| Step |
Hyp |
Ref |
Expression |
| 1 |
|
oveq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑛 ↑ 𝑗 ) = ( 𝑚 ↑ 𝑗 ) ) |
| 2 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑀 ↑ 𝑛 ) = ( 𝑀 ↑ 𝑚 ) ) |
| 3 |
1 2
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) = ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ) |
| 4 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( ! ‘ 𝑛 ) = ( ! ‘ 𝑚 ) ) |
| 5 |
4
|
oveq2d |
⊢ ( 𝑛 = 𝑚 → ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) ) |
| 6 |
3 5
|
breq12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) ) ) |
| 7 |
6
|
cbvralvw |
⊢ ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) ) |
| 8 |
|
nnre |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ ) |
| 9 |
|
1re |
⊢ 1 ∈ ℝ |
| 10 |
|
lelttric |
⊢ ( ( 𝑛 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝑛 ≤ 1 ∨ 1 < 𝑛 ) ) |
| 11 |
8 9 10
|
sylancl |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 ≤ 1 ∨ 1 < 𝑛 ) ) |
| 12 |
11
|
ancli |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 ∈ ℕ ∧ ( 𝑛 ≤ 1 ∨ 1 < 𝑛 ) ) ) |
| 13 |
|
andi |
⊢ ( ( 𝑛 ∈ ℕ ∧ ( 𝑛 ≤ 1 ∨ 1 < 𝑛 ) ) ↔ ( ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) ∨ ( 𝑛 ∈ ℕ ∧ 1 < 𝑛 ) ) ) |
| 14 |
12 13
|
sylib |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) ∨ ( 𝑛 ∈ ℕ ∧ 1 < 𝑛 ) ) ) |
| 15 |
|
nnge1 |
⊢ ( 𝑛 ∈ ℕ → 1 ≤ 𝑛 ) |
| 16 |
|
letri3 |
⊢ ( ( 𝑛 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝑛 = 1 ↔ ( 𝑛 ≤ 1 ∧ 1 ≤ 𝑛 ) ) ) |
| 17 |
8 9 16
|
sylancl |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 = 1 ↔ ( 𝑛 ≤ 1 ∧ 1 ≤ 𝑛 ) ) ) |
| 18 |
17
|
biimpar |
⊢ ( ( 𝑛 ∈ ℕ ∧ ( 𝑛 ≤ 1 ∧ 1 ≤ 𝑛 ) ) → 𝑛 = 1 ) |
| 19 |
18
|
anassrs |
⊢ ( ( ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) ∧ 1 ≤ 𝑛 ) → 𝑛 = 1 ) |
| 20 |
15 19
|
mpidan |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) → 𝑛 = 1 ) |
| 21 |
|
oveq1 |
⊢ ( 𝑛 = 1 → ( 𝑛 − 1 ) = ( 1 − 1 ) ) |
| 22 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
| 23 |
21 22
|
eqtrdi |
⊢ ( 𝑛 = 1 → ( 𝑛 − 1 ) = 0 ) |
| 24 |
20 23
|
syl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) → ( 𝑛 − 1 ) = 0 ) |
| 25 |
|
faclbnd4lem3 |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑛 − 1 ) = 0 ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) |
| 26 |
24 25
|
sylan2 |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) |
| 27 |
26
|
a1d |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 28 |
|
1nn |
⊢ 1 ∈ ℕ |
| 29 |
|
nnsub |
⊢ ( ( 1 ∈ ℕ ∧ 𝑛 ∈ ℕ ) → ( 1 < 𝑛 ↔ ( 𝑛 − 1 ) ∈ ℕ ) ) |
| 30 |
28 29
|
mpan |
⊢ ( 𝑛 ∈ ℕ → ( 1 < 𝑛 ↔ ( 𝑛 − 1 ) ∈ ℕ ) ) |
| 31 |
30
|
biimpa |
⊢ ( ( 𝑛 ∈ ℕ ∧ 1 < 𝑛 ) → ( 𝑛 − 1 ) ∈ ℕ ) |
| 32 |
|
oveq1 |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( 𝑚 ↑ 𝑗 ) = ( ( 𝑛 − 1 ) ↑ 𝑗 ) ) |
| 33 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( 𝑀 ↑ 𝑚 ) = ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) |
| 34 |
32 33
|
oveq12d |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) = ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ) |
| 35 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( ! ‘ 𝑚 ) = ( ! ‘ ( 𝑛 − 1 ) ) ) |
| 36 |
35
|
oveq2d |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) = ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) |
| 37 |
34 36
|
breq12d |
⊢ ( 𝑚 = ( 𝑛 − 1 ) → ( ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) ↔ ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 38 |
37
|
rspcv |
⊢ ( ( 𝑛 − 1 ) ∈ ℕ → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 39 |
31 38
|
syl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 1 < 𝑛 ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 40 |
