Step |
Hyp |
Ref |
Expression |
1 |
|
breprexp.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
2 |
|
breprexp.s |
⊢ ( 𝜑 → 𝑆 ∈ ℕ0 ) |
3 |
|
breprexp.z |
⊢ ( 𝜑 → 𝑍 ∈ ℂ ) |
4 |
|
breprexpnat.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ ) |
5 |
|
breprexpnat.p |
⊢ 𝑃 = Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) |
6 |
|
breprexpnat.r |
⊢ 𝑅 = ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) |
7 |
|
fvex |
⊢ ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ∈ V |
8 |
7
|
fconst |
⊢ ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) : ( 0 ..^ 𝑆 ) ⟶ { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } |
9 |
|
nnex |
⊢ ℕ ∈ V |
10 |
|
indf |
⊢ ( ( ℕ ∈ V ∧ 𝐴 ⊆ ℕ ) → ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) : ℕ ⟶ { 0 , 1 } ) |
11 |
9 4 10
|
sylancr |
⊢ ( 𝜑 → ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) : ℕ ⟶ { 0 , 1 } ) |
12 |
|
0cn |
⊢ 0 ∈ ℂ |
13 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
14 |
|
prssi |
⊢ ( ( 0 ∈ ℂ ∧ 1 ∈ ℂ ) → { 0 , 1 } ⊆ ℂ ) |
15 |
12 13 14
|
mp2an |
⊢ { 0 , 1 } ⊆ ℂ |
16 |
|
fss |
⊢ ( ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) : ℕ ⟶ { 0 , 1 } ∧ { 0 , 1 } ⊆ ℂ ) → ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) : ℕ ⟶ ℂ ) |
17 |
11 15 16
|
sylancl |
⊢ ( 𝜑 → ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) : ℕ ⟶ ℂ ) |
18 |
|
cnex |
⊢ ℂ ∈ V |
19 |
18 9
|
elmap |
⊢ ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ∈ ( ℂ ↑m ℕ ) ↔ ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) : ℕ ⟶ ℂ ) |
20 |
17 19
|
sylibr |
⊢ ( 𝜑 → ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ∈ ( ℂ ↑m ℕ ) ) |
21 |
7
|
snss |
⊢ ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ∈ ( ℂ ↑m ℕ ) ↔ { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ⊆ ( ℂ ↑m ℕ ) ) |
22 |
20 21
|
sylib |
⊢ ( 𝜑 → { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ⊆ ( ℂ ↑m ℕ ) ) |
23 |
|
fss |
⊢ ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) : ( 0 ..^ 𝑆 ) ⟶ { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ∧ { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ⊆ ( ℂ ↑m ℕ ) ) → ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) : ( 0 ..^ 𝑆 ) ⟶ ( ℂ ↑m ℕ ) ) |
24 |
8 22 23
|
sylancr |
⊢ ( 𝜑 → ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) : ( 0 ..^ 𝑆 ) ⟶ ( ℂ ↑m ℕ ) ) |
25 |
1 2 3 24
|
breprexp |
⊢ ( 𝜑 → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) Σ 𝑏 ∈ ( 1 ... 𝑁 ) ( ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) = Σ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
26 |
7
|
fvconst2 |
⊢ ( 𝑎 ∈ ( 0 ..^ 𝑆 ) → ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) = ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ) |
27 |
26
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) ∧ 𝑏 ∈ ( 1 ... 𝑁 ) ) → ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) = ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ) |
28 |
27
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) ∧ 𝑏 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ 𝑏 ) = ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ 𝑏 ) ) |
29 |
28
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) ∧ 𝑏 ∈ ( 1 ... 𝑁 ) ) → ( ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) = ( ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) ) |
30 |
29
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → Σ 𝑏 ∈ ( 1 ... 𝑁 ) ( ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) = Σ 𝑏 ∈ ( 1 ... 𝑁 ) ( ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) ) |
31 |
9
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ℕ ∈ V ) |
32 |
|
fzfi |
⊢ ( 1 ... 𝑁 ) ∈ Fin |
33 |
32
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( 1 ... 𝑁 ) ∈ Fin ) |
34 |
|
fz1ssnn |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ |
35 |
34
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( 1 ... 𝑁 ) ⊆ ℕ ) |
36 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → 𝐴 ⊆ ℕ ) |
37 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) ∧ 𝑏 ∈ ( 1 ... 𝑁 ) ) → 𝑍 ∈ ℂ ) |
38 |
|
nnssnn0 |
⊢ ℕ ⊆ ℕ0 |
39 |
34 38
|
sstri |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ0 |
