 = 𝐷 → ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) = ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) |
| 130 |
128 129
|
oveq12d |
⊢ ( 𝑐 = 𝐷 → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) = ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) |
| 131 |
125 130
|
ifbieq2d |
⊢ ( 𝑐 = 𝐷 → if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) = if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) ) |
| 132 |
115
|
breq2d |
⊢ ( 𝑐 = 𝐷 → ( 𝑃 < ( 𝑐 ‘ 𝑗 ) ↔ 𝑃 < ( 𝐷 ‘ 𝑗 ) ) ) |
| 133 |
115
|
oveq2d |
⊢ ( 𝑐 = 𝐷 → ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) = ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) |
| 134 |
133
|
fveq2d |
⊢ ( 𝑐 = 𝐷 → ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) = ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) |
| 135 |
134
|
oveq2d |
⊢ ( 𝑐 = 𝐷 → ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) = ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
| 136 |
133
|
oveq2d |
⊢ ( 𝑐 = 𝐷 → ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) = ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) |
| 137 |
135 136
|
oveq12d |
⊢ ( 𝑐 = 𝐷 → ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) = ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
| 138 |
132 137
|
ifbieq2d |
⊢ ( 𝑐 = 𝐷 → if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) = if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
| 139 |
138
|
prodeq2ad |
⊢ ( 𝑐 = 𝐷 → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) = ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) |
| 140 |
131 139
|
oveq12d |
⊢ ( 𝑐 = 𝐷 → ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) = ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) |
| 141 |
123 140
|
oveq12d |
⊢ ( 𝑐 = 𝐷 → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) = ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) ) |
| 142 |
17 18 19 58 120 141
|
fsumsplit1 |
⊢ ( 𝜑 → Σ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) = ( ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) + Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) ) ) |
| 143 |
33 74
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) ∈ ℕ0 ) |
| 144 |
143
|
faccld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ∈ ℕ ) |
| 145 |
144
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ∈ ℂ ) |
| 146 |
76
|
fveq2d |
⊢ ( 𝑗 = 0 → ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ! ‘ ( 𝐷 ‘ 0 ) ) ) |
| 147 |
72 145 146
|
fprod1p |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ( ! ‘ ( 𝐷 ‘ 0 ) ) · ∏ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) ) |
| 148 |
83
|
fveq2d |
⊢ ( 𝜑 → ( ! ‘ ( 𝐷 ‘ 0 ) ) = ( ! ‘ ( 𝑃 − 1 ) ) ) |
| 149 |
85
|
prodeq1i |
⊢ ∏ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) |
| 150 |
149
|
a1i |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) |
| 151 |
102
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ! ‘ 0 ) ) |
| 152 |
|
fac0 |
⊢ ( ! ‘ 0 ) = 1 |
| 153 |
151 152
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = 1 ) |
| 154 |
153
|
prodeq2dv |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = ∏ 𝑗 ∈ ( 1 ... 𝑀 ) 1 ) |
| 155 |
|
prod1 |
⊢ ( ( ( 1 ... 𝑀 ) ⊆ ( ℤ≥ ‘ 𝐴 ) ∨ ( 1 ... 𝑀 ) ∈ Fin ) → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) 1 = 1 ) |
| 156 |
105 155
|
mp1i |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) 1 = 1 ) |
| 157 |
150 154 156
|
3eqtrd |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = 1 ) |
| 158 |
148 157
|
oveq12d |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝐷 ‘ 0 ) ) · ∏ 𝑗 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · 1 ) ) |
| 159 |
13
|
faccld |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℕ ) |
| 160 |
159
|
nncnd |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℂ ) |
| 161 |
160
|
mulridd |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) · 1 ) = ( ! ‘ ( 𝑃 − 1 ) ) ) |
| 162 |
147 158 161
|
3eqtrd |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) = ( ! ‘ ( 𝑃 − 1 ) ) ) |
| 163 |
162
|
oveq2d |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
| 164 |
159
|
nnne0d |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ≠ 0 ) |
| 165 |
160 164
|
dividd |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) = 1 ) |
| 166 |
163 165
|
