| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nnrp |
⊢ ( 𝐷 ∈ ℕ → 𝐷 ∈ ℝ+ ) |
| 2 |
1
|
ad2antrr |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐷 ∈ ℝ+ ) |
| 3 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 4 |
3
|
a1i |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 1 ∈ ℝ+ ) |
| 5 |
2 4
|
rpaddcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 + 1 ) ∈ ℝ+ ) |
| 6 |
5
|
rpsqrtcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ ( 𝐷 + 1 ) ) ∈ ℝ+ ) |
| 7 |
6
|
rpred |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ ( 𝐷 + 1 ) ) ∈ ℝ ) |
| 8 |
2
|
rpsqrtcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ 𝐷 ) ∈ ℝ+ ) |
| 9 |
8
|
rpred |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ 𝐷 ) ∈ ℝ ) |
| 10 |
|
nn0re |
⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ ) |
| 11 |
10
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → 𝐴 ∈ ℝ ) |
| 12 |
11
|
ad2antlr |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ∈ ℝ ) |
| 13 |
|
nn0re |
⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ ) |
| 14 |
13
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → 𝐵 ∈ ℝ ) |
| 15 |
14
|
ad2antlr |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐵 ∈ ℝ ) |
| 16 |
9 15
|
remulcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) · 𝐵 ) ∈ ℝ ) |
| 17 |
2
|
rpred |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐷 ∈ ℝ ) |
| 18 |
|
1re |
⊢ 1 ∈ ℝ |
| 19 |
18
|
a1i |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 1 ∈ ℝ ) |
| 20 |
15
|
resqcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐵 ↑ 2 ) ∈ ℝ ) |
| 21 |
19 20
|
resubcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 1 − ( 𝐵 ↑ 2 ) ) ∈ ℝ ) |
| 22 |
17 21
|
remulcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ∈ ℝ ) |
| 23 |
|
0red |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 0 ∈ ℝ ) |
| 24 |
17 23
|
remulcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · 0 ) ∈ ℝ ) |
| 25 |
12
|
resqcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ↑ 2 ) ∈ ℝ ) |
| 26 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
| 27 |
26
|
a1i |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 1 ↑ 2 ) = 1 ) |
| 28 |
|
nnge1 |
⊢ ( 𝐵 ∈ ℕ → 1 ≤ 𝐵 ) |
| 29 |
28
|
adantl |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 ∈ ℕ ) → 1 ≤ 𝐵 ) |
| 30 |
|
simplrl |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ) |
| 31 |
|
oveq1 |
⊢ ( 𝐵 = 0 → ( 𝐵 ↑ 2 ) = ( 0 ↑ 2 ) ) |
| 32 |
31
|
adantl |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐵 ↑ 2 ) = ( 0 ↑ 2 ) ) |
| 33 |
|
sq0 |
⊢ ( 0 ↑ 2 ) = 0 |
| 34 |
32 33
|
eqtrdi |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐵 ↑ 2 ) = 0 ) |
| 35 |
34
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐷 · ( 𝐵 ↑ 2 ) ) = ( 𝐷 · 0 ) ) |
| 36 |
2
|
rpcnd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐷 ∈ ℂ ) |
| 37 |
36
|
adantr |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → 𝐷 ∈ ℂ ) |
| 38 |
37
|
mul01d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐷 · 0 ) = 0 ) |
| 39 |
35 38
|
eqtrd |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐷 · ( 𝐵 ↑ 2 ) ) = 0 ) |
| 40 |
39
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = ( ( 𝐴 ↑ 2 ) − 0 ) ) |
| 41 |
|
simplrr |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) |
| 42 |
12
|
recnd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐴 ∈ ℂ ) |
| 43 |
42
|
sqcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
| 44 |
43
|
adantr |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
| 45 |
44
|
subid1d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( ( 𝐴 ↑ 2 ) − 0 ) = ( 𝐴 ↑ 2 ) ) |
| 46 |
40 41 45
|
3eqtr3d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → 1 = ( 𝐴 ↑ 2 ) ) |
| 47 |
26 46
|