40 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) ∧ 𝑏 ∈ ( 1 ... 𝑁 ) ) → 𝑏 ∈ ( 1 ... 𝑁 ) ) |
41 |
39 40
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) ∧ 𝑏 ∈ ( 1 ... 𝑁 ) ) → 𝑏 ∈ ℕ0 ) |
42 |
37 41
|
expcld |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) ∧ 𝑏 ∈ ( 1 ... 𝑁 ) ) → ( 𝑍 ↑ 𝑏 ) ∈ ℂ ) |
43 |
31 33 35 36 42
|
indsumin |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → Σ 𝑏 ∈ ( 1 ... 𝑁 ) ( ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) = Σ 𝑏 ∈ ( ( 1 ... 𝑁 ) ∩ 𝐴 ) ( 𝑍 ↑ 𝑏 ) ) |
44 |
|
incom |
⊢ ( ( 1 ... 𝑁 ) ∩ 𝐴 ) = ( 𝐴 ∩ ( 1 ... 𝑁 ) ) |
45 |
44
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( ( 1 ... 𝑁 ) ∩ 𝐴 ) = ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ) |
46 |
45
|
sumeq1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → Σ 𝑏 ∈ ( ( 1 ... 𝑁 ) ∩ 𝐴 ) ( 𝑍 ↑ 𝑏 ) = Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ) |
47 |
30 43 46
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → Σ 𝑏 ∈ ( 1 ... 𝑁 ) ( ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) = Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ) |
48 |
47
|
prodeq2dv |
⊢ ( 𝜑 → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) Σ 𝑏 ∈ ( 1 ... 𝑁 ) ( ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) = ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ) |
49 |
|
fzofi |
⊢ ( 0 ..^ 𝑆 ) ∈ Fin |
50 |
49
|
a1i |
⊢ ( 𝜑 → ( 0 ..^ 𝑆 ) ∈ Fin ) |
51 |
|
inss2 |
⊢ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 ) |
52 |
|
ssfi |
⊢ ( ( ( 1 ... 𝑁 ) ∈ Fin ∧ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ⊆ ( 1 ... 𝑁 ) ) → ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ∈ Fin ) |
53 |
32 51 52
|
mp2an |
⊢ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ∈ Fin |
54 |
53
|
a1i |
⊢ ( 𝜑 → ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ∈ Fin ) |
55 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ) → 𝑍 ∈ ℂ ) |
56 |
51 39
|
sstri |
⊢ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ⊆ ℕ0 |
57 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ) → 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ) |
58 |
56 57
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ) → 𝑏 ∈ ℕ0 ) |
59 |
55 58
|
expcld |
⊢ ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ) → ( 𝑍 ↑ 𝑏 ) ∈ ℂ ) |
60 |
54 59
|
fsumcl |
⊢ ( 𝜑 → Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ∈ ℂ ) |
61 |
|
fprodconst |
⊢ ( ( ( 0 ..^ 𝑆 ) ∈ Fin ∧ Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ∈ ℂ ) → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) = ( Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ↑ ( ♯ ‘ ( 0 ..^ 𝑆 ) ) ) ) |
62 |
50 60 61
|
syl2anc |
⊢ ( 𝜑 → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) = ( Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ↑ ( ♯ ‘ ( 0 ..^ 𝑆 ) ) ) ) |
63 |
|
hashfzo0 |
⊢ ( 𝑆 ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ 𝑆 ) ) = 𝑆 ) |
64 |
2 63
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 0 ..^ 𝑆 ) ) = 𝑆 ) |
65 |
64
|
oveq2d |
⊢ ( 𝜑 → ( Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ↑ ( ♯ ‘ ( 0 ..^ 𝑆 ) ) ) = ( Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ↑ 𝑆 ) ) |
66 |
48 62 65
|
3eqtrd |
⊢ ( 𝜑 → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) Σ 𝑏 ∈ ( 1 ... 𝑁 ) ( ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ 𝑏 ) · ( 𝑍 ↑ 𝑏 ) ) = ( Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ↑ 𝑆 ) ) |
67 |
34
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ( 1 ... 𝑁 ) ⊆ ℕ ) |
68 |
|
fzssz |
⊢ ( 0 ... ( 𝑆 · 𝑁 ) ) ⊆ ℤ |
69 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) |
70 |
68 69
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → 𝑚 ∈ ℤ ) |
71 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → 𝑆 ∈ ℕ0 ) |
72 |
32
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ( 1 ... 𝑁 ) ∈ Fin ) |
73 |
67 70 71 72
|
reprfi |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ∈ Fin ) |
74 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → 𝑍 ∈ ℂ ) |
75 |
|
fz0ssnn0 |
⊢ ( 0 ... ( 𝑆 · 𝑁 ) ) ⊆ ℕ0 |
76 |
75 69
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → 𝑚 ∈ ℕ0 ) |
77 |
74 76
|
