Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
⊢ ( 𝐾 ∈ ℕ0 ↔ ( 𝐾 ∈ ℕ ∨ 𝐾 = 0 ) ) |
2 |
|
0exp |
⊢ ( 𝐾 ∈ ℕ → ( 0 ↑ 𝐾 ) = 0 ) |
3 |
2
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ ) → ( 0 ↑ 𝐾 ) = 0 ) |
4 |
|
nnnn0 |
⊢ ( 𝐾 ∈ ℕ → 𝐾 ∈ ℕ0 ) |
5 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
6 |
|
nn0sqcl |
⊢ ( 𝐾 ∈ ℕ0 → ( 𝐾 ↑ 2 ) ∈ ℕ0 ) |
7 |
|
nn0expcl |
⊢ ( ( 2 ∈ ℕ0 ∧ ( 𝐾 ↑ 2 ) ∈ ℕ0 ) → ( 2 ↑ ( 𝐾 ↑ 2 ) ) ∈ ℕ0 ) |
8 |
5 6 7
|
sylancr |
⊢ ( 𝐾 ∈ ℕ0 → ( 2 ↑ ( 𝐾 ↑ 2 ) ) ∈ ℕ0 ) |
9 |
8
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( 2 ↑ ( 𝐾 ↑ 2 ) ) ∈ ℕ0 ) |
10 |
|
nn0addcl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( 𝑀 + 𝐾 ) ∈ ℕ0 ) |
11 |
|
nn0expcl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ ( 𝑀 + 𝐾 ) ∈ ℕ0 ) → ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ∈ ℕ0 ) |
12 |
10 11
|
syldan |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ∈ ℕ0 ) |
13 |
9 12
|
nn0mulcld |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ∈ ℕ0 ) |
14 |
4 13
|
sylan2 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ ) → ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ∈ ℕ0 ) |
15 |
14
|
nn0ge0d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ ) → 0 ≤ ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ) |
16 |
3 15
|
eqbrtrd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ ) → ( 0 ↑ 𝐾 ) ≤ ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ) |
17 |
|
1nn |
⊢ 1 ∈ ℕ |
18 |
|
elnn0 |
⊢ ( 𝑀 ∈ ℕ0 ↔ ( 𝑀 ∈ ℕ ∨ 𝑀 = 0 ) ) |
19 |
|
nnnn0 |
⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℕ0 ) |
20 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
21 |
|
nn0addcl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 0 ∈ ℕ0 ) → ( 𝑀 + 0 ) ∈ ℕ0 ) |
22 |
19 20 21
|
sylancl |
⊢ ( 𝑀 ∈ ℕ → ( 𝑀 + 0 ) ∈ ℕ0 ) |
23 |
|
nnexpcl |
⊢ ( ( 𝑀 ∈ ℕ ∧ ( 𝑀 + 0 ) ∈ ℕ0 ) → ( 𝑀 ↑ ( 𝑀 + 0 ) ) ∈ ℕ ) |
24 |
22 23
|
mpdan |
⊢ ( 𝑀 ∈ ℕ → ( 𝑀 ↑ ( 𝑀 + 0 ) ) ∈ ℕ ) |
25 |
|
id |
⊢ ( 𝑀 = 0 → 𝑀 = 0 ) |
26 |
|
oveq1 |
⊢ ( 𝑀 = 0 → ( 𝑀 + 0 ) = ( 0 + 0 ) ) |
27 |
|
00id |
⊢ ( 0 + 0 ) = 0 |
28 |
26 27
|
eqtrdi |
⊢ ( 𝑀 = 0 → ( 𝑀 + 0 ) = 0 ) |
29 |
25 28
|
oveq12d |
⊢ ( 𝑀 = 0 → ( 𝑀 ↑ ( 𝑀 + 0 ) ) = ( 0 ↑ 0 ) ) |
30 |
|
0exp0e1 |
⊢ ( 0 ↑ 0 ) = 1 |
31 |
29 30
|
eqtrdi |
⊢ ( 𝑀 = 0 → ( 𝑀 ↑ ( 𝑀 + 0 ) ) = 1 ) |
32 |
31 17
|
eqeltrdi |
⊢ ( 𝑀 = 0 → ( 𝑀 ↑ ( 𝑀 + 0 ) ) ∈ ℕ ) |
33 |
24 32
|
jaoi |
⊢ ( ( 𝑀 ∈ ℕ ∨ 𝑀 = 0 ) → ( 𝑀 ↑ ( 𝑀 + 0 ) ) ∈ ℕ ) |
34 |
18 33
|
sylbi |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 ↑ ( 𝑀 + 0 ) ) ∈ ℕ ) |
35 |
|
nnmulcl |
⊢ ( ( 1 ∈ ℕ ∧ ( 𝑀 ↑ ( 𝑀 + 0 ) ) ∈ ℕ ) → ( 1 · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ∈ ℕ ) |
36 |
17 34 35
|
sylancr |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ∈ ℕ ) |
37 |
36
|
nnge1d |
⊢ ( 𝑀 ∈ ℕ0 → 1 ≤ ( 1 · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 = 0 ) → 1 ≤ ( 1 · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ) |
39 |
|
oveq2 |
⊢ ( 𝐾 = 0 → ( 0 ↑ 𝐾 ) = ( 0 ↑ 0 ) ) |
40 |
39 30
|
eqtrdi |
⊢ ( 𝐾 = 0 → ( 0 ↑ 𝐾 ) = 1 ) |
41 |
|
sq0i |
⊢ ( 𝐾 = 0 → ( 𝐾 ↑ 2 ) = 0 ) |
42 |
41
|
oveq2d |
⊢ ( 𝐾 = 0 → ( 2 ↑ ( 𝐾 ↑ 2 ) ) = ( 2 ↑ 0 ) ) |
43 |
|
2cn |
⊢ 2 ∈ ℂ |
44 |
|
exp0 |
⊢ ( 2 ∈ ℂ → ( 2 ↑ 0 ) = 1 ) |
45 |
43 44
|
ax-mp |
⊢ ( 2 ↑ 0 ) = 1 |
46 |
42 45
|
eqtrdi |
⊢ ( 𝐾 = 0 → ( 2 ↑ ( 𝐾 ↑ 2 ) ) = 1 ) |
47 |
|
oveq2 |
⊢ ( 𝐾 = 0 → ( 𝑀 + 𝐾 ) = ( 𝑀 + 0 ) ) |
48 |
47
|
oveq2d |
