Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑎 = 1 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 1 ) ) |
2 |
|
1z |
⊢ 1 ∈ ℤ |
3 |
|
seq1 |
⊢ ( 1 ∈ ℤ → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 1 ) = ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) ) |
4 |
2 3
|
ax-mp |
⊢ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 1 ) = ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) |
5 |
1 4
|
eqtrdi |
⊢ ( 𝑎 = 1 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) ) |
6 |
|
oveq1 |
⊢ ( 𝑎 = 1 → ( 𝑎 + 1 ) = ( 1 + 1 ) ) |
7 |
|
oveq1 |
⊢ ( 𝑎 = 1 → ( 𝑎 + ( 𝑀 + 1 ) ) = ( 1 + ( 𝑀 + 1 ) ) ) |
8 |
6 7
|
oveq12d |
⊢ ( 𝑎 = 1 → ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) = ( ( 1 + 1 ) / ( 1 + ( 𝑀 + 1 ) ) ) ) |
9 |
8
|
oveq2d |
⊢ ( 𝑎 = 1 → ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) = ( ( 𝑀 + 1 ) · ( ( 1 + 1 ) / ( 1 + ( 𝑀 + 1 ) ) ) ) ) |
10 |
5 9
|
eqeq12d |
⊢ ( 𝑎 = 1 → ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ↔ ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) = ( ( 𝑀 + 1 ) · ( ( 1 + 1 ) / ( 1 + ( 𝑀 + 1 ) ) ) ) ) ) |
11 |
10
|
imbi2d |
⊢ ( 𝑎 = 1 → ( ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) = ( ( 𝑀 + 1 ) · ( ( 1 + 1 ) / ( 1 + ( 𝑀 + 1 ) ) ) ) ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑎 = 𝑘 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) ) |
13 |
|
oveq1 |
⊢ ( 𝑎 = 𝑘 → ( 𝑎 + 1 ) = ( 𝑘 + 1 ) ) |
14 |
|
oveq1 |
⊢ ( 𝑎 = 𝑘 → ( 𝑎 + ( 𝑀 + 1 ) ) = ( 𝑘 + ( 𝑀 + 1 ) ) ) |
15 |
13 14
|
oveq12d |
⊢ ( 𝑎 = 𝑘 → ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) = ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) |
16 |
15
|
oveq2d |
⊢ ( 𝑎 = 𝑘 → ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) |
17 |
12 16
|
eqeq12d |
⊢ ( 𝑎 = 𝑘 → ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ↔ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) ) |
18 |
17
|
imbi2d |
⊢ ( 𝑎 = 𝑘 → ( ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) ) ) |
19 |
|
fveq2 |
⊢ ( 𝑎 = ( 𝑘 + 1 ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) |
20 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝑘 + 1 ) → ( 𝑎 + 1 ) = ( ( 𝑘 + 1 ) + 1 ) ) |
21 |
|
oveq1 |
⊢ ( 𝑎 = ( 𝑘 + 1 ) → ( 𝑎 + ( 𝑀 + 1 ) ) = ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) |
22 |
20 21
|
oveq12d |
⊢ ( 𝑎 = ( 𝑘 + 1 ) → ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) = ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) |
23 |
22
|
oveq2d |
⊢ ( 𝑎 = ( 𝑘 + 1 ) → ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) |
24 |
19 23
|
eqeq12d |
⊢ ( 𝑎 = ( 𝑘 + 1 ) → ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ↔ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) ) |
25 |
24
|
imbi2d |
⊢ ( 𝑎 = ( 𝑘 + 1 ) → ( ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) ) ) |
26 |
|
fveq2 |
⊢ ( 𝑎 = 𝑏 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) ) |
27 |
|
oveq1 |
⊢ ( 𝑎 = 𝑏 → ( 𝑎 + 1 ) = ( 𝑏 + 1 ) ) |
28 |
|
oveq1 |
⊢ ( 𝑎 = 𝑏 → ( 𝑎 + ( 𝑀 + 1 ) ) = ( 𝑏 + ( 𝑀 + 1 ) ) ) |
29 |
27 28
|
oveq12d |
⊢ ( 𝑎 = 𝑏 → ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) = ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) |
30 |
29
|
oveq2d |
⊢ ( 𝑎 = 𝑏 → ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) |
31 |
26 30
|
eqeq12d |
⊢ ( 𝑎 = 𝑏 → ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ↔ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) ) |
32 |
31
|
imbi2d |
⊢ ( 𝑎 = 𝑏 → ( ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑎 ) = ( ( 𝑀 + 1 ) · ( ( 𝑎 + 1 ) / ( 𝑎 + ( 𝑀 + 1 ) ) ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) ) ) |
33 |
|
1nn |
⊢ 1 ∈ ℕ |
34 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 𝑀 / 𝑛 ) = ( 𝑀 / 1 ) ) |
35 |
34
|
