Step |
Hyp |
Ref |
Expression |
1 |
|
fprodsubrecnncnvlem.k |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
fprodsubrecnncnvlem.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
3 |
|
fprodsubrecnncnvlem.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
4 |
|
fprodsubrecnncnvlem.s |
⊢ 𝑆 = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) ) |
5 |
|
fprodsubrecnncnvlem.f |
⊢ 𝐹 = ( 𝑥 ∈ ℂ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) ) |
6 |
|
fprodsubrecnncnvlem.g |
⊢ 𝐺 = ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) |
7 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
8 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
9 |
1 2 3 5
|
fprodsub2cncf |
⊢ ( 𝜑 → 𝐹 ∈ ( ℂ –cn→ ℂ ) ) |
10 |
|
1rp |
⊢ 1 ∈ ℝ+ |
11 |
10
|
a1i |
⊢ ( 𝑛 ∈ ℕ → 1 ∈ ℝ+ ) |
12 |
|
nnrp |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ+ ) |
13 |
11 12
|
rpdivcld |
⊢ ( 𝑛 ∈ ℕ → ( 1 / 𝑛 ) ∈ ℝ+ ) |
14 |
13
|
rpcnd |
⊢ ( 𝑛 ∈ ℕ → ( 1 / 𝑛 ) ∈ ℂ ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1 / 𝑛 ) ∈ ℂ ) |
16 |
15 6
|
fmptd |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ℂ ) |
17 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
18 |
|
divcnv |
⊢ ( 1 ∈ ℂ → ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ⇝ 0 ) |
19 |
17 18
|
syl |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ⇝ 0 ) |
20 |
6
|
a1i |
⊢ ( 𝜑 → 𝐺 = ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ) |
21 |
20
|
breq1d |
⊢ ( 𝜑 → ( 𝐺 ⇝ 0 ↔ ( 𝑛 ∈ ℕ ↦ ( 1 / 𝑛 ) ) ⇝ 0 ) ) |
22 |
19 21
|
mpbird |
⊢ ( 𝜑 → 𝐺 ⇝ 0 ) |
23 |
|
0cnd |
⊢ ( 𝜑 → 0 ∈ ℂ ) |
24 |
7 8 9 16 22 23
|
climcncf |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐺 ) ⇝ ( 𝐹 ‘ 0 ) ) |
25 |
|
nfv |
⊢ Ⅎ 𝑘 𝑥 ∈ ℂ |
26 |
1 25
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑥 ∈ ℂ ) |
27 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → 𝐴 ∈ Fin ) |
28 |
3
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
29 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) ∧ 𝑘 ∈ 𝐴 ) → 𝑥 ∈ ℂ ) |
30 |
28 29
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 − 𝑥 ) ∈ ℂ ) |
31 |
26 27 30
|
fprodclf |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℂ ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) ∈ ℂ ) |
32 |
31 5
|
fmptd |
⊢ ( 𝜑 → 𝐹 : ℂ ⟶ ℂ ) |
33 |
|
fcompt |
⊢ ( ( 𝐹 : ℂ ⟶ ℂ ∧ 𝐺 : ℕ ⟶ ℂ ) → ( 𝐹 ∘ 𝐺 ) = ( 𝑛 ∈ ℕ ↦ ( 𝐹 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
34 |
32 16 33
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐺 ) = ( 𝑛 ∈ ℕ ↦ ( 𝐹 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
35 |
4
|
a1i |
⊢ ( 𝜑 → 𝑆 = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) ) ) |
36 |
|
id |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ ) |
37 |
6
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) ∈ ℂ ) → ( 𝐺 ‘ 𝑛 ) = ( 1 / 𝑛 ) ) |
38 |
36 14 37
|
syl2anc |
⊢ ( 𝑛 ∈ ℕ → ( 𝐺 ‘ 𝑛 ) = ( 1 / 𝑛 ) ) |
39 |
38
|
fveq2d |
⊢ ( 𝑛 ∈ ℕ → ( 𝐹 ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 𝐹 ‘ ( 1 / 𝑛 ) ) ) |
40 |
39
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 𝐹 ‘ ( 1 / 𝑛 ) ) ) |
41 |
|
oveq2 |
⊢ ( 𝑥 = ( 1 / 𝑛 ) → ( 𝐵 − 𝑥 ) = ( 𝐵 − ( 1 / 𝑛 ) ) ) |
42 |
41
|
prodeq2ad |
⊢ ( 𝑥 = ( 1 / 𝑛 ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) = ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) ) |
43 |
|
prodex |
⊢ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) ∈ V |
44 |
43
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) ∈ V ) |
45 |
5 42 15 44
|
fvmptd3 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ ( 1 / 𝑛 ) ) = ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) ) |
46 |
40 45
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) = ( 𝐹 ‘ ( 𝐺 ‘ 𝑛 ) ) ) |
47 |
46
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − ( 1 / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ( 𝐹 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
48 |
35 47
|
eqtrd |
⊢ ( 𝜑 → 𝑆 = ( 𝑛 ∈ ℕ ↦ ( 𝐹 ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
49 |
34 48
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐹 ∘ 𝐺 ) = 𝑆 ) |
50 |
5
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ℂ ↦ ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) ) ) |
51 |
|
nfv |
⊢ Ⅎ 𝑘 𝑥 = 0 |
52 |
1 51
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑥 = 0 ) |
53 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 𝐵 − 𝑥 ) = ( 𝐵 − 0 ) ) |
54 |
53
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 0 ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 − 𝑥 ) = ( 𝐵 − 0 ) ) |
55 |
3
|
subid1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 − 0 ) = 𝐵 ) |
56 |
55
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 0 ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 − 0 ) = 𝐵 ) |
57 |
54 56
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 = 0 ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 − 𝑥 ) = 𝐵 ) |
58 |
57
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 = 0 ) → ( 𝑘 ∈ 𝐴 → ( 𝐵 − 𝑥 ) = 𝐵 ) ) |
59 |
52 58
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑥 = 0 ) → ∀ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) = 𝐵 ) |
60 |
59
|
prodeq2d |
⊢ ( ( 𝜑 ∧ 𝑥 = 0 ) → ∏ 𝑘 ∈ 𝐴 ( 𝐵 − 𝑥 ) = ∏ 𝑘 ∈ 𝐴 𝐵 ) |
61 |
|
prodex |
⊢ ∏ 𝑘 ∈ 𝐴 𝐵 ∈ V |
62 |
61
|
a1i |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ V ) |
63 |
50 60 23 62
|
fvmptd |
⊢ ( 𝜑 → ( 𝐹 ‘ 0 ) = ∏ 𝑘 ∈ 𝐴 𝐵 ) |
64 |
49 63
|
breq12d |
⊢ ( 𝜑 → ( ( 𝐹 ∘ 𝐺 ) ⇝ ( 𝐹 ‘ 0 ) ↔ 𝑆 ⇝ ∏ 𝑘 ∈ 𝐴 𝐵 ) ) |
65 |
24 64
|
mpbid |
⊢ ( 𝜑 → 𝑆 ⇝ ∏ 𝑘 ∈ 𝐴 𝐵 ) |