| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pell14qrrp |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐴 ∈ ℝ+ ) |
| 2 |
|
pellfundrp |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( PellFund ‘ 𝐷 ) ∈ ℝ+ ) |
| 3 |
2
|
adantr |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( PellFund ‘ 𝐷 ) ∈ ℝ+ ) |
| 4 |
|
pellfundne1 |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( PellFund ‘ 𝐷 ) ≠ 1 ) |
| 5 |
4
|
adantr |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( PellFund ‘ 𝐷 ) ≠ 1 ) |
| 6 |
|
reglogcl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( PellFund ‘ 𝐷 ) ∈ ℝ+ ∧ ( PellFund ‘ 𝐷 ) ≠ 1 ) → ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ∈ ℝ ) |
| 7 |
1 3 5 6
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ∈ ℝ ) |
| 8 |
7
|
flcld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℤ ) |
| 9 |
|
pell14qrre |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐴 ∈ ℝ ) |
| 10 |
9
|
recnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐴 ∈ ℂ ) |
| 11 |
3 8
|
rpexpcld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ℝ+ ) |
| 12 |
11
|
rpcnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ℂ ) |
| 13 |
8
|
znegcld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℤ ) |
| 14 |
3 13
|
rpexpcld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ℝ+ ) |
| 15 |
14
|
rpcnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ℂ ) |
| 16 |
14
|
rpne0d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ≠ 0 ) |
| 17 |
|
simpl |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐷 ∈ ( ℕ ∖ ◻NN ) ) |
| 18 |
|
pell1qrss14 |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( Pell1QR ‘ 𝐷 ) ⊆ ( Pell14QR ‘ 𝐷 ) ) |
| 19 |
|
pellfundex |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( PellFund ‘ 𝐷 ) ∈ ( Pell1QR ‘ 𝐷 ) ) |
| 20 |
18 19
|
sseldd |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → ( PellFund ‘ 𝐷 ) ∈ ( Pell14QR ‘ 𝐷 ) ) |
| 21 |
20
|
adantr |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( PellFund ‘ 𝐷 ) ∈ ( Pell14QR ‘ 𝐷 ) ) |
| 22 |
|
pell14qrexpcl |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ ( PellFund ‘ 𝐷 ) ∈ ( Pell14QR ‘ 𝐷 ) ∧ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℤ ) → ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ( Pell14QR ‘ 𝐷 ) ) |
| 23 |
17 21 13 22
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ( Pell14QR ‘ 𝐷 ) ) |
| 24 |
|
pell14qrmulcl |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ∧ ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ∈ ( Pell14QR ‘ 𝐷 ) ) |
| 25 |
23 24
|
mpd3an3 |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ∈ ( Pell14QR ‘ 𝐷 ) ) |
| 26 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 27 |
26
|
a1i |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 1 ∈ ℝ+ ) |
| 28 |
|
modge0 |
⊢ ( ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ∈ ℝ ∧ 1 ∈ ℝ+ ) → 0 ≤ ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) mod 1 ) ) |
| 29 |
7 27 28
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 0 ≤ ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) mod 1 ) ) |
| 30 |
7
|
recnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ∈ ℂ ) |
| 31 |
8
|
zcnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℂ ) |
| 32 |
30 31
|
negsubd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) − ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 33 |
|
modfrac |
⊢ ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ∈ ℝ → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) mod 1 ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) − ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 34 |
7 33
|
syl |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) mod 1 ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) − ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 35 |
32 34
|
eqtr4d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) mod 1 ) ) |
| 36 |
29 35
|
breqtrrd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 0 ≤ ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 37 |
|
reglog1 |
⊢ ( ( ( PellFund ‘ 𝐷 ) ∈ ℝ+ ∧ ( PellFund ‘ 𝐷 ) ≠ 1 ) → ( ( log ‘ 1 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = 0 ) |
| 38 |
3 5 37
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ 1 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = 0 ) |
| 39 |
|
reglogmul |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ∈ ℝ+ ∧ ( ( PellFund ‘ 𝐷 ) ∈ ℝ+ ∧ ( PellFund ‘ 𝐷 ) ≠ 1 ) ) → ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + ( ( log ‘ ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 40 |
1 14 3 5 39
|
syl112anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + ( ( log ‘ ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 41 |
|
reglogexpbas |
⊢ ( ( - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℤ ∧ ( ( PellFund ‘ 𝐷 ) ∈ ℝ+ ∧ ( PellFund ‘ 𝐷 ) ≠ 1 ) ) → ( ( log ‘ ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 42 |
13 3 5 41
|
syl12anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 43 |
42
|
oveq2d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + ( ( log ‘ ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 44 |
40 43
|
eqtrd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 45 |
36 38 44
|