oveq2d |
⊢ ( 𝑛 = 1 → ( 1 + ( 𝑀 / 𝑛 ) ) = ( 1 + ( 𝑀 / 1 ) ) ) |
36 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 1 / 𝑛 ) = ( 1 / 1 ) ) |
37 |
36
|
oveq2d |
⊢ ( 𝑛 = 1 → ( 1 + ( 1 / 𝑛 ) ) = ( 1 + ( 1 / 1 ) ) ) |
38 |
35 37
|
oveq12d |
⊢ ( 𝑛 = 1 → ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) = ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) ) |
39 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( ( 𝑀 + 1 ) / 𝑛 ) = ( ( 𝑀 + 1 ) / 1 ) ) |
40 |
39
|
oveq2d |
⊢ ( 𝑛 = 1 → ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) = ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) ) |
41 |
38 40
|
oveq12d |
⊢ ( 𝑛 = 1 → ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) = ( ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) ) ) |
42 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) = ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) |
43 |
|
ovex |
⊢ ( ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) ) ∈ V |
44 |
41 42 43
|
fvmpt |
⊢ ( 1 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) = ( ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) ) ) |
45 |
33 44
|
ax-mp |
⊢ ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) = ( ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) ) |
46 |
|
nn0cn |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ ) |
47 |
46
|
div1d |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 / 1 ) = 𝑀 ) |
48 |
47
|
oveq2d |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + ( 𝑀 / 1 ) ) = ( 1 + 𝑀 ) ) |
49 |
|
1div1e1 |
⊢ ( 1 / 1 ) = 1 |
50 |
49
|
oveq2i |
⊢ ( 1 + ( 1 / 1 ) ) = ( 1 + 1 ) |
51 |
50
|
a1i |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + ( 1 / 1 ) ) = ( 1 + 1 ) ) |
52 |
48 51
|
oveq12d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) = ( ( 1 + 𝑀 ) · ( 1 + 1 ) ) ) |
53 |
|
nn0p1nn |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 + 1 ) ∈ ℕ ) |
54 |
53
|
nncnd |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 + 1 ) ∈ ℂ ) |
55 |
54
|
div1d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( 𝑀 + 1 ) / 1 ) = ( 𝑀 + 1 ) ) |
56 |
55
|
oveq2d |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) = ( 1 + ( 𝑀 + 1 ) ) ) |
57 |
52 56
|
oveq12d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) ) = ( ( ( 1 + 𝑀 ) · ( 1 + 1 ) ) / ( 1 + ( 𝑀 + 1 ) ) ) ) |
58 |
|
1cnd |
⊢ ( 𝑀 ∈ ℕ0 → 1 ∈ ℂ ) |
59 |
58 46
|
addcomd |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + 𝑀 ) = ( 𝑀 + 1 ) ) |
60 |
59
|
oveq1d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( 1 + 𝑀 ) · ( 1 + 1 ) ) = ( ( 𝑀 + 1 ) · ( 1 + 1 ) ) ) |
61 |
60
|
oveq1d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( ( 1 + 𝑀 ) · ( 1 + 1 ) ) / ( 1 + ( 𝑀 + 1 ) ) ) = ( ( ( 𝑀 + 1 ) · ( 1 + 1 ) ) / ( 1 + ( 𝑀 + 1 ) ) ) ) |
62 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
63 |
62 62
|
addcli |
⊢ ( 1 + 1 ) ∈ ℂ |
64 |
63
|
a1i |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + 1 ) ∈ ℂ ) |
65 |
33
|
a1i |
⊢ ( 𝑀 ∈ ℕ0 → 1 ∈ ℕ ) |
66 |
65 53
|
nnaddcld |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + ( 𝑀 + 1 ) ) ∈ ℕ ) |
67 |
66
|
nncnd |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + ( 𝑀 + 1 ) ) ∈ ℂ ) |
68 |
66
|
nnne0d |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 + ( 𝑀 + 1 ) ) ≠ 0 ) |
69 |
54 64 67 68
|
divassd |
⊢ ( 𝑀 ∈ ℕ0 → ( ( ( 𝑀 + 1 ) · ( 1 + 1 ) ) / ( 1 + ( 𝑀 + 1 ) ) ) = ( ( 𝑀 + 1 ) · ( ( 1 + 1 ) / ( 1 + ( 𝑀 + 1 ) ) ) ) ) |
70 |
57 61 69
|
3eqtrd |
⊢ ( 𝑀 ∈ ℕ0 → ( ( ( 1 + ( 𝑀 / 1 ) ) · ( 1 + ( 1 / 1 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 1 ) ) ) = ( ( 𝑀 + 1 ) · ( ( 1 + 1 ) / ( 1 + ( 𝑀 + 1 ) ) ) ) ) |
71 |
45 70
|
syl5eq |
⊢ ( 𝑀 ∈ ℕ0 → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ 1 ) = ( ( 𝑀 + 1 ) · ( ( 1 + 1 ) / ( 1 + ( 𝑀 + 1 ) ) ) ) ) |
