Proof of Theorem pellfund14
Step | Hyp | Ref
| Expression |
1 | | pell14qrrp 40385 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈
ℝ+) |
2 | | pellfundrp 40413 |
. . . . 5
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ∈
ℝ+) |
3 | 2 | adantr 484 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈
ℝ+) |
4 | | pellfundne1 40414 |
. . . . 5
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ≠ 1) |
5 | 4 | adantr 484 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 1) |
6 | | reglogcl 40415 |
. . . 4
⊢ ((𝐴 ∈ ℝ+
∧ (PellFund‘𝐷)
∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘𝐴) /
(log‘(PellFund‘𝐷))) ∈ ℝ) |
7 | 1, 3, 5, 6 | syl3anc 1373 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈
ℝ) |
8 | 7 | flcld 13373 |
. 2
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) ∈ ℤ) |
9 | | pell14qrre 40382 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ) |
10 | 9 | recnd 10861 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ) |
11 | 3, 8 | rpexpcld 13814 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈
ℝ+) |
12 | 11 | rpcnd 12630 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈ ℂ) |
13 | 8 | znegcld 12284 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → -(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) ∈ ℤ) |
14 | 3, 13 | rpexpcld 13814 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈
ℝ+) |
15 | 14 | rpcnd 12630 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈ ℂ) |
16 | 14 | rpne0d 12633 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ≠ 0) |
17 | | simpl 486 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐷 ∈ (ℕ ∖
◻NN)) |
18 | | pell1qrss14 40393 |
. . . . . . . . 9
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷)) |
19 | | pellfundex 40411 |
. . . . . . . . 9
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷)) |
20 | 18, 19 | sseldd 3902 |
. . . . . . . 8
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
21 | 20 | adantr 484 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
22 | | pell14qrexpcl 40392 |
. . . . . . 7
⊢ ((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 1373 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈ (Pell14QR‘𝐷)) |
24 | | pell14qrmulcl 40388 |
. . . . . 6
⊢ ((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 1464 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) ∈ (Pell14QR‘𝐷)) |
26 | | 1rp 12590 |
. . . . . . . . . 10
⊢ 1 ∈
ℝ+ |
27 | 26 | a1i 11 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ∈
ℝ+) |
28 | | modge0 13452 |
. . . . . . . . 9
⊢
((((log‘𝐴) /
(log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈
ℝ+) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1)) |
29 | 7, 27, 28 | syl2anc 587 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) /
(log‘(PellFund‘𝐷))) mod 1)) |
30 | 7 | recnd 10861 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈
ℂ) |
31 | 8 | zcnd 12283 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) ∈ ℂ) |
32 | 30, 31 | negsubd 11195 |
. . . . . . . . 9
⊢ ((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 13457 |
. . . . . . . . . 10
⊢
(((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 17 |
. . . . . . . . 9
⊢ ((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 2780 |
. . . . . . . 8
⊢ ((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 5081 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) /
(log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) |
37 | | reglog1 40421 |
. . . . . . . 8
⊢
(((PellFund‘𝐷)
∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘1) /
(log‘(PellFund‘𝐷))) = 0) |
38 | 3, 5, 37 | syl2anc 587 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) /
(log‘(PellFund‘𝐷))) = 0) |
39 | | reglogmul 40418 |
. . . . . . . . 9
⊢ ((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 1376 |
. . . . . . . 8
⊢ ((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 40422 |
. . . . . . . . . 10
⊢
((-(⌊‘((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 837 |
. . . . . . . . 9
⊢ ((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 7229 |
. . . . . . . 8
⊢ ((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 2777 |
. . . . . . 7
⊢ ((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 5085 |
. . . . . 6
⊢ ((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 12644 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) ∈
ℝ+) |
47 | | pellfundgt1 40408 |
. . . . . . . 8
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → 1 < (PellFund‘𝐷)) |
48 | 47 | adantr 484 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 < (PellFund‘𝐷)) |
49 | | reglogleb 40417 |
. . . . . . 7
⊢ (((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 839 |
. . . . . 6
⊢ ((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 260 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ≤ (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))))) |
52 | | modlt 13453 |
. . . . . . . . 9
⊢
((((log‘𝐴) /
(log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈
ℝ+) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) <
1) |
53 | 7, 27, 52 | syl2anc 587 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) <
1) |
54 | 35, 53 | eqbrtrd 5075 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) +
-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) < 1) |
55 | | reglogbas 40420 |
. . . . . . . 8
⊢
(((PellFund‘𝐷)
∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) →
((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1) |
56 | 3, 5, 55 | syl2anc 587 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘(PellFund‘𝐷)) /
(log‘(PellFund‘𝐷))) = 1) |
57 | 54, 44, 56 | 3brtr4d 5085 |
. . . . . 6
⊢ ((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 40416 |
. . . . . . 7
⊢ ((((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 839 |
. . . . . 6
⊢ ((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 260 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) < (PellFund‘𝐷)) |
61 | | pellfund14gap 40412 |
. . . . 5
⊢ ((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 1376 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) = 1) |
63 | 31 | negidd 11179 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) = 0) |
64 | 63 | oveq2d 7229 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑((⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) = ((PellFund‘𝐷)↑0)) |
65 | 3 | rpcnd 12630 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℂ) |
66 | 3 | rpne0d 12633 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 0) |
67 | | expaddz 13679 |
. . . . . 6
⊢
((((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 839 |
. . . . 5
⊢ ((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 13710 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑0) = 1) |
70 | 64, 68, 69 | 3eqtr3rd 2786 |
. . . 4
⊢ ((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 2777 |
. . 3
⊢ ((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 11468 |
. 2
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) |
73 | | oveq2 7221 |
. . 3
⊢ (𝑥 =
(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) →
((PellFund‘𝐷)↑𝑥) = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) |
74 | 73 | rspceeqv 3552 |
. 2
⊢
(((⌊‘((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 587 |
1
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) |