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 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
5 |
|
prod0 |
⊢ ∏ 𝑘 ∈ ∅ 𝐵 = 1 |
6 |
|
prod0 |
⊢ ∏ 𝑘 ∈ ∅ 𝐶 = 1 |
7 |
5 6
|
oveq12i |
⊢ ( ∏ 𝑘 ∈ ∅ 𝐵 · ∏ 𝑘 ∈ ∅ 𝐶 ) = ( 1 · 1 ) |
8 |
|
prod0 |
⊢ ∏ 𝑘 ∈ ∅ ( 𝐵 · 𝐶 ) = 1 |
9 |
4 7 8
|
3eqtr4ri |
⊢ ∏ 𝑘 ∈ ∅ ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ ∅ 𝐵 · ∏ 𝑘 ∈ ∅ 𝐶 ) |
10 |
|
prodeq1 |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ∏ 𝑘 ∈ ∅ ( 𝐵 · 𝐶 ) ) |
11 |
|
prodeq1 |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ ∅ 𝐵 ) |
12 |
|
prodeq1 |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ ∅ 𝐶 ) |
13 |
11 12
|
oveq12d |
⊢ ( 𝐴 = ∅ → ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) = ( ∏ 𝑘 ∈ ∅ 𝐵 · ∏ 𝑘 ∈ ∅ 𝐶 ) ) |
14 |
9 10 13
|
3eqtr4a |
⊢ ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) ) |
15 |
14
|
a1i |
⊢ ( 𝜑 → ( 𝐴 = ∅ → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
16 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ℕ ) |
17 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
18 |
16 17
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ♯ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 1 ) ) |
19 |
2
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
21 |
|
f1of |
⊢ ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
22 |
21
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) |
23 |
|
fco |
⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
24 |
20 22 23
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
25 |
24
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) ∈ ℂ ) |
26 |
3
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ℂ ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ℂ ) |
28 |
|
fco |
⊢ ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ ℂ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
29 |
27 22 28
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ ℂ ) |
30 |
29
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) ∈ ℂ ) |
31 |
22
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( 𝑓 ‘ 𝑛 ) ∈ 𝐴 ) |
32 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ 𝐴 ) |
33 |
2 3
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 · 𝐶 ) ∈ ℂ ) |
34 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) = ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) |
35 |
34
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ ( 𝐵 · 𝐶 ) ∈ ℂ ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑘 ) = ( 𝐵 · 𝐶 ) ) |
36 |
32 33 35
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑘 ) = ( 𝐵 · 𝐶 ) ) |
37 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
38 |
37
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝐵 ∈ ℂ ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = 𝐵 ) |
39 |
32 2 38
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = 𝐵 ) |
40 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) |
41 |
40
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝐶 ∈ ℂ ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) = 𝐶 ) |
42 |
32 3 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) = 𝐶 ) |
43 |
39 42
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) · ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) = ( 𝐵 · 𝐶 ) ) |
44 |
36 43
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) · ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) ) |
45 |
44
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑘 ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) · ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) ) |
46 |
45
|
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 ) ) ) |
47 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) |
48 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) |
49 |
|
nfcv |
⊢ Ⅎ 𝑘 · |
50 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) |
51 |
48 49 50
|
nfov |
⊢ Ⅎ 𝑘 ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) · ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
52 |
47 51
|
nfeq |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) · ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
53 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
54 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
55 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
56 |
54 55
|
oveq12d |
⊢ ( 𝑘 = ( 𝑓 ‘ 𝑛 ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) · ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑘 ) ) = ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) · ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) ) |
57 |
53 56
|
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 ) ) ) ) ) |
58 |
52 57
|
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 ) ) ) ) ) |
59 |
31 46 58
|
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 ) ) ) ) |
60 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
61 |
22 60
|
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 ) ) ) |
62 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
63 |
22 62
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
64 |
|
fvco3 |
⊢ ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) ⟶ 𝐴 ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
65 |
22 64
|
sylan |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑛 ∈ ( 1 ... ( ♯ ‘ 𝐴 ) ) ) → ( ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ‘ 𝑛 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
66 |
63 65
|
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 ) ) ) ) |
67 |
59 61 66
|
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 ) ) ) |
68 |
18 25 30 67
|
prodfmul |
⊢ ( ( 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 ) ) ) ) |
69 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
70 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) |
71 |
33
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) : 𝐴 ⟶ ℂ ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) : 𝐴 ⟶ ℂ ) |
73 |
72
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑚 ) ∈ ℂ ) |
74 |
69 16 70 73 61
|
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 ) ) ) |
75 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
76 |
20
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) ∈ ℂ ) |
77 |
75 16 70 76 63
|
fprod |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
78 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑓 ‘ 𝑛 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ ( 𝑓 ‘ 𝑛 ) ) ) |
79 |
27
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ∧ 𝑚 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) ∈ ℂ ) |
80 |
78 16 70 79 65
|
fprod |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) = ( seq 1 ( · , ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝐴 ) ) ) |
81 |
77 80
|
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 ) ) ) ) |
82 |
68 74 81
|
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 ) ) ) |
83 |
|
prodfc |
⊢ ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ‘ 𝑚 ) = ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) |
84 |
|
prodfc |
⊢ ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) = ∏ 𝑘 ∈ 𝐴 𝐵 |
85 |
|
prodfc |
⊢ ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) = ∏ 𝑘 ∈ 𝐴 𝐶 |
86 |
84 85
|
oveq12i |
⊢ ( ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑚 ) · ∏ 𝑚 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐶 ) ‘ 𝑚 ) ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) |
87 |
82 83 86
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) ) |
88 |
87
|
expr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
89 |
88
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝐴 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
90 |
89
|
expimpd |
⊢ ( 𝜑 → ( ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) ) ) |
91 |
|
fz1f1o |
⊢ ( 𝐴 ∈ Fin → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
92 |
1 91
|
syl |
⊢ ( 𝜑 → ( 𝐴 = ∅ ∨ ( ( ♯ ‘ 𝐴 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝐴 ) ) –1-1-onto→ 𝐴 ) ) ) |
93 |
15 90 92
|
mpjaod |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = ( ∏ 𝑘 ∈ 𝐴 𝐵 · ∏ 𝑘 ∈ 𝐴 𝐶 ) ) |