72 |
|
seqp1 |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 1 ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
73 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
74 |
72 73
|
eleq2s |
⊢ ( 𝑘 ∈ ℕ → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
75 |
74
|
adantr |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
76 |
75
|
adantr |
⊢ ( ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) ∧ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
77 |
|
oveq1 |
⊢ ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) → ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
78 |
77
|
adantl |
⊢ ( ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) ∧ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) → ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) ) |
79 |
|
peano2nn |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ ) |
80 |
|
oveq2 |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 𝑀 / 𝑛 ) = ( 𝑀 / ( 𝑘 + 1 ) ) ) |
81 |
80
|
oveq2d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 1 + ( 𝑀 / 𝑛 ) ) = ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) ) |
82 |
|
oveq2 |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 1 / 𝑛 ) = ( 1 / ( 𝑘 + 1 ) ) ) |
83 |
82
|
oveq2d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 1 + ( 1 / 𝑛 ) ) = ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) |
84 |
81 83
|
oveq12d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) = ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) |
85 |
|
oveq2 |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( ( 𝑀 + 1 ) / 𝑛 ) = ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) |
86 |
85
|
oveq2d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) = ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) |
87 |
84 86
|
oveq12d |
⊢ ( 𝑛 = ( 𝑘 + 1 ) → ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) = ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) |
88 |
|
ovex |
⊢ ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ∈ V |
89 |
87 42 88
|
fvmpt |
⊢ ( ( 𝑘 + 1 ) ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) |
90 |
79 89
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) |
91 |
90
|
oveq2d |
⊢ ( 𝑘 ∈ ℕ → ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) ) |
92 |
91
|
adantr |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) ) |
93 |
53
|
adantl |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 + 1 ) ∈ ℕ ) |
94 |
93
|
nncnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 + 1 ) ∈ ℂ ) |
95 |
79
|
adantr |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℕ ) |
96 |
95
|
nnrpd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℝ+ ) |
97 |
|
simpl |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 𝑘 ∈ ℕ ) |
98 |
97
|
nnrpd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 𝑘 ∈ ℝ+ ) |
99 |
93
|
nnrpd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 + 1 ) ∈ ℝ+ ) |
100 |
98 99
|
rpaddcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + ( 𝑀 + 1 ) ) ∈ ℝ+ ) |
101 |
96 100
|
rpdivcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ∈ ℝ+ ) |
102 |
101
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ∈ ℂ ) |
103 |
|
1cnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 1 ∈ ℂ ) |
104 |
|
nn0re |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ ) |
105 |
104
|
adantl |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 𝑀 ∈ ℝ ) |
106 |
105 95
|
nndivred |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 / ( 𝑘 + 1 ) ) ∈ ℝ ) |
107 |
106
|
recnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 / ( 𝑘 + 1 ) ) ∈ ℂ ) |
108 |
103 107
|