3brtr4d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ 1 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ≤ ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) |
| 46 |
1 14
|
rpmulcld |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ∈ ℝ+ ) |
| 47 |
|
pellfundgt1 |
⊢ ( 𝐷 ∈ ( ℕ ∖ ◻NN ) → 1 < ( PellFund ‘ 𝐷 ) ) |
| 48 |
47
|
adantr |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 1 < ( PellFund ‘ 𝐷 ) ) |
| 49 |
|
reglogleb |
⊢ ( ( ( 1 ∈ ℝ+ ∧ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ∈ ℝ+ ) ∧ ( ( PellFund ‘ 𝐷 ) ∈ ℝ+ ∧ 1 < ( PellFund ‘ 𝐷 ) ) ) → ( 1 ≤ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ↔ ( ( log ‘ 1 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ≤ ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 50 |
27 46 3 48 49
|
syl22anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 1 ≤ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ↔ ( ( log ‘ 1 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ≤ ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 51 |
45 50
|
mpbird |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 1 ≤ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) |
| 52 |
|
modlt |
⊢ ( ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ∈ ℝ ∧ 1 ∈ ℝ+ ) → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) mod 1 ) < 1 ) |
| 53 |
7 27 52
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) mod 1 ) < 1 ) |
| 54 |
35 53
|
eqbrtrd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) < 1 ) |
| 55 |
|
reglogbas |
⊢ ( ( ( PellFund ‘ 𝐷 ) ∈ ℝ+ ∧ ( PellFund ‘ 𝐷 ) ≠ 1 ) → ( ( log ‘ ( PellFund ‘ 𝐷 ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = 1 ) |
| 56 |
3 5 55
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ ( PellFund ‘ 𝐷 ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) = 1 ) |
| 57 |
54 44 56
|
3brtr4d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) < ( ( log ‘ ( PellFund ‘ 𝐷 ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) |
| 58 |
|
reglogltb |
⊢ ( ( ( ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ∈ ℝ+ ∧ ( PellFund ‘ 𝐷 ) ∈ ℝ+ ) ∧ ( ( PellFund ‘ 𝐷 ) ∈ ℝ+ ∧ 1 < ( PellFund ‘ 𝐷 ) ) ) → ( ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) < ( PellFund ‘ 𝐷 ) ↔ ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) < ( ( log ‘ ( PellFund ‘ 𝐷 ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 59 |
46 3 3 48 58
|
syl22anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) < ( PellFund ‘ 𝐷 ) ↔ ( ( log ‘ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) < ( ( log ‘ ( PellFund ‘ 𝐷 ) ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) |
| 60 |
57 59
|
mpbird |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) < ( PellFund ‘ 𝐷 ) ) |
| 61 |
|
pellfund14gap |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ∈ ( Pell14QR ‘ 𝐷 ) ∧ ( 1 ≤ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ∧ ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) < ( PellFund ‘ 𝐷 ) ) ) → ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) = 1 ) |
| 62 |
17 25 51 60 61
|
syl112anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) = 1 ) |
| 63 |
31
|
negidd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) = 0 ) |
| 64 |
63
|
oveq2d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) = ( ( PellFund ‘ 𝐷 ) ↑ 0 ) ) |
| 65 |
3
|
rpcnd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( PellFund ‘ 𝐷 ) ∈ ℂ ) |
| 66 |
3
|
rpne0d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( PellFund ‘ 𝐷 ) ≠ 0 ) |
| 67 |
|
expaddz |
⊢ ( ( ( ( PellFund ‘ 𝐷 ) ∈ ℂ ∧ ( PellFund ‘ 𝐷 ) ≠ 0 ) ∧ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℤ ∧ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℤ ) ) → ( ( PellFund ‘ 𝐷 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) = ( ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) |
| 68 |
65 66 8 13 67
|
syl22anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) + - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) = ( ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) |
| 69 |
65
|
exp0d |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( ( PellFund ‘ 𝐷 ) ↑ 0 ) = 1 ) |
| 70 |
64 68 69
|
3eqtr3rd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 1 = ( ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) |
| 71 |
62 70
|
eqtrd |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ( 𝐴 · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) = ( ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) · ( ( PellFund ‘ 𝐷 ) ↑ - ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) ) |
| 72 |
10 12 15 16 71
|
mulcan2ad |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → 𝐴 = ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 73 |
|
oveq2 |
⊢ ( 𝑥 = ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) → ( ( PellFund ‘ 𝐷 ) ↑ 𝑥 ) = ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) |
| 74 |
73
|
rspceeqv |
⊢ ( ( ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ∈ ℤ ∧ 𝐴 = ( ( PellFund ‘ 𝐷 ) ↑ ( ⌊ ‘ ( ( log ‘ 𝐴 ) / ( log ‘ ( PellFund ‘ 𝐷 ) ) ) ) ) ) → ∃ 𝑥 ∈ ℤ 𝐴 = ( ( PellFund ‘ 𝐷 ) ↑ 𝑥 ) ) |
| 75 |
8 72 74
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ℕ ∖ ◻NN ) ∧ 𝐴 ∈ ( Pell14QR ‘ 𝐷 ) ) → ∃ 𝑥 ∈ ℤ 𝐴 = ( ( PellFund ‘ 𝐷 ) ↑ 𝑥 ) ) |