eqtrd |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) = 1 ) |
| 167 |
13
|
nn0red |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℝ ) |
| 168 |
83 167
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐷 ‘ 0 ) ∈ ℝ ) |
| 169 |
168 167
|
lttri3d |
⊢ ( 𝜑 → ( ( 𝐷 ‘ 0 ) = ( 𝑃 − 1 ) ↔ ( ¬ ( 𝐷 ‘ 0 ) < ( 𝑃 − 1 ) ∧ ¬ ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) ) ) ) |
| 170 |
83 169
|
mpbid |
⊢ ( 𝜑 → ( ¬ ( 𝐷 ‘ 0 ) < ( 𝑃 − 1 ) ∧ ¬ ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) ) ) |
| 171 |
170
|
simprd |
⊢ ( 𝜑 → ¬ ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) ) |
| 172 |
171
|
iffalsed |
⊢ ( 𝜑 → if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) = ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) |
| 173 |
83
|
eqcomd |
⊢ ( 𝜑 → ( 𝑃 − 1 ) = ( 𝐷 ‘ 0 ) ) |
| 174 |
112 173
|
subeq0bd |
⊢ ( 𝜑 → ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) = 0 ) |
| 175 |
174
|
fveq2d |
⊢ ( 𝜑 → ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) = ( ! ‘ 0 ) ) |
| 176 |
175 152
|
eqtrdi |
⊢ ( 𝜑 → ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) = 1 ) |
| 177 |
176
|
oveq2d |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) / 1 ) ) |
| 178 |
160
|
div1d |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) / 1 ) = ( ! ‘ ( 𝑃 − 1 ) ) ) |
| 179 |
177 178
|
eqtrd |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) = ( ! ‘ ( 𝑃 − 1 ) ) ) |
| 180 |
174
|
oveq2d |
⊢ ( 𝜑 → ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) = ( 0 ↑ 0 ) ) |
| 181 |
|
0cnd |
⊢ ( 𝜑 → 0 ∈ ℂ ) |
| 182 |
181
|
exp0d |
⊢ ( 𝜑 → ( 0 ↑ 0 ) = 1 ) |
| 183 |
180 182
|
eqtrd |
⊢ ( 𝜑 → ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) = 1 ) |
| 184 |
179 183
|
oveq12d |
⊢ ( 𝜑 → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · 1 ) ) |
| 185 |
172 184 161
|
3eqtrd |
⊢ ( 𝜑 → if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) = ( ! ‘ ( 𝑃 − 1 ) ) ) |
| 186 |
|
fzssre |
⊢ ( 0 ... ( 𝑃 − 1 ) ) ⊆ ℝ |
| 187 |
67
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝐷 : ( 0 ... 𝑀 ) ⟶ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 188 |
51
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑗 ∈ ( 0 ... 𝑀 ) ) |
| 189 |
187 188
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 190 |
186 189
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) ∈ ℝ ) |
| 191 |
1
|
nnred |
⊢ ( 𝜑 → 𝑃 ∈ ℝ ) |
| 192 |
191
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑃 ∈ ℝ ) |
| 193 |
1
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑃 ) |
| 194 |
15 191 193
|
ltled |
⊢ ( 𝜑 → 0 ≤ 𝑃 ) |
| 195 |
194
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 0 ≤ 𝑃 ) |
| 196 |
102 195
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) ≤ 𝑃 ) |
| 197 |
190 192 196
|
lensymd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ¬ 𝑃 < ( 𝐷 ‘ 𝑗 ) ) |
| 198 |
197
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) |
| 199 |
102
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) = ( 𝑃 − 0 ) ) |
| 200 |
110
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑃 ∈ ℂ ) |
| 201 |
200
|
subid1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑃 − 0 ) = 𝑃 ) |
| 202 |
199 201
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) = 𝑃 ) |
| 203 |
202
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) = ( ! ‘ 𝑃 ) ) |
| 204 |
203
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) = ( ( ! ‘ 𝑃 ) / ( ! ‘ 𝑃 ) ) ) |
| 205 |
1
|
nnnn0d |
⊢ ( 𝜑 → 𝑃 ∈ ℕ0 ) |
| 206 |
205
|
faccld |
⊢ ( 𝜑 → ( ! ‘ 𝑃 ) ∈ ℕ ) |
| 207 |
206
|
nncnd |
⊢ ( 𝜑 → ( ! ‘ 𝑃 ) ∈ ℂ ) |
| 208 |
206
|
nnne0d |
⊢ ( 𝜑 → ( ! ‘ 𝑃 ) ≠ 0 ) |
| 209 |
207 208
|
dividd |
⊢ ( 𝜑 → ( ( ! ‘ 𝑃 ) / ( ! ‘ 𝑃 ) ) = 1 ) |
| 210 |
209
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ! ‘ 𝑃 ) / ( ! ‘ 𝑃 ) ) = 1 ) |
| 211 |
204 210
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) = 1 ) |
| 212 |
|
df-neg |
⊢ - 𝑗 = ( 0 − 𝑗 ) |
| 213 |
212
|
eqcomi |
⊢ ( 0 − 𝑗 ) = - 𝑗 |
| 214 |
213
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 0 − 𝑗 ) = - 𝑗 ) |
| 215 |
214 202
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) = ( - 𝑗 ↑ 𝑃 ) ) |
| 216 |
211 215
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) = ( 1 · ( - 𝑗 ↑ 𝑃 ) ) ) |
| 217 |
92
|
znegcld |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → - 𝑗 ∈ ℤ ) |
| 218 |
217
|
zcnd |
⊢ ( 𝑗 ∈ ( 1 ... 𝑀 ) → - 𝑗 ∈ ℂ ) |
