Step |
Hyp |
Ref |
Expression |
1 |
|
1rp |
⊢ 1 ∈ ℝ+ |
2 |
1
|
a1i |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 1 ∈ ℝ+ ) |
3 |
|
nnrp |
⊢ ( 𝐵 ∈ ℕ → 𝐵 ∈ ℝ+ ) |
4 |
3
|
rpreccld |
⊢ ( 𝐵 ∈ ℕ → ( 1 / 𝐵 ) ∈ ℝ+ ) |
5 |
4
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 / 𝐵 ) ∈ ℝ+ ) |
6 |
2 5
|
rpaddcld |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( 1 / 𝐵 ) ) ∈ ℝ+ ) |
7 |
6
|
rpcnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( 1 / 𝐵 ) ) ∈ ℂ ) |
8 |
|
simpl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 𝑀 ∈ ℕ0 ) |
9 |
7 8
|
expp1d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 1 + ( 1 / 𝐵 ) ) ↑ ( 𝑀 + 1 ) ) = ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) · ( 1 + ( 1 / 𝐵 ) ) ) ) |
10 |
1
|
a1i |
⊢ ( 𝐵 ∈ ℕ → 1 ∈ ℝ+ ) |
11 |
10 4
|
rpaddcld |
⊢ ( 𝐵 ∈ ℕ → ( 1 + ( 1 / 𝐵 ) ) ∈ ℝ+ ) |
12 |
|
nn0z |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ ) |
13 |
|
rpexpcl |
⊢ ( ( ( 1 + ( 1 / 𝐵 ) ) ∈ ℝ+ ∧ 𝑀 ∈ ℤ ) → ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) ∈ ℝ+ ) |
14 |
11 12 13
|
syl2anr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) ∈ ℝ+ ) |
15 |
14
|
rpcnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) ∈ ℂ ) |
16 |
|
1cnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 1 ∈ ℂ ) |
17 |
|
nn0nndivcl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 𝑀 / 𝐵 ) ∈ ℝ ) |
18 |
17
|
recnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 𝑀 / 𝐵 ) ∈ ℂ ) |
19 |
16 18
|
addcomd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( 𝑀 / 𝐵 ) ) = ( ( 𝑀 / 𝐵 ) + 1 ) ) |
20 |
|
nn0ge0div |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 0 ≤ ( 𝑀 / 𝐵 ) ) |
21 |
17 20
|
ge0p1rpd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 𝑀 / 𝐵 ) + 1 ) ∈ ℝ+ ) |
22 |
19 21
|
eqeltrd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( 𝑀 / 𝐵 ) ) ∈ ℝ+ ) |
23 |
22
|
rpcnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( 𝑀 / 𝐵 ) ) ∈ ℂ ) |
24 |
22
|
rpne0d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( 𝑀 / 𝐵 ) ) ≠ 0 ) |
25 |
15 23 24
|
divcan1d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( 1 + ( 𝑀 / 𝐵 ) ) ) = ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) ) |
26 |
25
|
oveq1d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( 1 + ( 1 / 𝐵 ) ) ) = ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) · ( 1 + ( 1 / 𝐵 ) ) ) ) |
27 |
14 22
|
rpdivcld |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) ∈ ℝ+ ) |
28 |
27
|
rpcnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) ∈ ℂ ) |
29 |
28 23 7
|
mulassd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( 1 + ( 1 / 𝐵 ) ) ) = ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) ) ) |
30 |
9 26 29
|
3eqtr2d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 1 + ( 1 / 𝐵 ) ) ↑ ( 𝑀 + 1 ) ) = ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) ) ) |
31 |
30
|
oveq1d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ ( 𝑀 + 1 ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ) = ( ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ) ) |
32 |
22 6
|
rpmulcld |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) ∈ ℝ+ ) |
33 |
32
|
rpcnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) ∈ ℂ ) |
34 |
|
nn0p1nn |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 + 1 ) ∈ ℕ ) |
35 |
34
|
nnrpd |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 + 1 ) ∈ ℝ+ ) |
36 |
35
|
adantr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 𝑀 + 1 ) ∈ ℝ+ ) |
37 |
3
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → 𝐵 ∈ ℝ+ ) |
38 |
36 37
|
rpdivcld |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( 𝑀 + 1 ) / 𝐵 ) ∈ ℝ+ ) |
39 |
2 38
|
rpaddcld |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ∈ ℝ+ ) |
40 |
39
|
rpcnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ∈ ℂ ) |
41 |
39
|
rpne0d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ≠ 0 ) |
42 |
28 33 40 41
|
divassd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ) = ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ) ) ) |
43 |
31 42
|
eqtrd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ ( 𝑀 + 1 ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ) = ( ( ( ( 1 + ( 1 / 𝐵 ) ) ↑ 𝑀 ) / ( 1 + ( 𝑀 / 𝐵 ) ) ) · ( ( ( 1 + ( 𝑀 / 𝐵 ) ) · ( 1 + ( 1 / 𝐵 ) ) ) / ( 1 + ( ( 𝑀 + 1 ) / 𝐵 ) ) ) ) ) |