eqtr2id |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐴 ↑ 2 ) = ( 1 ↑ 2 ) ) |
| 48 |
|
nn0ge0 |
⊢ ( 𝐴 ∈ ℕ0 → 0 ≤ 𝐴 ) |
| 49 |
48
|
adantr |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → 0 ≤ 𝐴 ) |
| 50 |
49
|
ad2antlr |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 0 ≤ 𝐴 ) |
| 51 |
|
0le1 |
⊢ 0 ≤ 1 |
| 52 |
51
|
a1i |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 0 ≤ 1 ) |
| 53 |
|
sq11 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ) → ( ( 𝐴 ↑ 2 ) = ( 1 ↑ 2 ) ↔ 𝐴 = 1 ) ) |
| 54 |
12 50 19 52 53
|
syl22anc |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝐴 ↑ 2 ) = ( 1 ↑ 2 ) ↔ 𝐴 = 1 ) ) |
| 55 |
54
|
adantr |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( ( 𝐴 ↑ 2 ) = ( 1 ↑ 2 ) ↔ 𝐴 = 1 ) ) |
| 56 |
47 55
|
mpbid |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → 𝐴 = 1 ) |
| 57 |
|
simpr |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → 𝐵 = 0 ) |
| 58 |
57
|
oveq2d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( ( √ ‘ 𝐷 ) · 𝐵 ) = ( ( √ ‘ 𝐷 ) · 0 ) ) |
| 59 |
8
|
rpcnd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ 𝐷 ) ∈ ℂ ) |
| 60 |
59
|
adantr |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( √ ‘ 𝐷 ) ∈ ℂ ) |
| 61 |
60
|
mul01d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( ( √ ‘ 𝐷 ) · 0 ) = 0 ) |
| 62 |
58 61
|
eqtrd |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( ( √ ‘ 𝐷 ) · 𝐵 ) = 0 ) |
| 63 |
56 62
|
oveq12d |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) = ( 1 + 0 ) ) |
| 64 |
|
1p0e1 |
⊢ ( 1 + 0 ) = 1 |
| 65 |
63 64
|
eqtrdi |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) = 1 ) |
| 66 |
30 65
|
breqtrd |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → 1 < 1 ) |
| 67 |
18
|
ltnri |
⊢ ¬ 1 < 1 |
| 68 |
|
pm2.24 |
⊢ ( 1 < 1 → ( ¬ 1 < 1 → 1 ≤ 𝐵 ) ) |
| 69 |
66 67 68
|
mpisyl |
⊢ ( ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) ∧ 𝐵 = 0 ) → 1 ≤ 𝐵 ) |
| 70 |
|
simplrr |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐵 ∈ ℕ0 ) |
| 71 |
|
elnn0 |
⊢ ( 𝐵 ∈ ℕ0 ↔ ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) ) |
| 72 |
70 71
|
sylib |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐵 ∈ ℕ ∨ 𝐵 = 0 ) ) |
| 73 |
29 69 72
|
mpjaodan |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 1 ≤ 𝐵 ) |
| 74 |
|
nn0ge0 |
⊢ ( 𝐵 ∈ ℕ0 → 0 ≤ 𝐵 ) |
| 75 |
74
|
adantl |
⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → 0 ≤ 𝐵 ) |
| 76 |
75
|
ad2antlr |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 0 ≤ 𝐵 ) |
| 77 |
19 15 52 76
|
le2sqd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 1 ≤ 𝐵 ↔ ( 1 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) ) |
| 78 |
73 77
|
mpbid |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 1 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) |
| 79 |
27 78
|
eqbrtrrd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 1 ≤ ( 𝐵 ↑ 2 ) ) |
| 80 |
19 20
|
suble0d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 1 − ( 𝐵 ↑ 2 ) ) ≤ 0 ↔ 1 ≤ ( 𝐵 ↑ 2 ) ) ) |
| 81 |
79 80
|
mpbird |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 1 − ( 𝐵 ↑ 2 ) ) ≤ 0 ) |
| 82 |
21 23 2
|
lemul2d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 1 − ( 𝐵 ↑ 2 ) ) ≤ 0 ↔ ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ≤ ( 𝐷 · 0 ) ) ) |
| 83 |
81 82
|
mpbid |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ≤ ( 𝐷 · 0 ) ) |
| 84 |
22 24 25 83
|
leadd2dd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝐴 ↑ 2 ) + ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ) ≤ ( ( 𝐴 ↑ 2 ) + ( 𝐷 · 0 ) ) ) |
| 85 |
5
|
rpcnd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 + 1 ) ∈ ℂ ) |
| 86 |
85
|
sqsqrtd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ ( 𝐷 + 1 ) ) ↑ 2 ) = ( 𝐷 + 1 ) ) |