39
|
adantl |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ ( 𝑛 ∈ ℕ ∧ 1 < 𝑛 ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 41 |
27 40
|
jaodan |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ ( ( 𝑛 ∈ ℕ ∧ 𝑛 ≤ 1 ) ∨ ( 𝑛 ∈ ℕ ∧ 1 < 𝑛 ) ) ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 42 |
14 41
|
sylan2 |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) ) ) |
| 43 |
|
faclbnd4lem2 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) → ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 44 |
43
|
3expa |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) → ( ( ( ( 𝑛 − 1 ) ↑ 𝑗 ) · ( 𝑀 ↑ ( 𝑛 − 1 ) ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ ( 𝑛 − 1 ) ) ) → ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 45 |
42 44
|
syld |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 46 |
45
|
ralrimdva |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) → ( ∀ 𝑚 ∈ ℕ ( ( 𝑚 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑚 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑚 ) ) → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 47 |
7 46
|
biimtrid |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑗 ∈ ℕ0 ) → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 48 |
47
|
expcom |
⊢ ( 𝑗 ∈ ℕ0 → ( 𝑀 ∈ ℕ0 → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) ) |
| 49 |
48
|
a2d |
⊢ ( 𝑗 ∈ ℕ0 → ( ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) ) → ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) ) |
| 50 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
| 51 |
|
faclbnd3 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( 𝑀 ↑ 𝑛 ) ≤ ( ( 𝑀 ↑ 𝑀 ) · ( ! ‘ 𝑛 ) ) ) |
| 52 |
50 51
|
sylan2 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( 𝑀 ↑ 𝑛 ) ≤ ( ( 𝑀 ↑ 𝑀 ) · ( ! ‘ 𝑛 ) ) ) |
| 53 |
|
nncn |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ ) |
| 54 |
53
|
exp0d |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 ↑ 0 ) = 1 ) |
| 55 |
54
|
oveq1d |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) = ( 1 · ( 𝑀 ↑ 𝑛 ) ) ) |
| 56 |
55
|
adantl |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) = ( 1 · ( 𝑀 ↑ 𝑛 ) ) ) |
| 57 |
|
nn0cn |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ ) |
| 58 |
|
expcl |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑀 ↑ 𝑛 ) ∈ ℂ ) |
| 59 |
57 50 58
|
syl2an |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( 𝑀 ↑ 𝑛 ) ∈ ℂ ) |
| 60 |
59
|
mullidd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( 1 · ( 𝑀 ↑ 𝑛 ) ) = ( 𝑀 ↑ 𝑛 ) ) |
| 61 |
56 60
|
eqtrd |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) = ( 𝑀 ↑ 𝑛 ) ) |
| 62 |
|
sq0 |
⊢ ( 0 ↑ 2 ) = 0 |
| 63 |
62
|
oveq2i |
⊢ ( 2 ↑ ( 0 ↑ 2 ) ) = ( 2 ↑ 0 ) |
| 64 |
|
2cn |
⊢ 2 ∈ ℂ |
| 65 |
|
exp0 |
⊢ ( 2 ∈ ℂ → ( 2 ↑ 0 ) = 1 ) |
| 66 |
64 65
|
ax-mp |
⊢ ( 2 ↑ 0 ) = 1 |
| 67 |
63 66
|
eqtri |
⊢ ( 2 ↑ ( 0 ↑ 2 ) ) = 1 |
| 68 |
67
|
a1i |
⊢ ( 𝑀 ∈ ℕ0 → ( 2 ↑ ( 0 ↑ 2 ) ) = 1 ) |
| 69 |
57
|
addridd |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 + 0 ) = 𝑀 ) |
| 70 |
69
|
oveq2d |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 ↑ ( 𝑀 + 0 ) ) = ( 𝑀 ↑ 𝑀 ) ) |
| 71 |
68 70
|
oveq12d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) = ( 1 · ( 𝑀 ↑ 𝑀 ) ) ) |
| 72 |
|
expcl |
⊢ ( ( 𝑀 ∈ ℂ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 ↑ 𝑀 ) ∈ ℂ ) |
| 73 |
57 72
|
mpancom |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 ↑ 𝑀 ) ∈ ℂ ) |
| 74 |
73
|
mullidd |
⊢ ( 𝑀 ∈ ℕ0 → ( 1 · ( 𝑀 ↑ 𝑀 ) ) = ( 𝑀 ↑ 𝑀 ) ) |
| 75 |
71 74
|
eqtrd |