| 219 |
218
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → - 𝑗 ∈ ℂ ) |
| 220 |
205
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → 𝑃 ∈ ℕ0 ) |
| 221 |
219 220
|
expcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( - 𝑗 ↑ 𝑃 ) ∈ ℂ ) |
| 222 |
221
|
mullidd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( 1 · ( - 𝑗 ↑ 𝑃 ) ) = ( - 𝑗 ↑ 𝑃 ) ) |
| 223 |
198 216 222
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ( - 𝑗 ↑ 𝑃 ) ) |
| 224 |
223
|
prodeq2dv |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) = ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) |
| 225 |
185 224
|
oveq12d |
⊢ ( 𝜑 → ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) ) |
| 226 |
166 225
|
oveq12d |
⊢ ( 𝜑 → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) = ( 1 · ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) ) ) |
| 227 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ Fin ) |
| 228 |
|
zexpcl |
⊢ ( ( - 𝑗 ∈ ℤ ∧ 𝑃 ∈ ℕ0 ) → ( - 𝑗 ↑ 𝑃 ) ∈ ℤ ) |
| 229 |
217 205 228
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑀 ) ) → ( - 𝑗 ↑ 𝑃 ) ∈ ℤ ) |
| 230 |
227 229
|
fprodzcl |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ∈ ℤ ) |
| 231 |
230
|
zcnd |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ∈ ℂ ) |
| 232 |
160 231
|
mulcld |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) ∈ ℂ ) |
| 233 |
232
|
mullidd |
⊢ ( 𝜑 → ( 1 · ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) ) |
| 234 |
226 233
|
eqtrd |
⊢ ( 𝜑 → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) ) |
| 235 |
|
eldifi |
⊢ ( 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) → 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) |
| 236 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 0 ∈ ( 0 ... 𝑀 ) ) |
| 237 |
44 236
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( 𝑐 ‘ 0 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 238 |
235 237
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑐 ‘ 0 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 239 |
186 238
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑐 ‘ 0 ) ∈ ℝ ) |
| 240 |
167
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑃 − 1 ) ∈ ℝ ) |
| 241 |
|
elfzle2 |
⊢ ( ( 𝑐 ‘ 0 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) → ( 𝑐 ‘ 0 ) ≤ ( 𝑃 − 1 ) ) |
| 242 |
238 241
|
syl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑐 ‘ 0 ) ≤ ( 𝑃 − 1 ) ) |
| 243 |
239 240 242
|
lensymd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ¬ ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) ) |
| 244 |
243
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) = ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) |
| 245 |
13
|
nn0zd |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℤ ) |
| 246 |
245
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑃 − 1 ) ∈ ℤ ) |
| 247 |
238
|
elfzelzd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑐 ‘ 0 ) ∈ ℤ ) |
| 248 |
246 247
|
zsubcld |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ∈ ℤ ) |
| 249 |
44
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 𝑐 Fn ( 0 ... 𝑀 ) ) |
| 250 |
249
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) → 𝑐 Fn ( 0 ... 𝑀 ) ) |
| 251 |
67
|
ffnd |
⊢ ( 𝜑 → 𝐷 Fn ( 0 ... 𝑀 ) ) |
| 252 |
251
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) → 𝐷 Fn ( 0 ... 𝑀 ) ) |
| 253 |
|
fveq2 |
⊢ ( 𝑗 = 0 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 0 ) ) |
| 254 |
253
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 = 0 ) → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 0 ) ) |
| 255 |
|
id |
⊢ ( ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) → ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) |
| 256 |
255
|
eqcomd |
⊢ ( ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) → ( 𝑐 ‘ 0 ) = ( 𝑃 − 1 ) ) |
| 257 |
256
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 = 0 ) → ( 𝑐 ‘ 0 ) = ( 𝑃 − 1 ) ) |
| 258 |
76
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 = 0 ) → ( 𝐷 ‘ 𝑗 ) = ( 𝐷 ‘ 0 ) ) |
| 259 |
83
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 = 0 ) → ( 𝐷 ‘ 0 ) = ( 𝑃 − 1 ) ) |
| 260 |
258 259
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑗 = 0 ) → ( 𝑃 − 1 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 261 |
260
