Metamath Proof Explorer


Theorem sqrlearg

Description: The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypothesis sqrlearg.1 ( 𝜑𝐴 ∈ ℝ )
Assertion sqrlearg ( 𝜑 → ( ( 𝐴 ↑ 2 ) ≤ 𝐴𝐴 ∈ ( 0 [,] 1 ) ) )

Proof

Step Hyp Ref Expression
1 sqrlearg.1 ( 𝜑𝐴 ∈ 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 ( 𝜑𝐴 ∈ 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 ( ( 𝜑𝐴 ≤ 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 ) ≤ 𝐴𝐴 ∈ ( 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 ( ( 𝜑𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 𝐴 ) ≤ ( 𝐴 · 1 ) )
53 22 adantr ( ( 𝜑𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 𝐴 ) = ( 𝐴 ↑ 2 ) )
54 19 adantr ( ( 𝜑𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 · 1 ) = 𝐴 )
55 53 54 breq12d ( ( 𝜑𝐴 ∈ ( 0 [,] 1 ) ) → ( ( 𝐴 · 𝐴 ) ≤ ( 𝐴 · 1 ) ↔ ( 𝐴 ↑ 2 ) ≤ 𝐴 ) )
56 52 55 mpbid ( ( 𝜑𝐴 ∈ ( 0 [,] 1 ) ) → ( 𝐴 ↑ 2 ) ≤ 𝐴 )
57 56 ex ( 𝜑 → ( 𝐴 ∈ ( 0 [,] 1 ) → ( 𝐴 ↑ 2 ) ≤ 𝐴 ) )
58 41 57 impbid ( 𝜑 → ( ( 𝐴 ↑ 2 ) ≤ 𝐴𝐴 ∈ ( 0 [,] 1 ) ) )