⊢ ( 𝑀 ∈ ℕ0 → ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) = ( 𝑀 ↑ 𝑀 ) ) |
| 76 |
75
|
oveq1d |
⊢ ( 𝑀 ∈ ℕ0 → ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( 𝑀 ↑ 𝑀 ) · ( ! ‘ 𝑛 ) ) ) |
| 77 |
76
|
adantr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( 𝑀 ↑ 𝑀 ) · ( ! ‘ 𝑛 ) ) ) |
| 78 |
52 61 77
|
3brtr4d |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑛 ∈ ℕ ) → ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) ) |
| 79 |
78
|
ralrimiva |
⊢ ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) ) |
| 80 |
|
oveq2 |
⊢ ( 𝑚 = 0 → ( 𝑛 ↑ 𝑚 ) = ( 𝑛 ↑ 0 ) ) |
| 81 |
80
|
oveq1d |
⊢ ( 𝑚 = 0 → ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) = ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) ) |
| 82 |
|
oveq1 |
⊢ ( 𝑚 = 0 → ( 𝑚 ↑ 2 ) = ( 0 ↑ 2 ) ) |
| 83 |
82
|
oveq2d |
⊢ ( 𝑚 = 0 → ( 2 ↑ ( 𝑚 ↑ 2 ) ) = ( 2 ↑ ( 0 ↑ 2 ) ) ) |
| 84 |
|
oveq2 |
⊢ ( 𝑚 = 0 → ( 𝑀 + 𝑚 ) = ( 𝑀 + 0 ) ) |
| 85 |
84
|
oveq2d |
⊢ ( 𝑚 = 0 → ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) = ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) |
| 86 |
83 85
|
oveq12d |
⊢ ( 𝑚 = 0 → ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) = ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) ) |
| 87 |
86
|
oveq1d |
⊢ ( 𝑚 = 0 → ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) ) |
| 88 |
81 87
|
breq12d |
⊢ ( 𝑚 = 0 → ( ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 89 |
88
|
ralbidv |
⊢ ( 𝑚 = 0 → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 90 |
89
|
imbi2d |
⊢ ( 𝑚 = 0 → ( ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 0 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 0 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 0 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) ) |
| 91 |
|
oveq2 |
⊢ ( 𝑚 = 𝑗 → ( 𝑛 ↑ 𝑚 ) = ( 𝑛 ↑ 𝑗 ) ) |
| 92 |
91
|
oveq1d |
⊢ ( 𝑚 = 𝑗 → ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) = ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ) |
| 93 |
|
oveq1 |
⊢ ( 𝑚 = 𝑗 → ( 𝑚 ↑ 2 ) = ( 𝑗 ↑ 2 ) ) |
| 94 |
93
|
oveq2d |
⊢ ( 𝑚 = 𝑗 → ( 2 ↑ ( 𝑚 ↑ 2 ) ) = ( 2 ↑ ( 𝑗 ↑ 2 ) ) ) |
| 95 |
|
oveq2 |
⊢ ( 𝑚 = 𝑗 → ( 𝑀 + 𝑚 ) = ( 𝑀 + 𝑗 ) ) |
| 96 |
95
|
oveq2d |
⊢ ( 𝑚 = 𝑗 → ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) = ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) |
| 97 |
94 96
|
oveq12d |
⊢ ( 𝑚 = 𝑗 → ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) = ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) ) |
| 98 |
97
|
oveq1d |
⊢ ( 𝑚 = 𝑗 → ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) ) |
| 99 |
92 98
|
breq12d |
⊢ ( 𝑚 = 𝑗 → ( ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 100 |
99
|
ralbidv |
⊢ ( 𝑚 = 𝑗 → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 101 |
100
|
imbi2d |
⊢ ( 𝑚 = 𝑗 → ( ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑗 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑗 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑗 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) ) |
| 102 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( 𝑛 ↑ 𝑚 ) = ( 𝑛 ↑ ( 𝑗 + 1 ) ) ) |
| 103 |
102
|
oveq1d |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) = ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ) |
| 104 |
|
oveq1 |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( 𝑚 ↑ 2 ) = ( ( 𝑗 + 1 ) ↑ 2 ) ) |
| 105 |
104
|
oveq2d |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( 2 ↑ ( 𝑚 ↑ 2 ) ) = ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) ) |
| 106 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( 𝑀 + 𝑚 ) = ( 𝑀 + ( 𝑗 + 1 ) ) ) |
