Step |
Hyp |
Ref |
Expression |
1 |
|
fprodmul.1 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
2 |
|
fprodmul.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
3 |
|
fprodmul.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
4 |
|
fproddiv.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ≠ 0 ) |
5 |
|
1div1e1 |
⊢ ( 1 / 1 ) = 1 |
6 |
5
|
eqcomi |
⊢ 1 = ( 1 / 1 ) |
7 |
|
prodeq1 |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ∏ 𝑘 ∈ ∅ ( 𝐵 / 𝐶 ) ) |
8 |
|
prod0 |
⊢ ∏ 𝑘 ∈ ∅ ( 𝐵 / 𝐶 ) = 1 |
9 |
7 8
|
eqtrdi |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = 1 ) |
10 |
|
prodeq1 |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ ∅ 𝐵 ) |
11 |
|
prod0 |
⊢ ∏ 𝑘 ∈ ∅ 𝐵 = 1 |
12 |
10 11
|
eqtrdi |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = 1 ) |
13 |
|
prodeq1 |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ ∅ 𝐶 ) |
14 |
|
prod0 |
⊢ ∏ 𝑘 ∈ ∅ 𝐶 = 1 |
15 |
13 14
|
eqtrdi |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐶 = 1 ) |
16 |
12 15
|
oveq12d |
⊢ ( 𝐴 = ∅ → ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) = ( 1 / 1 ) ) |
17 |
6 9 16
|
3eqtr4a |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) ) |
18 |
17
|
a1i |
⊢ ( 𝜑 → ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
19 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
20 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
21 |
19 20
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 1 ) ) |
22 |
2
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
23 |
|
f1of |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
24 |
23
|
adantl |
⊢ ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
25 |
|
fco |
⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
26 |
22 24 25
|
syl2an |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
27 |
26
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) ∈ ℂ ) |
28 |
3
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ℂ ) |
29 |
|
fco |
⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ℂ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
30 |
28 24 29
|
syl2an |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
31 |
30
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) ∈ ℂ ) |
32 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
33 |
32 23
|
syl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
34 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
35 |
33 34
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
36 |
33
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( 𝑓 ‘ 𝑛 ) ∈ 𝐴 ) |
37 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ 𝐴 ) |
38 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) |
39 |
38
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) = 𝐶 ) |
40 |
37 3 39
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) = 𝐶 ) |
41 |
40 4
|
eqnetrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ≠ 0 ) |
42 |
41
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ≠ 0 ) |
43 |
42
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ≠ 0 ) |
44 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) |
45 |
|
nfcv |
⊢ Ⅎ 𝑘 0 |
46 |
44 45
|
nfne |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ≠ 0 |
47 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
48 |
47
|
neeq1d |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ≠ 0 ↔ ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ≠ 0 ) ) |
49 |
46 48
|
rspc |
⊢ ( ( 𝑓 ‘ 𝑛 ) ∈ 𝐴 → ( ∀ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ≠ 0 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ≠ 0 ) ) |
50 |
36 43 49
|
sylc |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ≠ 0 ) |
51 |
35 50
|
eqnetrd |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) ≠ 0 ) |
52 |
2 3 4
|
divcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 / 𝐶 ) ∈ ℂ ) |
53 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) = ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) |
54 |
53
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ ( 𝐵 / 𝐶 ) ∈ ℂ ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( 𝐵 / 𝐶 ) ) |
55 |
37 52 54
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( 𝐵 / 𝐶 ) ) |
56 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
57 |
56
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = 𝐵 ) |
58 |
37 2 57
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = 𝐵 ) |
59 |
58 40
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) = ( 𝐵 / 𝐶 ) ) |
60 |
55 59
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) ) |
61 |
60
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) ) |
62 |
61
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) ) |
63 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) |
64 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) |
65 |
|
nfcv |
⊢ Ⅎ 𝑘 / |
66 |
64 65 44
|
nfov |
⊢ Ⅎ 𝑘 ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
67 |
63 66
|
nfeq |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
68 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
69 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
70 |
69 47
|
oveq12d |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
71 |
68 70
|
eqeq12d |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) ↔ ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) ) |
72 |
67 71
|
rspc |
⊢ ( ( 𝑓 ‘ 𝑛 ) ∈ 𝐴 → ( ∀ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) ) |
73 |
36 62 72
|
sylc |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
74 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
75 |
33 74
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
76 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
77 |
33 76
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
78 |
77 35
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) / ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) / ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
79 |
73 75 78
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) / ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) ) ) |
80 |
21 27 31 51 79
|
prodfdiv |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) = ( ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) / ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) ) |
81 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
82 |
52
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) : 𝐴 ⟶ ℂ ) |
83 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) : 𝐴 ⟶ ℂ ) |
84 |
83
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑚 ) ∈ ℂ ) |
85 |
81 19 32 84 75
|
fprod |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑚 ) = ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
86 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
87 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
88 |
87
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ∈ ℂ ) |
89 |
86 19 32 88 77
|
fprod |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
90 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
91 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ℂ ) |
92 |
91
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) ∈ ℂ ) |
93 |
90 19 32 92 35
|
fprod |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) = ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
94 |
89 93
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) / ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) ) = ( ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) / ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) ) |
95 |
80 85 94
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑚 ) = ( ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) / ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) ) ) |
96 |
|
prodfc |
⊢ ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 / 𝐶 ) ) ‘ 𝑚 ) = ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) |
97 |
|
prodfc |
⊢ ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ∏ 𝑘 ∈ 𝐴 𝐵 |
98 |
|
prodfc |
⊢ ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) = ∏ 𝑘 ∈ 𝐴 𝐶 |
99 |
97 98
|
oveq12i |
⊢ ( ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) / ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) |
100 |
95 96 99
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) ) |
101 |
100
|
expr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
102 |
101
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
103 |
102
|
expimpd |
⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
104 |
|
fz1f1o |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
105 |
1 104
|
syl |
⊢ ( 𝜑 → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
106 |
18 103 105
|
mpjaod |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 ( 𝐵 / 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 / ∏ 𝑘 ∈ 𝐴 𝐶 ) ) |