⊢ ( 𝐾 = 0 → ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) = ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) |
49 |
46 48
|
oveq12d |
⊢ ( 𝐾 = 0 → ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) = ( 1 · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ) |
50 |
40 49
|
breq12d |
⊢ ( 𝐾 = 0 → ( ( 0 ↑ 𝐾 ) ≤ ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ↔ 1 ≤ ( 1 · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ) ) |
51 |
50
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 = 0 ) → ( ( 0 ↑ 𝐾 ) ≤ ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ↔ 1 ≤ ( 1 · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ) ) |
52 |
38 51
|
mpbird |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 = 0 ) → ( 0 ↑ 𝐾 ) ≤ ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ) |
53 |
16 52
|
jaodan |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ ( 𝐾 ∈ ℕ ∨ 𝐾 = 0 ) ) → ( 0 ↑ 𝐾 ) ≤ ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ) |
54 |
1 53
|
sylan2b |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( 0 ↑ 𝐾 ) ≤ ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ) |
55 |
|
nn0cn |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ ) |
56 |
55
|
exp0d |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 ↑ 0 ) = 1 ) |
57 |
56
|
oveq2d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( 0 ↑ 𝐾 ) · ( 𝑀 ↑ 0 ) ) = ( ( 0 ↑ 𝐾 ) · 1 ) ) |
58 |
|
nn0expcl |
⊢ ( ( 0 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( 0 ↑ 𝐾 ) ∈ ℕ0 ) |
59 |
20 58
|
mpan |
⊢ ( 𝐾 ∈ ℕ0 → ( 0 ↑ 𝐾 ) ∈ ℕ0 ) |
60 |
59
|
nn0cnd |
⊢ ( 𝐾 ∈ ℕ0 → ( 0 ↑ 𝐾 ) ∈ ℂ ) |
61 |
60
|
mulid1d |
⊢ ( 𝐾 ∈ ℕ0 → ( ( 0 ↑ 𝐾 ) · 1 ) = ( 0 ↑ 𝐾 ) ) |
62 |
57 61
|
sylan9eq |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( ( 0 ↑ 𝐾 ) · ( 𝑀 ↑ 0 ) ) = ( 0 ↑ 𝐾 ) ) |
63 |
13
|
nn0cnd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ∈ ℂ ) |
64 |
63
|
mulid1d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · 1 ) = ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ) |
65 |
54 62 64
|
3brtr4d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) → ( ( 0 ↑ 𝐾 ) · ( 𝑀 ↑ 0 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · 1 ) ) |
66 |
65
|
adantr |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑁 = 0 ) → ( ( 0 ↑ 𝐾 ) · ( 𝑀 ↑ 0 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · 1 ) ) |
67 |
|
oveq1 |
⊢ ( 𝑁 = 0 → ( 𝑁 ↑ 𝐾 ) = ( 0 ↑ 𝐾 ) ) |
68 |
|
oveq2 |
⊢ ( 𝑁 = 0 → ( 𝑀 ↑ 𝑁 ) = ( 𝑀 ↑ 0 ) ) |
69 |
67 68
|
oveq12d |
⊢ ( 𝑁 = 0 → ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) = ( ( 0 ↑ 𝐾 ) · ( 𝑀 ↑ 0 ) ) ) |
70 |
|
fveq2 |
⊢ ( 𝑁 = 0 → ( ! ‘ 𝑁 ) = ( ! ‘ 0 ) ) |
71 |
|
fac0 |
⊢ ( ! ‘ 0 ) = 1 |
72 |
70 71
|
eqtrdi |
⊢ ( 𝑁 = 0 → ( ! ‘ 𝑁 ) = 1 ) |
73 |
72
|
oveq2d |
⊢ ( 𝑁 = 0 → ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) = ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · 1 ) ) |
74 |
69 73
|
breq12d |
⊢ ( 𝑁 = 0 → ( ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ↔ ( ( 0 ↑ 𝐾 ) · ( 𝑀 ↑ 0 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · 1 ) ) ) |
75 |
74
|
adantl |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑁 = 0 ) → ( ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ↔ ( ( 0 ↑ 𝐾 ) · ( 𝑀 ↑ 0 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · 1 ) ) ) |
76 |
66 75
|
mpbird |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0 ) ∧ 𝑁 = 0 ) → ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ) |