addcomd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) = ( ( 𝑀 / ( 𝑘 + 1 ) ) + 1 ) ) |
109 |
|
nn0ge0 |
⊢ ( 𝑀 ∈ ℕ0 → 0 ≤ 𝑀 ) |
110 |
109
|
adantl |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 0 ≤ 𝑀 ) |
111 |
105 96 110
|
divge0d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 0 ≤ ( 𝑀 / ( 𝑘 + 1 ) ) ) |
112 |
106 111
|
ge0p1rpd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑀 / ( 𝑘 + 1 ) ) + 1 ) ∈ ℝ+ ) |
113 |
108 112
|
eqeltrd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) ∈ ℝ+ ) |
114 |
|
1rp |
⊢ 1 ∈ ℝ+ |
115 |
114
|
a1i |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 1 ∈ ℝ+ ) |
116 |
96
|
rpreccld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 / ( 𝑘 + 1 ) ) ∈ ℝ+ ) |
117 |
115 116
|
rpaddcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ∈ ℝ+ ) |
118 |
113 117
|
rpmulcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ∈ ℝ+ ) |
119 |
99 96
|
rpdivcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ∈ ℝ+ ) |
120 |
115 119
|
rpaddcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ∈ ℝ+ ) |
121 |
118 120
|
rpdivcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ∈ ℝ+ ) |
122 |
121
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ∈ ℂ ) |
123 |
94 102 122
|
mulassd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) ) ) |
124 |
101 118
|
rpmulcld |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ∈ ℝ+ ) |
125 |
124
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ∈ ℂ ) |
126 |
120
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ∈ ℂ ) |
127 |
95
|
nncnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℂ ) |
128 |
120
|
rpne0d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ≠ 0 ) |
129 |
95
|
nnne0d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + 1 ) ≠ 0 ) |
130 |
125 126 127 128 129
|
divcan5d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) / ( ( 𝑘 + 1 ) · ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) = ( ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) |
131 |
118
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ∈ ℂ ) |
132 |
127 102 131
|
mul12d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) = ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 𝑘 + 1 ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) ) |
133 |
113
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) ∈ ℂ ) |
134 |
117
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ∈ ℂ ) |
135 |
127 133 134
|
mulassd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) |
136 |
127 103 107
|
adddid |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) ) = ( ( ( 𝑘 + 1 ) · 1 ) + ( ( 𝑘 + 1 ) · ( 𝑀 / ( 𝑘 + 1 ) ) ) ) ) |
137 |
127
|
mulid1d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · 1 ) = ( 𝑘 + 1 ) ) |
138 |
|
simpr |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 𝑀 ∈ ℕ0 ) |
139 |
138
|
nn0cnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 𝑀 ∈ ℂ ) |
140 |
139 127 129
|
divcan2d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 𝑀 / ( 𝑘 + 1 ) ) ) = 𝑀 ) |
141 |
137 140
|
oveq12d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · 1 ) + ( ( 𝑘 + 1 ) · ( 𝑀 / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) + 𝑀 ) ) |
142 |
97
|
nncnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → 𝑘 ∈ ℂ ) |
143 |
142 103 139
|
addassd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) + 𝑀 ) = ( 𝑘 + ( 1 + 𝑀 ) ) ) |
144 |
103 139
|
addcomd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 + 𝑀 ) = ( 𝑀 + 1 ) ) |
145 |
144
|
oveq2d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + ( 1 + 𝑀 ) ) = ( 𝑘 + ( 𝑀 + 1 ) ) ) |
146 |
143 145
|
eqtrd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) + 𝑀 ) = ( 𝑘 + ( 𝑀 + 1 ) ) ) |
147 |
136 141 146
|