expcld |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ( 𝑍 ↑ 𝑚 ) ∈ ℂ ) |
78 |
49
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → ( 0 ..^ 𝑆 ) ∈ Fin ) |
79 |
11
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) : ℕ ⟶ { 0 , 1 } ) |
80 |
34
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → ( 1 ... 𝑁 ) ⊆ ℕ ) |
81 |
70
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → 𝑚 ∈ ℤ ) |
82 |
71
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → 𝑆 ∈ ℕ0 ) |
83 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) |
84 |
80 81 82 83
|
reprf |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → 𝑐 : ( 0 ..^ 𝑆 ) ⟶ ( 1 ... 𝑁 ) ) |
85 |
84
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( 𝑐 ‘ 𝑎 ) ∈ ( 1 ... 𝑁 ) ) |
86 |
34 85
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( 𝑐 ‘ 𝑎 ) ∈ ℕ ) |
87 |
79 86
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ∈ { 0 , 1 } ) |
88 |
15 87
|
sselid |
⊢ ( ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ∈ ℂ ) |
89 |
78 88
|
fprodcl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ∈ ℂ ) |
90 |
73 77 89
|
fsummulc1 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ( Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) = Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
91 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → 𝐴 ⊆ ℕ ) |
92 |
91 70 71 72 67
|
hashreprin |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) = Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |
93 |
92
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ( ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) · ( 𝑍 ↑ 𝑚 ) ) = ( Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
94 |
26
|
fveq1d |
⊢ ( 𝑎 ∈ ( 0 ..^ 𝑆 ) → ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) = ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |
95 |
94
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑎 ∈ ( 0 ..^ 𝑆 ) ) → ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) = ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |
96 |
95
|
prodeq2dv |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) = ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |
97 |
96
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) = ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |
98 |
97
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) ∧ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ) → ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) = ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
99 |
98
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) = Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
100 |
90 93 99
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ) → Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) = ( ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
101 |
100
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) Σ 𝑐 ∈ ( ( 1 ... 𝑁 ) ( repr ‘ 𝑆 ) 𝑚 ) ( ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( ( 0 ..^ 𝑆 ) × { ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) } ) ‘ 𝑎 ) ‘ ( 𝑐 ‘ 𝑎 ) ) · ( 𝑍 ↑ 𝑚 ) ) = Σ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ( ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
102 |
25 66 101
|
3eqtr3d |
⊢ ( 𝜑 → ( Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ↑ 𝑆 ) = Σ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ( ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
103 |
5
|
oveq1i |
⊢ ( 𝑃 ↑ 𝑆 ) = ( Σ 𝑏 ∈ ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( 𝑍 ↑ 𝑏 ) ↑ 𝑆 ) |
104 |
6
|
oveq1i |
⊢ ( 𝑅 · ( 𝑍 ↑ 𝑚 ) ) = ( ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) · ( 𝑍 ↑ 𝑚 ) ) |
105 |
104
|
a1i |
⊢ ( 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) → ( 𝑅 · ( 𝑍 ↑ 𝑚 ) ) = ( ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) · ( 𝑍 ↑ 𝑚 ) ) ) |
106 |
105
|
sumeq2i |
⊢ Σ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ( 𝑅 · ( 𝑍 ↑ 𝑚 ) ) = Σ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ( ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑁 ) ) ( repr ‘ 𝑆 ) 𝑚 ) ) · ( 𝑍 ↑ 𝑚 ) ) |
107 |
102 103 106
|
3eqtr4g |
⊢ ( 𝜑 → ( 𝑃 ↑ 𝑆 ) = Σ 𝑚 ∈ ( 0 ... ( 𝑆 · 𝑁 ) ) ( 𝑅 · ( 𝑍 ↑ 𝑚 ) ) ) |