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 = 0 ) → ( 𝑃 − 1 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 262 |
254 257 261
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 = 0 ) → ( 𝑐 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 263 |
262
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 = 0 ) → ( 𝑐 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 264 |
263
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 = 0 ) → ( 𝑐 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 265 |
26
|
ad4antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) |
| 266 |
167
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑃 − 1 ) ∈ ℝ ) |
| 267 |
167
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑃 − 1 ) ∈ ℝ ) |
| 268 |
44
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → 𝑐 : ( 0 ... 𝑀 ) ⟶ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 269 |
50
|
sseli |
⊢ ( 𝑘 ∈ ( 1 ... 𝑀 ) → 𝑘 ∈ ( 0 ... 𝑀 ) ) |
| 270 |
269
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → 𝑘 ∈ ( 0 ... 𝑀 ) ) |
| 271 |
268 270
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 272 |
33 271
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ℕ0 ) |
| 273 |
47 272
|
fsumnn0cl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ∈ ℕ0 ) |
| 274 |
273
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ∈ ℝ ) |
| 275 |
274
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ∈ ℝ ) |
| 276 |
|
0red |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 0 ∈ ℝ ) |
| 277 |
44
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 278 |
186 277
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ∈ ℝ ) |
| 279 |
278
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑐 ‘ 𝑗 ) ∈ ℝ ) |
| 280 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) |
| 281 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 𝑐 ‘ 𝑗 ) |
| 282 |
|
fzfid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 1 ... 𝑀 ) ∈ Fin ) |
| 283 |
|
simp-4l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ) |
| 284 |
73 271
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ℂ ) |
| 285 |
283 284
|
sylancom |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) ∧ 𝑘 ∈ ( 1 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ℂ ) |
| 286 |
|
1zzd |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 1 ∈ ℤ ) |
| 287 |
|
elfzel2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℤ ) |
| 288 |
287
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 𝑀 ∈ ℤ ) |
| 289 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℤ ) |
| 290 |
289
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 𝑗 ∈ ℤ ) |
| 291 |
|
elfznn0 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℕ0 ) |
| 292 |
291
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 𝑗 ∈ ℕ0 ) |
| 293 |
|
neqne |
⊢ ( ¬ 𝑗 = 0 → 𝑗 ≠ 0 ) |
| 294 |
293
|
adantl |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 𝑗 ≠ 0 ) |
| 295 |
|
elnnne0 |
⊢ ( 𝑗 ∈ ℕ ↔ ( 𝑗 ∈ ℕ0 ∧ 𝑗 ≠ 0 ) ) |
| 296 |
292 294 295
|
sylanbrc |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 𝑗 ∈ ℕ ) |
| 297 |
296
|
nnge1d |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 1 ≤ 𝑗 ) |
| 298 |
|
elfzle2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ≤ 𝑀 ) |
| 299 |
298
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 𝑗 ≤ 𝑀 ) |
| 300 |
286 288 290 297 299
|
elfzd |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) → 𝑗 ∈ ( 1 ... 𝑀 ) ) |
| 301 |
300
|
adantr |
⊢ ( ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 𝑗 ∈ ( 1 ... 𝑀 ) ) |
| 302 |
301
|
adantlll |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 𝑗 ∈ ( 1 ... 𝑀 ) ) |
| 303 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝑐 ‘ 𝑘 ) = ( 𝑐 ‘ 𝑗 ) ) |
| 304 |
280 281 282 285 302 303
|
fsumsplit1 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) = ( ( 𝑐 ‘ 𝑗 ) + Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 305 |
304
|
eqcomd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( ( 𝑐 ‘ 𝑗 ) + Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) |
| 306 |
305 275
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( ( 𝑐 ‘ 𝑗 ) + Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) ∈ ℝ ) |
| 307 |
|
elfzle1 |
⊢ ( ( 𝑐 ‘ 𝑗 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) → 0 ≤ ( 𝑐 ‘ 𝑗 ) ) |
| 308 |
277 307