3eqtrd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) ) = ( 𝑘 + ( 𝑀 + 1 ) ) ) |
148 |
147
|
oveq1d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + ( 𝑀 + 1 ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) |
149 |
135 148
|
eqtr3d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) = ( ( 𝑘 + ( 𝑀 + 1 ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) |
150 |
149
|
oveq2d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 𝑘 + 1 ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) = ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 𝑘 + ( 𝑀 + 1 ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) |
151 |
100
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + ( 𝑀 + 1 ) ) ∈ ℂ ) |
152 |
102 151 134
|
mulassd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) = ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 𝑘 + ( 𝑀 + 1 ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) |
153 |
100
|
rpne0d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑘 + ( 𝑀 + 1 ) ) ≠ 0 ) |
154 |
127 151 153
|
divcan1d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( 𝑘 + ( 𝑀 + 1 ) ) ) = ( 𝑘 + 1 ) ) |
155 |
154
|
oveq1d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) |
156 |
116
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( 1 / ( 𝑘 + 1 ) ) ∈ ℂ ) |
157 |
127 103 156
|
adddid |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) = ( ( ( 𝑘 + 1 ) · 1 ) + ( ( 𝑘 + 1 ) · ( 1 / ( 𝑘 + 1 ) ) ) ) ) |
158 |
103 127 129
|
divcan2d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 1 / ( 𝑘 + 1 ) ) ) = 1 ) |
159 |
137 158
|
oveq12d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · 1 ) + ( ( 𝑘 + 1 ) · ( 1 / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) + 1 ) ) |
160 |
155 157 159
|
3eqtrd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) + 1 ) ) |
161 |
152 160
|
eqtr3d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 𝑘 + ( 𝑀 + 1 ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) = ( ( 𝑘 + 1 ) + 1 ) ) |
162 |
132 150 161
|
3eqtrd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑘 + 1 ) + 1 ) ) |
163 |
119
|
rpcnd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ∈ ℂ ) |
164 |
127 103 163
|
adddid |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) = ( ( ( 𝑘 + 1 ) · 1 ) + ( ( 𝑘 + 1 ) · ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) |
165 |
94 127 129
|
divcan2d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) = ( 𝑀 + 1 ) ) |
166 |
137 165
|
oveq12d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · 1 ) + ( ( 𝑘 + 1 ) · ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) |
167 |
164 166
|
eqtrd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑘 + 1 ) · ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) = ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) |
168 |
162 167
|
oveq12d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) · ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) ) / ( ( 𝑘 + 1 ) · ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) = ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) |
169 |
102 131 126 128
|
divassd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) = ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) ) |
170 |
130 168 169
|
3eqtr3rd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) = ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) |
171 |
170
|
oveq2d |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) · ( ( ( 1 + ( 𝑀 / ( 𝑘 + 1 ) ) ) · ( 1 + ( 1 / ( 𝑘 + 1 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / ( 𝑘 + 1 ) ) ) ) ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) |
172 |
92 123 171
|
3eqtrd |
⊢ ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) → ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) |
