Step |
Hyp |
Ref |
Expression |
1 |
|
sqrlearg.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
0re |
⊢ 0 ∈ ℝ |
3 |
2
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → 0 ∈ ℝ ) |
4 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 1 ) → ¬ 𝐴 ≤ 1 ) |
5 |
|
1red |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 1 ) → 1 ∈ ℝ ) |
6 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 1 ) → 𝐴 ∈ ℝ ) |
7 |
5 6
|
ltnled |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 1 ) → ( 1 < 𝐴 ↔ ¬ 𝐴 ≤ 1 ) ) |
8 |
4 7
|
mpbird |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 1 ) → 1 < 𝐴 ) |
9 |
|
1red |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 1 ∈ ℝ ) |
10 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 𝐴 ∈ ℝ ) |
11 |
2
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 0 ∈ ℝ ) |
12 |
|
0lt1 |
⊢ 0 < 1 |
13 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 0 < 1 ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 1 < 𝐴 ) |
15 |
11 9 10 13 14
|
lttrd |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 0 < 𝐴 ) |
16 |
10 15
|
elrpd |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 𝐴 ∈ ℝ+ ) |
17 |
9 10 16 14
|
ltmul2dd |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → ( 𝐴 · 1 ) < ( 𝐴 · 𝐴 ) ) |
18 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
19 |
18
|
mulid1d |
⊢ ( 𝜑 → ( 𝐴 · 1 ) = 𝐴 ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → ( 𝐴 · 1 ) = 𝐴 ) |
21 |
18
|
sqvald |
⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) ) |
22 |
21
|
eqcomd |
⊢ ( 𝜑 → ( 𝐴 · 𝐴 ) = ( 𝐴 ↑ 2 ) ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → ( 𝐴 · 𝐴 ) = ( 𝐴 ↑ 2 ) ) |
24 |
20 23
|
breq12d |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → ( ( 𝐴 · 1 ) < ( 𝐴 · 𝐴 ) ↔ 𝐴 < ( 𝐴 ↑ 2 ) ) ) |
25 |
17 24
|
mpbid |
⊢ ( ( 𝜑 ∧ 1 < 𝐴 ) → 𝐴 < ( 𝐴 ↑ 2 ) ) |
26 |
8 25
|
syldan |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 ≤ 1 ) → 𝐴 < ( 𝐴 ↑ 2 ) ) |
27 |
26
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) ∧ ¬ 𝐴 ≤ 1 ) → 𝐴 < ( 𝐴 ↑ 2 ) ) |
28 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → ( 𝐴 ↑ 2 ) ≤ 𝐴 ) |
29 |
1
|
resqcld |
⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) ∈ ℝ ) |
30 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → ( 𝐴 ↑ 2 ) ∈ ℝ ) |
31 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → 𝐴 ∈ ℝ ) |
32 |
30 31
|
lenltd |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → ( ( 𝐴 ↑ 2 ) ≤ 𝐴 ↔ ¬ 𝐴 < ( 𝐴 ↑ 2 ) ) ) |
33 |
28 32
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → ¬ 𝐴 < ( 𝐴 ↑ 2 ) ) |
34 |
33
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) ∧ ¬ 𝐴 ≤ 1 ) → ¬ 𝐴 < ( 𝐴 ↑ 2 ) ) |
35 |
27 34
|
condan |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → 𝐴 ≤ 1 ) |
36 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝐴 ≤ 1 ) → 1 ∈ ℝ ) |
37 |
35 36
|
syldan |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → 1 ∈ ℝ ) |
38 |
31
|
sqge0d |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → 0 ≤ ( 𝐴 ↑ 2 ) ) |
39 |
3 30 31 38 28
|
letrd |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → 0 ≤ 𝐴 ) |
40 |
3 37 31 39 35
|
eliccd |
⊢ ( ( 𝜑 ∧ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) → 𝐴 ∈ ( 0 [,] 1 ) ) |
41 |
40
|
ex |
⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) ≤ 𝐴 → 𝐴 ∈ ( 0 [,] 1 ) ) ) |
42 |
|
unitssre |
⊢ ( 0 [,] 1 ) ⊆ ℝ |
43 |
42
|
sseli |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → 𝐴 ∈ ℝ ) |
44 |
|
1red |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → 1 ∈ ℝ ) |
45 |
|
0xr |
⊢ 0 ∈ ℝ* |
46 |
45
|
a1i |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → 0 ∈ ℝ* ) |
47 |
44
|
rexrd |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → 1 ∈ ℝ* ) |
48 |
|
id |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → 𝐴 ∈ ( 0 [,] 1 ) ) |
49 |
46 47 48
|
iccgelbd |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → 0 ≤ 𝐴 ) |
50 |
46 47 48
|
iccleubd |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → 𝐴 ≤ 1 ) |
51 |
43 44 43 49 50
|
lemul2ad |
⊢ ( 𝐴 ∈ ( 0 [,] 1 ) → ( 𝐴 · 𝐴 ) ≤ ( 𝐴 · 1 ) ) |
52 |
51
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 𝐴 ) ≤ ( 𝐴 · 1 ) ) |
53 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 𝐴 ) = ( 𝐴 ↑ 2 ) ) |
54 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 1 ) = 𝐴 ) |
55 |
53 54
|
breq12d |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( ( 𝐴 · 𝐴 ) ≤ ( 𝐴 · 1 ) ↔ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) ) |
56 |
52 55
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 ↑ 2 ) ≤ 𝐴 ) |
57 |
56
|
ex |
⊢ ( 𝜑 → ( 𝐴 ∈ ( 0 [,] 1 ) → ( 𝐴 ↑ 2 ) ≤ 𝐴 ) ) |
58 |
41 57
|
impbid |
⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) ≤ 𝐴 ↔ 𝐴 ∈ ( 0 [,] 1 ) ) ) |