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 0 ≤ ( 𝑐 ‘ 𝑗 ) ) |
| 309 |
308
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 0 ≤ ( 𝑐 ‘ 𝑗 ) ) |
| 310 |
|
neqne |
⊢ ( ¬ ( 𝑐 ‘ 𝑗 ) = 0 → ( 𝑐 ‘ 𝑗 ) ≠ 0 ) |
| 311 |
310
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑐 ‘ 𝑗 ) ≠ 0 ) |
| 312 |
276 279 309 311
|
leneltd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 0 < ( 𝑐 ‘ 𝑗 ) ) |
| 313 |
|
diffi |
⊢ ( ( 1 ... 𝑀 ) ∈ Fin → ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ∈ Fin ) |
| 314 |
104 313
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ∈ Fin ) |
| 315 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) → 𝑘 ∈ ( 1 ... 𝑀 ) ) |
| 316 |
315
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ) → 𝑘 ∈ ( 1 ... 𝑀 ) ) |
| 317 |
50 316
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ) → 𝑘 ∈ ( 0 ... 𝑀 ) ) |
| 318 |
44
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) ) |
| 319 |
186 318
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ℝ ) |
| 320 |
317 319
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ℝ ) |
| 321 |
|
elfzle1 |
⊢ ( ( 𝑐 ‘ 𝑘 ) ∈ ( 0 ... ( 𝑃 − 1 ) ) → 0 ≤ ( 𝑐 ‘ 𝑘 ) ) |
| 322 |
318 321
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 0 ≤ ( 𝑐 ‘ 𝑘 ) ) |
| 323 |
317 322
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ) → 0 ≤ ( 𝑐 ‘ 𝑘 ) ) |
| 324 |
314 320 323
|
fsumge0 |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → 0 ≤ Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) |
| 325 |
324
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 0 ≤ Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) |
| 326 |
314 320
|
fsumrecl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) → Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ∈ ℝ ) |
| 327 |
326
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ∈ ℝ ) |
| 328 |
278 327
|
addge01d |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 0 ≤ Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ↔ ( 𝑐 ‘ 𝑗 ) ≤ ( ( 𝑐 ‘ 𝑗 ) + Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) ) ) |
| 329 |
325 328
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ≤ ( ( 𝑐 ‘ 𝑗 ) + Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 330 |
329
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑐 ‘ 𝑗 ) ≤ ( ( 𝑐 ‘ 𝑗 ) + Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 331 |
276 279 306 312 330
|
ltletrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 0 < ( ( 𝑐 ‘ 𝑗 ) + Σ 𝑘 ∈ ( ( 1 ... 𝑀 ) ∖ { 𝑗 } ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 332 |
331 305
|
breqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 0 < Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) |
| 333 |
275 332
|
elrpd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ∈ ℝ+ ) |
| 334 |
267 333
|
ltaddrpd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑃 − 1 ) < ( ( 𝑃 − 1 ) + Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 335 |
334
|
adantl3r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑃 − 1 ) < ( ( 𝑃 − 1 ) + Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 336 |
|
fveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝑐 ‘ 𝑗 ) = ( 𝑐 ‘ 𝑘 ) ) |
| 337 |
336
|
cbvsumv |
⊢ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) |
| 338 |
337
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) |
| 339 |
72
|
ad5antr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
| 340 |
|
simp-5l |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ) |
| 341 |
73 318
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ℂ ) |
| 342 |
340 341
|
sylancom |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑘 ) ∈ ℂ ) |
| 343 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( 𝑐 ‘ 𝑘 ) = ( 𝑐 ‘ 0 ) ) |
| 344 |
339 342 343
|
fsum1p |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) = ( ( 𝑐 ‘ 0 ) + Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 345 |
256
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑐 ‘ 0 ) = ( 𝑃 − 1 ) ) |
| 346 |
85
|
sumeq1i |
⊢ Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) = Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) |
| 347 |
346
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) = Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) |
| 348 |
345 347
|
