Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑗 = 0 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 0 ) ) |
2 |
|
oveq2 |
⊢ ( 𝑗 = 0 → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ 0 ) ) |
3 |
1 2
|
breq12d |
⊢ ( 𝑗 = 0 → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ↔ ( 𝐴 ↑ 0 ) ≤ ( 𝐵 ↑ 0 ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑗 = 0 → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 0 ) ≤ ( 𝐵 ↑ 0 ) ) ) ) |
5 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 𝑘 ) ) |
6 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ 𝑘 ) ) |
7 |
5 6
|
breq12d |
⊢ ( 𝑗 = 𝑘 → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ↔ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) ) |
8 |
7
|
imbi2d |
⊢ ( 𝑗 = 𝑘 → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ ( 𝑘 + 1 ) ) ) |
10 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) |
11 |
9 10
|
breq12d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ↔ ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑗 = 𝑁 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 𝑁 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑗 = 𝑁 → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ 𝑁 ) ) |
15 |
13 14
|
breq12d |
⊢ ( 𝑗 = 𝑁 → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ↔ ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐵 ↑ 𝑁 ) ) ) |
16 |
15
|
imbi2d |
⊢ ( 𝑗 = 𝑁 → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐵 ↑ 𝑗 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐵 ↑ 𝑁 ) ) ) ) |
17 |
|
recn |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ ) |
18 |
|
recn |
⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ ) |
19 |
|
exp0 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 0 ) = 1 ) |
20 |
19
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑ 0 ) = 1 ) |
21 |
|
1le1 |
⊢ 1 ≤ 1 |
22 |
20 21
|
eqbrtrdi |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑ 0 ) ≤ 1 ) |
23 |
|
exp0 |
⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ↑ 0 ) = 1 ) |
24 |
23
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 ↑ 0 ) = 1 ) |
25 |
22 24
|
breqtrrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑ 0 ) ≤ ( 𝐵 ↑ 0 ) ) |
26 |
17 18 25
|
syl2an |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ↑ 0 ) ≤ ( 𝐵 ↑ 0 ) ) |
27 |
26
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 0 ) ≤ ( 𝐵 ↑ 0 ) ) |
28 |
|
reexpcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℝ ) |
29 |
28
|
ad4ant14 |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℝ ) |
30 |
|
simplll |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝐴 ∈ ℝ ) |
31 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
32 |
|
simplrl |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → 0 ≤ 𝐴 ) |
33 |
|
expge0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴 ) → 0 ≤ ( 𝐴 ↑ 𝑘 ) ) |
34 |
30 31 32 33
|
syl3anc |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → 0 ≤ ( 𝐴 ↑ 𝑘 ) ) |
35 |
|
reexpcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ 𝑘 ) ∈ ℝ ) |
36 |
35
|
ad4ant24 |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ 𝑘 ) ∈ ℝ ) |
37 |
29 34 36
|
jca31 |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ↑ 𝑘 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 ↑ 𝑘 ) ) ∧ ( 𝐵 ↑ 𝑘 ) ∈ ℝ ) ) |
38 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → 𝐴 ∈ ℝ ) |
39 |
|
simpl |
⊢ ( ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) → 0 ≤ 𝐴 ) |
40 |
38 39
|
anim12i |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) |
41 |
40
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) |
42 |
|
simpllr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝐵 ∈ ℝ ) |
43 |
37 41 42
|
jca32 |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( ( 𝐴 ↑ 𝑘 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 ↑ 𝑘 ) ) ∧ ( 𝐵 ↑ 𝑘 ) ∈ ℝ ) ∧ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ ) ) ) |
44 |
43
|
adantr |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) → ( ( ( ( 𝐴 ↑ 𝑘 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 ↑ 𝑘 ) ) ∧ ( 𝐵 ↑ 𝑘 ) ∈ ℝ ) ∧ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ ) ) ) |
45 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝐴 ≤ 𝐵 ) |
46 |
45
|
anim1ci |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) → ( ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ∧ 𝐴 ≤ 𝐵 ) ) |
47 |
|
lemul12a |
⊢ ( ( ( ( ( 𝐴 ↑ 𝑘 ) ∈ ℝ ∧ 0 ≤ ( 𝐴 ↑ 𝑘 ) ) ∧ ( 𝐵 ↑ 𝑘 ) ∈ ℝ ) ∧ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ ) ) → ( ( ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ≤ ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) ) |
48 |
44 46 47
|
sylc |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) → ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ≤ ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
49 |
|
expp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
50 |
17 49
|
sylan |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
51 |
50
|
ad5ant14 |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
52 |
|
expp1 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
53 |
18 52
|
sylan |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
54 |
53
|
ad5ant24 |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) → ( 𝐵 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐵 ↑ 𝑘 ) · 𝐵 ) ) |
55 |
48 51 54
|
3brtr4d |
⊢ ( ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) |
56 |
55
|
ex |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) |
57 |
56
|
expcom |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) |
58 |
57
|
a2d |
⊢ ( 𝑘 ∈ ℕ0 → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐵 ↑ 𝑘 ) ) → ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐵 ↑ ( 𝑘 + 1 ) ) ) ) ) |
59 |
4 8 12 16 27 58
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐵 ↑ 𝑁 ) ) ) |
60 |
59
|
exp4c |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → ( ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐵 ↑ 𝑁 ) ) ) ) ) |
61 |
60
|
com3l |
⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → ( 𝑁 ∈ ℕ0 → ( ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐵 ↑ 𝑁 ) ) ) ) ) |
62 |
61
|
3imp1 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐵 ↑ 𝑁 ) ) |