173 |
172
|
adantr |
⊢ ( ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) ∧ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) → ( ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) · ( ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ‘ ( 𝑘 + 1 ) ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) |
174 |
76 78 173
|
3eqtrd |
⊢ ( ( ( 𝑘 ∈ ℕ ∧ 𝑀 ∈ ℕ0 ) ∧ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) |
175 |
174
|
exp31 |
⊢ ( 𝑘 ∈ ℕ → ( 𝑀 ∈ ℕ0 → ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) ) ) |
176 |
175
|
a2d |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑘 ) = ( ( 𝑀 + 1 ) · ( ( 𝑘 + 1 ) / ( 𝑘 + ( 𝑀 + 1 ) ) ) ) ) → ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ ( 𝑘 + 1 ) ) = ( ( 𝑀 + 1 ) · ( ( ( 𝑘 + 1 ) + 1 ) / ( ( 𝑘 + 1 ) + ( 𝑀 + 1 ) ) ) ) ) ) ) |
177 |
11 18 25 32 71 176
|
nnind |
⊢ ( 𝑏 ∈ ℕ → ( 𝑀 ∈ ℕ0 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) ) |
178 |
177
|
impcom |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑏 ∈ ℕ ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) |
179 |
|
oveq1 |
⊢ ( 𝑥 = 𝑏 → ( 𝑥 + 1 ) = ( 𝑏 + 1 ) ) |
180 |
|
oveq1 |
⊢ ( 𝑥 = 𝑏 → ( 𝑥 + ( 𝑀 + 1 ) ) = ( 𝑏 + ( 𝑀 + 1 ) ) ) |
181 |
179 180
|
oveq12d |
⊢ ( 𝑥 = 𝑏 → ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) = ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) |
182 |
181
|
oveq2d |
⊢ ( 𝑥 = 𝑏 → ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) |
183 |
|
eqid |
⊢ ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) = ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) |
184 |
|
ovex |
⊢ ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ∈ V |
185 |
182 183 184
|
fvmpt |
⊢ ( 𝑏 ∈ ℕ → ( ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) |
186 |
185
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑏 ∈ ℕ ) → ( ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑀 + 1 ) · ( ( 𝑏 + 1 ) / ( 𝑏 + ( 𝑀 + 1 ) ) ) ) ) |
187 |
178 186
|
eqtr4d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑏 ∈ ℕ ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ‘ 𝑏 ) ) |
188 |
187
|
ralrimiva |
⊢ ( 𝑀 ∈ ℕ0 → ∀ 𝑏 ∈ ℕ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ‘ 𝑏 ) ) |
189 |
|
seqfn |
⊢ ( 1 ∈ ℤ → seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) Fn ( ℤ≥ ‘ 1 ) ) |
190 |
2 189
|
ax-mp |
⊢ seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) Fn ( ℤ≥ ‘ 1 ) |
191 |
73
|
fneq2i |
⊢ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) Fn ℕ ↔ seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) Fn ( ℤ≥ ‘ 1 ) ) |
192 |
190 191
|
mpbir |
⊢ seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) Fn ℕ |
193 |
|
ovex |
⊢ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ∈ V |
194 |
193 183
|
fnmpti |
⊢ ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) Fn ℕ |
195 |
|
eqfnfv |
⊢ ( ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) Fn ℕ ∧ ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) Fn ℕ ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) = ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ℕ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ‘ 𝑏 ) ) ) |
196 |
192 194 195
|
mp2an |
⊢ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) = ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ↔ ∀ 𝑏 ∈ ℕ ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) ‘ 𝑏 ) = ( ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ‘ 𝑏 ) ) |
197 |
188 196
|
sylibr |
⊢ ( 𝑀 ∈ ℕ0 → seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ( ( ( 1 + ( 𝑀 / 𝑛 ) ) · ( 1 + ( 1 / 𝑛 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝑛 ) ) ) ) ) = ( 𝑥 ∈ ℕ ↦ ( ( 𝑀 + 1 ) · ( ( 𝑥 + 1 ) / ( 𝑥 + ( 𝑀 + 1 ) ) ) ) ) ) |