oveq12d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( ( 𝑐 ‘ 0 ) + Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) = ( ( 𝑃 − 1 ) + Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) ) |
| 349 |
338 344 348
|
3eqtrrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( ( 𝑃 − 1 ) + Σ 𝑘 ∈ ( 1 ... 𝑀 ) ( 𝑐 ‘ 𝑘 ) ) = Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) ) |
| 350 |
335 349
|
breqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ( 𝑃 − 1 ) < Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) ) |
| 351 |
266 350
|
gtned |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) ≠ ( 𝑃 − 1 ) ) |
| 352 |
351
|
neneqd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) ∧ ¬ ( 𝑐 ‘ 𝑗 ) = 0 ) → ¬ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = ( 𝑃 − 1 ) ) |
| 353 |
265 352
|
condan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → ( 𝑐 ‘ 𝑗 ) = 0 ) |
| 354 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → 𝑗 ∈ ( 0 ... 𝑀 ) ) |
| 355 |
33 66
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ∈ ℕ0 ) |
| 356 |
5
|
fvmpt2 |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ∈ ℕ0 ) → ( 𝐷 ‘ 𝑗 ) = if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ) |
| 357 |
354 355 356
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝐷 ‘ 𝑗 ) = if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ) |
| 358 |
357
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → ( 𝐷 ‘ 𝑗 ) = if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) ) |
| 359 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → ¬ 𝑗 = 0 ) |
| 360 |
359
|
iffalsed |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 0 ) = 0 ) |
| 361 |
358 360
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → 0 = ( 𝐷 ‘ 𝑗 ) ) |
| 362 |
361
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → 0 = ( 𝐷 ‘ 𝑗 ) ) |
| 363 |
362
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → 0 = ( 𝐷 ‘ 𝑗 ) ) |
| 364 |
353 363
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑗 = 0 ) → ( 𝑐 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 365 |
264 364
|
pm2.61dan |
⊢ ( ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) = ( 𝐷 ‘ 𝑗 ) ) |
| 366 |
250 252 365
|
eqfnfvd |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) → 𝑐 = 𝐷 ) |
| 367 |
235 366
|
sylanl2 |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) → 𝑐 = 𝐷 ) |
| 368 |
|
eldifsni |
⊢ ( 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) → 𝑐 ≠ 𝐷 ) |
| 369 |
368
|
neneqd |
⊢ ( 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) → ¬ 𝑐 = 𝐷 ) |
| 370 |
369
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) ∧ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) → ¬ 𝑐 = 𝐷 ) |
| 371 |
367 370
|
pm2.65da |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ¬ ( 𝑃 − 1 ) = ( 𝑐 ‘ 0 ) ) |
| 372 |
371
|
neqned |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑃 − 1 ) ≠ ( 𝑐 ‘ 0 ) ) |
| 373 |
239 240 242 372
|
leneltd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 𝑐 ‘ 0 ) < ( 𝑃 − 1 ) ) |
| 374 |
239 240
|
posdifd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( 𝑐 ‘ 0 ) < ( 𝑃 − 1 ) ↔ 0 < ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) |
| 375 |
373 374
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → 0 < ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) |
| 376 |
|
elnnz |
⊢ ( ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ∈ ℕ ↔ ( ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ∈ ℤ ∧ 0 < ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) |
| 377 |
248 375 376
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ∈ ℕ ) |
| 378 |
377
|
0expd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) = 0 ) |
| 379 |
378
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) = ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · 0 ) ) |
| 380 |
160
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℂ ) |
| 381 |
377
|
nnnn0d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ∈ ℕ0 ) |
| 382 |
381
|
faccld |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ∈ ℕ ) |
| 383 |
382
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ∈ ℂ ) |
| 384 |
382
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ≠ 0 ) |
| 385 |
380 383 384
|
divcld |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ∈ ℂ ) |
| 386 |
385
|
mul01d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · 0 ) = 0 ) |
| 387 |
244 379 386
