Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑗 = 𝑀 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 𝑀 ) ) |
2 |
1
|
breq1d |
⊢ ( 𝑗 = 𝑀 → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ↔ ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
3 |
2
|
imbi2d |
⊢ ( 𝑗 = 𝑀 → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
4 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 𝑘 ) ) |
5 |
4
|
breq1d |
⊢ ( 𝑗 = 𝑘 → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ↔ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
6 |
5
|
imbi2d |
⊢ ( 𝑗 = 𝑘 → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
7 |
|
oveq2 |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ ( 𝑘 + 1 ) ) ) |
8 |
7
|
breq1d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ↔ ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑗 = ( 𝑘 + 1 ) → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
10 |
|
oveq2 |
⊢ ( 𝑗 = 𝑁 → ( 𝐴 ↑ 𝑗 ) = ( 𝐴 ↑ 𝑁 ) ) |
11 |
10
|
breq1d |
⊢ ( 𝑗 = 𝑁 → ( ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ↔ ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
12 |
11
|
imbi2d |
⊢ ( 𝑗 = 𝑁 → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑗 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ↔ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
13 |
|
reexpcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑀 ) ∈ ℝ ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑀 ) ∈ ℝ ) |
15 |
14
|
leidd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑀 ) ≤ ( 𝐴 ↑ 𝑀 ) ) |
16 |
|
simprll |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 𝐴 ∈ ℝ ) |
17 |
|
1red |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 1 ∈ ℝ ) |
18 |
|
simprlr |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 𝑀 ∈ ℕ0 ) |
19 |
|
simpl |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
20 |
|
eluznn0 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
21 |
18 19 20
|
syl2anc |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 𝑘 ∈ ℕ0 ) |
22 |
|
reexpcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ 𝑘 ) ∈ ℝ ) |
23 |
16 21 22
|
syl2anc |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝐴 ↑ 𝑘 ) ∈ ℝ ) |
24 |
|
simprrl |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 0 ≤ 𝐴 ) |
25 |
|
expge0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴 ) → 0 ≤ ( 𝐴 ↑ 𝑘 ) ) |
26 |
16 21 24 25
|
syl3anc |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 0 ≤ ( 𝐴 ↑ 𝑘 ) ) |
27 |
|
simprrr |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 𝐴 ≤ 1 ) |
28 |
16 17 23 26 27
|
lemul2ad |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ≤ ( ( 𝐴 ↑ 𝑘 ) · 1 ) ) |
29 |
16
|
recnd |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → 𝐴 ∈ ℂ ) |
30 |
|
expp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
31 |
29 21 30
|
syl2anc |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) = ( ( 𝐴 ↑ 𝑘 ) · 𝐴 ) ) |
32 |
23
|
recnd |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝐴 ↑ 𝑘 ) ∈ ℂ ) |
33 |
32
|
mulid1d |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( ( 𝐴 ↑ 𝑘 ) · 1 ) = ( 𝐴 ↑ 𝑘 ) ) |
34 |
33
|
eqcomd |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝐴 ↑ 𝑘 ) = ( ( 𝐴 ↑ 𝑘 ) · 1 ) ) |
35 |
28 31 34
|
3brtr4d |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑘 ) ) |
36 |
|
peano2nn0 |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℕ0 ) |
37 |
21 36
|
syl |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝑘 + 1 ) ∈ ℕ0 ) |
38 |
|
reexpcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝑘 + 1 ) ∈ ℕ0 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ∈ ℝ ) |
39 |
16 37 38
|
syl2anc |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ∈ ℝ ) |
40 |
13
|
ad2antrl |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( 𝐴 ↑ 𝑀 ) ∈ ℝ ) |
41 |
|
letr |
⊢ ( ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) ∈ ℝ ∧ ( 𝐴 ↑ 𝑘 ) ∈ ℝ ∧ ( 𝐴 ↑ 𝑀 ) ∈ ℝ ) → ( ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑘 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐴 ↑ 𝑀 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
42 |
39 23 40 41
|
syl3anc |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( ( ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑘 ) ∧ ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐴 ↑ 𝑀 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
43 |
35 42
|
mpand |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) ) → ( ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐴 ↑ 𝑀 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
44 |
43
|
ex |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐴 ↑ 𝑀 ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
45 |
44
|
a2d |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑘 ) ≤ ( 𝐴 ↑ 𝑀 ) ) → ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ ( 𝑘 + 1 ) ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
46 |
3 6 9 12 15 45
|
uzind4i |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
47 |
46
|
expd |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) → ( ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
48 |
47
|
com12 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) ) |
49 |
48
|
3impia |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) ) ) |
50 |
49
|
imp |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ ( 0 ≤ 𝐴 ∧ 𝐴 ≤ 1 ) ) → ( 𝐴 ↑ 𝑁 ) ≤ ( 𝐴 ↑ 𝑀 ) ) |