| 107 |
106
|
oveq2d |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) = ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) |
| 108 |
105 107
|
oveq12d |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) = ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) ) |
| 109 |
108
|
oveq1d |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) |
| 110 |
103 109
|
breq12d |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 111 |
110
|
ralbidv |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 112 |
111
|
imbi2d |
⊢ ( 𝑚 = ( 𝑗 + 1 ) → ( ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ ( 𝑗 + 1 ) ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( ( 𝑗 + 1 ) ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + ( 𝑗 + 1 ) ) ) ) · ( ! ‘ 𝑛 ) ) ) ) ) |
| 113 |
|
oveq2 |
⊢ ( 𝑚 = 𝐾 → ( 𝑛 ↑ 𝑚 ) = ( 𝑛 ↑ 𝐾 ) ) |
| 114 |
113
|
oveq1d |
⊢ ( 𝑚 = 𝐾 → ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) = ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ) |
| 115 |
|
oveq1 |
⊢ ( 𝑚 = 𝐾 → ( 𝑚 ↑ 2 ) = ( 𝐾 ↑ 2 ) ) |
| 116 |
115
|
oveq2d |
⊢ ( 𝑚 = 𝐾 → ( 2 ↑ ( 𝑚 ↑ 2 ) ) = ( 2 ↑ ( 𝐾 ↑ 2 ) ) ) |
| 117 |
|
oveq2 |
⊢ ( 𝑚 = 𝐾 → ( 𝑀 + 𝑚 ) = ( 𝑀 + 𝐾 ) ) |
| 118 |
117
|
oveq2d |
⊢ ( 𝑚 = 𝐾 → ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) = ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) |
| 119 |
116 118
|
oveq12d |
⊢ ( 𝑚 = 𝐾 → ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) = ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) ) |
| 120 |
119
|
oveq1d |
⊢ ( 𝑚 = 𝐾 → ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ) |
| 121 |
114 120
|
breq12d |
⊢ ( 𝑚 = 𝐾 → ( ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 122 |
121
|
ralbidv |
⊢ ( 𝑚 = 𝐾 → ( ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 123 |
122
|
imbi2d |
⊢ ( 𝑚 = 𝐾 → ( ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝑚 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝑚 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝑚 ) ) ) · ( ! ‘ 𝑛 ) ) ) ↔ ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) ) |
| 124 |
49 79 90 101 112 123
|
nn0indALT |
⊢ ( 𝐾 ∈ ℕ0 → ( 𝑀 ∈ ℕ0 → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ) ) |
| 125 |
124
|
imp |
⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ) |
| 126 |
|
oveq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑛 ↑ 𝐾 ) = ( 𝑁 ↑ 𝐾 ) ) |
| 127 |
|
oveq2 |
⊢ ( 𝑛 = 𝑁 → ( 𝑀 ↑ 𝑛 ) = ( 𝑀 ↑ 𝑁 ) ) |
| 128 |
126 127
|
oveq12d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) = ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ) |
| 129 |
|
fveq2 |
⊢ ( 𝑛 = 𝑁 → ( ! ‘ 𝑛 ) = ( ! ‘ 𝑁 ) ) |
| 130 |
129
|
oveq2d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) = ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ) |
| 131 |
128 130
|
breq12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ↔ ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ) ) |
| 132 |
131
|
rspcva |
⊢ ( ( 𝑁 ∈ ℕ ∧ ∀ 𝑛 ∈ ℕ ( ( 𝑛 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑛 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑛 ) ) ) → ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ) |
| 133 |
125 132
|
sylan2 |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐾 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) ) → ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ) |
| 134 |
133
|
3impb |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐾 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝑁 ↑ 𝐾 ) · ( 𝑀 ↑ 𝑁 ) ) ≤ ( ( ( 2 ↑ ( 𝐾 ↑ 2 ) ) · ( 𝑀 ↑ ( 𝑀 + 𝐾 ) ) ) · ( ! ‘ 𝑁 ) ) ) |