Step |
Hyp |
Ref |
Expression |
1 |
|
recgt0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( 1 / 𝐴 ) ) |
2 |
|
gt0ne0 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 𝐴 ≠ 0 ) |
3 |
|
rereccl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ∈ ℝ ) |
4 |
2 3
|
syldan |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 / 𝐴 ) ∈ ℝ ) |
5 |
|
1re |
⊢ 1 ∈ ℝ |
6 |
|
ltaddpos |
⊢ ( ( ( 1 / 𝐴 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( 0 < ( 1 / 𝐴 ) ↔ 1 < ( 1 + ( 1 / 𝐴 ) ) ) ) |
7 |
4 5 6
|
sylancl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 0 < ( 1 / 𝐴 ) ↔ 1 < ( 1 + ( 1 / 𝐴 ) ) ) ) |
8 |
1 7
|
mpbid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 1 < ( 1 + ( 1 / 𝐴 ) ) ) |
9 |
|
readdcl |
⊢ ( ( 1 ∈ ℝ ∧ ( 1 / 𝐴 ) ∈ ℝ ) → ( 1 + ( 1 / 𝐴 ) ) ∈ ℝ ) |
10 |
5 4 9
|
sylancr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 + ( 1 / 𝐴 ) ) ∈ ℝ ) |
11 |
|
0lt1 |
⊢ 0 < 1 |
12 |
|
0re |
⊢ 0 ∈ ℝ |
13 |
|
lttr |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ ∧ ( 1 + ( 1 / 𝐴 ) ) ∈ ℝ ) → ( ( 0 < 1 ∧ 1 < ( 1 + ( 1 / 𝐴 ) ) ) → 0 < ( 1 + ( 1 / 𝐴 ) ) ) ) |
14 |
12 5 10 13
|
mp3an12i |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( ( 0 < 1 ∧ 1 < ( 1 + ( 1 / 𝐴 ) ) ) → 0 < ( 1 + ( 1 / 𝐴 ) ) ) ) |
15 |
11 14
|
mpani |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < ( 1 + ( 1 / 𝐴 ) ) → 0 < ( 1 + ( 1 / 𝐴 ) ) ) ) |
16 |
8 15
|
mpd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < ( 1 + ( 1 / 𝐴 ) ) ) |
17 |
|
recgt1 |
⊢ ( ( ( 1 + ( 1 / 𝐴 ) ) ∈ ℝ ∧ 0 < ( 1 + ( 1 / 𝐴 ) ) ) → ( 1 < ( 1 + ( 1 / 𝐴 ) ) ↔ ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 1 ) ) |
18 |
10 16 17
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 < ( 1 + ( 1 / 𝐴 ) ) ↔ ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 1 ) ) |
19 |
8 18
|
mpbid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 1 ) |
20 |
|
ltaddpos |
⊢ ( ( 1 ∈ ℝ ∧ ( 1 / 𝐴 ) ∈ ℝ ) → ( 0 < 1 ↔ ( 1 / 𝐴 ) < ( ( 1 / 𝐴 ) + 1 ) ) ) |
21 |
5 4 20
|
sylancr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 0 < 1 ↔ ( 1 / 𝐴 ) < ( ( 1 / 𝐴 ) + 1 ) ) ) |
22 |
11 21
|
mpbii |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 / 𝐴 ) < ( ( 1 / 𝐴 ) + 1 ) ) |
23 |
4
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 / 𝐴 ) ∈ ℂ ) |
24 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
25 |
|
addcom |
⊢ ( ( ( 1 / 𝐴 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 1 / 𝐴 ) + 1 ) = ( 1 + ( 1 / 𝐴 ) ) ) |
26 |
23 24 25
|
sylancl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( ( 1 / 𝐴 ) + 1 ) = ( 1 + ( 1 / 𝐴 ) ) ) |
27 |
22 26
|
breqtrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 / 𝐴 ) < ( 1 + ( 1 / 𝐴 ) ) ) |
28 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 𝐴 ∈ ℝ ) |
29 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 0 < 𝐴 ) |
30 |
|
ltrec1 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) ∧ ( ( 1 + ( 1 / 𝐴 ) ) ∈ ℝ ∧ 0 < ( 1 + ( 1 / 𝐴 ) ) ) ) → ( ( 1 / 𝐴 ) < ( 1 + ( 1 / 𝐴 ) ) ↔ ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 𝐴 ) ) |
31 |
28 29 10 16 30
|
syl22anc |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( ( 1 / 𝐴 ) < ( 1 + ( 1 / 𝐴 ) ) ↔ ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 𝐴 ) ) |
32 |
27 31
|
mpbid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 𝐴 ) |
33 |
19 32
|
jca |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 1 ∧ ( 1 / ( 1 + ( 1 / 𝐴 ) ) ) < 𝐴 ) ) |