Step |
Hyp |
Ref |
Expression |
1 |
|
rmspecnonsq |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ( ℕ ∖ ◻NN ) ) |
2 |
|
eluzelz |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 𝐴 ∈ ℤ ) |
3 |
|
zsqcl |
⊢ ( 𝐴 ∈ ℤ → ( 𝐴 ↑ 2 ) ∈ ℤ ) |
4 |
2 3
|
syl |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 ↑ 2 ) ∈ ℤ ) |
5 |
4
|
zred |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 ↑ 2 ) ∈ ℝ ) |
6 |
|
1red |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 1 ∈ ℝ ) |
7 |
5 6
|
resubcld |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ℝ ) |
8 |
|
sq1 |
⊢ ( 1 ↑ 2 ) = 1 |
9 |
8
|
a1i |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 1 ↑ 2 ) = 1 ) |
10 |
|
eluz2b2 |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝐴 ∈ ℕ ∧ 1 < 𝐴 ) ) |
11 |
10
|
simprbi |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 1 < 𝐴 ) |
12 |
|
eluzelre |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 𝐴 ∈ ℝ ) |
13 |
|
0le1 |
⊢ 0 ≤ 1 |
14 |
13
|
a1i |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 0 ≤ 1 ) |
15 |
|
eluzge2nn0 |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 𝐴 ∈ ℕ0 ) |
16 |
15
|
nn0ge0d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 0 ≤ 𝐴 ) |
17 |
6 12 14 16
|
lt2sqd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 1 < 𝐴 ↔ ( 1 ↑ 2 ) < ( 𝐴 ↑ 2 ) ) ) |
18 |
11 17
|
mpbid |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 1 ↑ 2 ) < ( 𝐴 ↑ 2 ) ) |
19 |
9 18
|
eqbrtrrd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 1 < ( 𝐴 ↑ 2 ) ) |
20 |
6 5
|
posdifd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 1 < ( 𝐴 ↑ 2 ) ↔ 0 < ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
21 |
19 20
|
mpbid |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 0 < ( ( 𝐴 ↑ 2 ) − 1 ) ) |
22 |
7 21
|
elrpd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ℝ+ ) |
23 |
22
|
rpsqrtcld |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ∈ ℝ+ ) |
24 |
23
|
rpred |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ∈ ℝ ) |
25 |
24
|
recnd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ∈ ℂ ) |
26 |
25
|
mulid1d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) · 1 ) = ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
27 |
26
|
oveq2d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 + ( ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) · 1 ) ) = ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |
28 |
|
pell1qrss14 |
⊢ ( ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ( ℕ ∖ ◻NN ) → ( Pell1QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ⊆ ( Pell14QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
29 |
1 28
|
syl |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( Pell1QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ⊆ ( Pell14QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
30 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
31 |
30
|
a1i |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 1 ∈ ℕ0 ) |
32 |
8
|
oveq2i |
⊢ ( ( ( 𝐴 ↑ 2 ) − 1 ) · ( 1 ↑ 2 ) ) = ( ( ( 𝐴 ↑ 2 ) − 1 ) · 1 ) |
33 |
7
|
recnd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ℂ ) |
34 |
33
|
mulid1d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( ( 𝐴 ↑ 2 ) − 1 ) · 1 ) = ( ( 𝐴 ↑ 2 ) − 1 ) ) |
35 |
32 34
|
eqtrid |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( ( 𝐴 ↑ 2 ) − 1 ) · ( 1 ↑ 2 ) ) = ( ( 𝐴 ↑ 2 ) − 1 ) ) |
36 |
35
|
oveq2d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − ( ( ( 𝐴 ↑ 2 ) − 1 ) · ( 1 ↑ 2 ) ) ) = ( ( 𝐴 ↑ 2 ) − ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
37 |
5
|
recnd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 ↑ 2 ) ∈ ℂ ) |
38 |
|
1cnd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 1 ∈ ℂ ) |
39 |
37 38
|
nncand |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − ( ( 𝐴 ↑ 2 ) − 1 ) ) = 1 ) |
40 |
36 39
|
eqtrd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( 𝐴 ↑ 2 ) − ( ( ( 𝐴 ↑ 2 ) − 1 ) · ( 1 ↑ 2 ) ) ) = 1 ) |
41 |
|
pellqrexplicit |
⊢ ( ( ( ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ℕ0 ∧ 1 ∈ ℕ0 ) ∧ ( ( 𝐴 ↑ 2 ) − ( ( ( 𝐴 ↑ 2 ) − 1 ) · ( 1 ↑ 2 ) ) ) = 1 ) → ( 𝐴 + ( ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) · 1 ) ) ∈ ( Pell1QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
42 |
1 15 31 40 41
|
syl31anc |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 + ( ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) · 1 ) ) ∈ ( Pell1QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
43 |
29 42
|
sseldd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 + ( ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) · 1 ) ) ∈ ( Pell14QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
44 |
27 43
|
eqeltrrd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ∈ ( Pell14QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
45 |
6 24
|
readdcld |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 1 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ∈ ℝ ) |
46 |
12 24
|
readdcld |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ∈ ℝ ) |
47 |
6 23
|
ltaddrpd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 1 < ( 1 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |
48 |
6 12 24 11
|
ltadd1dd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 1 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) < ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |
49 |
6 45 46 47 48
|
lttrd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → 1 < ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |
50 |
|
pellfundlb |
⊢ ( ( ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ( ℕ ∖ ◻NN ) ∧ ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ∈ ( Pell14QR ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ∧ 1 < ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) → ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ≤ ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |
51 |
1 44 49 50
|
syl3anc |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ≤ ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |
52 |
37 38
|
npcand |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( ( 𝐴 ↑ 2 ) − 1 ) + 1 ) = ( 𝐴 ↑ 2 ) ) |
53 |
52
|
fveq2d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( √ ‘ ( ( ( 𝐴 ↑ 2 ) − 1 ) + 1 ) ) = ( √ ‘ ( 𝐴 ↑ 2 ) ) ) |
54 |
12 16
|
sqrtsqd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( √ ‘ ( 𝐴 ↑ 2 ) ) = 𝐴 ) |
55 |
53 54
|
eqtrd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( √ ‘ ( ( ( 𝐴 ↑ 2 ) − 1 ) + 1 ) ) = 𝐴 ) |
56 |
55
|
oveq1d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( √ ‘ ( ( ( 𝐴 ↑ 2 ) − 1 ) + 1 ) ) + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) = ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |
57 |
|
pellfundge |
⊢ ( ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ( ℕ ∖ ◻NN ) → ( ( √ ‘ ( ( ( 𝐴 ↑ 2 ) − 1 ) + 1 ) ) + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ≤ ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
58 |
1 57
|
syl |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( √ ‘ ( ( ( 𝐴 ↑ 2 ) − 1 ) + 1 ) ) + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ≤ ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
59 |
56 58
|
eqbrtrrd |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ≤ ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) |
60 |
|
pellfundre |
⊢ ( ( ( 𝐴 ↑ 2 ) − 1 ) ∈ ( ℕ ∖ ◻NN ) → ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ∈ ℝ ) |
61 |
1 60
|
syl |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ∈ ℝ ) |
62 |
61 46
|
letri3d |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) = ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ↔ ( ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ≤ ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ∧ ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ≤ ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) ) |
63 |
51 59 62
|
mpbir2and |
⊢ ( 𝐴 ∈ ( ℤ≥ ‘ 2 ) → ( PellFund ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) = ( 𝐴 + ( √ ‘ ( ( 𝐴 ↑ 2 ) − 1 ) ) ) ) |