| 87 |
|
simprr |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) |
| 88 |
87
|
eqcomd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 1 = ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) |
| 89 |
88
|
oveq2d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 + 1 ) = ( 𝐷 + ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) ) |
| 90 |
15
|
recnd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 𝐵 ∈ ℂ ) |
| 91 |
90
|
sqcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐵 ↑ 2 ) ∈ ℂ ) |
| 92 |
36 91
|
mulcld |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · ( 𝐵 ↑ 2 ) ) ∈ ℂ ) |
| 93 |
36 43 92
|
addsub12d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 + ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) = ( ( 𝐴 ↑ 2 ) + ( 𝐷 − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) ) |
| 94 |
19
|
recnd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 1 ∈ ℂ ) |
| 95 |
36 94 91
|
subdid |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) = ( ( 𝐷 · 1 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) |
| 96 |
36
|
mulridd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · 1 ) = 𝐷 ) |
| 97 |
96
|
oveq1d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝐷 · 1 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = ( 𝐷 − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) |
| 98 |
95 97
|
eqtr2d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ) |
| 99 |
98
|
oveq2d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝐴 ↑ 2 ) + ( 𝐷 − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) = ( ( 𝐴 ↑ 2 ) + ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ) ) |
| 100 |
93 99
|
eqtrd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 + ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) ) = ( ( 𝐴 ↑ 2 ) + ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ) ) |
| 101 |
86 89 100
|
3eqtrd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ ( 𝐷 + 1 ) ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) + ( 𝐷 · ( 1 − ( 𝐵 ↑ 2 ) ) ) ) ) |
| 102 |
36
|
mul01d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐷 · 0 ) = 0 ) |
| 103 |
102
|
oveq2d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝐴 ↑ 2 ) + ( 𝐷 · 0 ) ) = ( ( 𝐴 ↑ 2 ) + 0 ) ) |
| 104 |
43
|
addridd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( 𝐴 ↑ 2 ) + 0 ) = ( 𝐴 ↑ 2 ) ) |
| 105 |
103 104
|
eqtr2d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 𝐴 ↑ 2 ) = ( ( 𝐴 ↑ 2 ) + ( 𝐷 · 0 ) ) ) |
| 106 |
84 101 105
|
3brtr4d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ ( 𝐷 + 1 ) ) ↑ 2 ) ≤ ( 𝐴 ↑ 2 ) ) |
| 107 |
6
|
rpge0d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → 0 ≤ ( √ ‘ ( 𝐷 + 1 ) ) ) |
| 108 |
7 12 107 50
|
le2sqd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ ( 𝐷 + 1 ) ) ≤ 𝐴 ↔ ( ( √ ‘ ( 𝐷 + 1 ) ) ↑ 2 ) ≤ ( 𝐴 ↑ 2 ) ) ) |
| 109 |
106 108
|
mpbird |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ ( 𝐷 + 1 ) ) ≤ 𝐴 ) |
| 110 |
59
|
mulridd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) · 1 ) = ( √ ‘ 𝐷 ) ) |
| 111 |
19 15 8
|
lemul2d |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( 1 ≤ 𝐵 ↔ ( ( √ ‘ 𝐷 ) · 1 ) ≤ ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ) |
| 112 |
73 111
|
mpbid |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ 𝐷 ) · 1 ) ≤ ( ( √ ‘ 𝐷 ) · 𝐵 ) ) |
| 113 |
110 112
|
eqbrtrrd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( √ ‘ 𝐷 ) ≤ ( ( √ ‘ 𝐷 ) · 𝐵 ) ) |
| 114 |
7 9 12 16 109 113
|
le2addd |
⊢ ( ( ( 𝐷 ∈ ℕ ∧ ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) ) ∧ ( 1 < ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ∧ ( ( 𝐴 ↑ 2 ) − ( 𝐷 · ( 𝐵 ↑ 2 ) ) ) = 1 ) ) → ( ( √ ‘ ( 𝐷 + 1 ) ) + ( √ ‘ 𝐷 ) ) ≤ ( 𝐴 + ( ( √ ‘ 𝐷 ) · 𝐵 ) ) ) |