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) = 0 ) |
| 388 |
387
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) = ( 0 · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) |
| 389 |
235 55
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ∈ ℤ ) |
| 390 |
389
|
zcnd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ∈ ℂ ) |
| 391 |
390
|
mul02d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 0 · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) = 0 ) |
| 392 |
388 391
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) = 0 ) |
| 393 |
392
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) = ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · 0 ) ) |
| 394 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( 0 ... 𝑀 ) ∈ Fin ) |
| 395 |
33 277
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( 𝐶 ‘ ( 𝑃 − 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ∈ ℕ0 ) |
| 396 |
235 395
|
sylanl2 |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑐 ‘ 𝑗 ) ∈ ℕ0 ) |
| 397 |
396
|
faccld |
⊢ ( ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ∈ ℕ ) |
| 398 |
394 397
|
fprodnncl |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ∈ ℕ ) |
| 399 |
398
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ∈ ℂ ) |
| 400 |
398
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ≠ 0 ) |
| 401 |
380 399 400
|
divcld |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) ∈ ℂ ) |
| 402 |
401
|
mul01d |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · 0 ) = 0 ) |
| 403 |
393 402
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ) → ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) = 0 ) |
| 404 |
403
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) = Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) 0 ) |
| 405 |
|
diffi |
⊢ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∈ Fin → ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ∈ Fin ) |
| 406 |
19 405
|
syl |
⊢ ( 𝜑 → ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ∈ Fin ) |
| 407 |
406
|
olcd |
⊢ ( 𝜑 → ( ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ∈ Fin ) ) |
| 408 |
|
sumz |
⊢ ( ( ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ∈ Fin ) → Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) 0 = 0 ) |
| 409 |
407 408
|
syl |
⊢ ( 𝜑 → Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) 0 = 0 ) |
| 410 |
404 409
|
eqtrd |
⊢ ( 𝜑 → Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) = 0 ) |
| 411 |
234 410
|
oveq12d |
⊢ ( 𝜑 → ( ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) + Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) ) = ( ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) + 0 ) ) |
| 412 |
232
|
addridd |
⊢ ( 𝜑 → ( ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) + 0 ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) ) |
| 413 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
| 414 |
413 205 227 219
|
fprodexp |
⊢ ( 𝜑 → ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) = ( ∏ 𝑗 ∈ ( 1 ... 𝑀 ) - 𝑗 ↑ 𝑃 ) ) |
| 415 |
414
|
oveq2d |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( - 𝑗 ↑ 𝑃 ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · ( ∏ 𝑗 ∈ ( 1 ... 𝑀 ) - 𝑗 ↑ 𝑃 ) ) ) |
| 416 |
411 412 415
|
3eqtrd |
⊢ ( 𝜑 → ( ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝐷 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝐷 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝐷 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝐷 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝐷 ‘ 𝑗 ) ) ) ) ) ) ) + Σ 𝑐 ∈ ( ( 𝐶 ‘ ( 𝑃 − 1 ) ) ∖ { 𝐷 } ) ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ∏ 𝑗 ∈ ( 0 ... 𝑀 ) ( ! ‘ ( 𝑐 ‘ 𝑗 ) ) ) · ( if ( ( 𝑃 − 1 ) < ( 𝑐 ‘ 0 ) , 0 , ( ( ( ! ‘ ( 𝑃 − 1 ) ) / ( ! ‘ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) · ( 0 ↑ ( ( 𝑃 − 1 ) − ( 𝑐 ‘ 0 ) ) ) ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) if ( 𝑃 < ( 𝑐 ‘ 𝑗 ) , 0 , ( ( ( ! ‘ 𝑃 ) / ( ! ‘ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) · ( ( 0 − 𝑗 ) ↑ ( 𝑃 − ( 𝑐 ‘ 𝑗 ) ) ) ) ) ) ) ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · ( ∏ 𝑗 ∈ ( 1 ... 𝑀 ) - 𝑗 ↑ 𝑃 ) ) ) |
| 417 |
16 142 416
|
3eqtrd |
⊢ ( 𝜑 → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) = ( ( ! ‘ ( 𝑃 − 1 ) ) · ( ∏ 𝑗 ∈ ( 1 ... 𝑀 ) - 𝑗 ↑ 𝑃 ) ) ) |