Proof of Theorem pellfund14
| Step | Hyp | Ref
| Expression |
| 1 | | pell14qrrp 42871 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈
ℝ+) |
| 2 | | pellfundrp 42899 |
. . . . 5
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ∈
ℝ+) |
| 3 | 2 | adantr 480 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈
ℝ+) |
| 4 | | pellfundne1 42900 |
. . . . 5
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ≠ 1) |
| 5 | 4 | adantr 480 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 1) |
| 6 | | reglogcl 42901 |
. . . 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 13838 |
. 2
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) ∈ ℤ) |
| 9 | | pell14qrre 42868 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ) |
| 10 | 9 | recnd 11289 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ) |
| 11 | 3, 8 | rpexpcld 14286 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈
ℝ+) |
| 12 | 11 | rpcnd 13079 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈ ℂ) |
| 13 | 8 | znegcld 12724 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → -(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) ∈ ℤ) |
| 14 | 3, 13 | rpexpcld 14286 |
. . . 4
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈
ℝ+) |
| 15 | 14 | rpcnd 13079 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ∈ ℂ) |
| 16 | 14 | rpne0d 13082 |
. . 3
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) ≠ 0) |
| 17 | | simpl 482 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐷 ∈ (ℕ ∖
◻NN)) |
| 18 | | pell1qrss14 42879 |
. . . . . . . . 9
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷)) |
| 19 | | pellfundex 42897 |
. . . . . . . . 9
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷)) |
| 20 | 18, 19 | sseldd 3984 |
. . . . . . . 8
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
| 21 | 20 | adantr 480 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷)) |
| 22 | | pell14qrexpcl 42878 |
. . . . . . 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 42874 |
. . . . . 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 13038 |
. . . . . . . . . 10
⊢ 1 ∈
ℝ+ |
| 27 | 26 | a1i 11 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ∈
ℝ+) |
| 28 | | modge0 13919 |
. . . . . . . . 9
⊢
((((log‘𝐴) /
(log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈
ℝ+) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1)) |
| 29 | 7, 27, 28 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) /
(log‘(PellFund‘𝐷))) mod 1)) |
| 30 | 7 | recnd 11289 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈
ℂ) |
| 31 | 8 | zcnd 12723 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) ∈ ℂ) |
| 32 | 30, 31 | negsubd 11626 |
. . . . . . . . 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 13924 |
. . . . . . . . . 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 5171 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) /
(log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) |
| 37 | | reglog1 42907 |
. . . . . . . 8
⊢
(((PellFund‘𝐷)
∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘1) /
(log‘(PellFund‘𝐷))) = 0) |
| 38 | 3, 5, 37 | syl2anc 584 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) /
(log‘(PellFund‘𝐷))) = 0) |
| 39 | | reglogmul 42904 |
. . . . . . . . 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 42908 |
. . . . . . . . . 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 7447 |
. . . . . . . 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 5175 |
. . . . . 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 13093 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) ∈
ℝ+) |
| 47 | | pellfundgt1 42894 |
. . . . . . . 8
⊢ (𝐷 ∈ (ℕ ∖
◻NN) → 1 < (PellFund‘𝐷)) |
| 48 | 47 | adantr 480 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 < (PellFund‘𝐷)) |
| 49 | | reglogleb 42903 |
. . . . . . 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 257 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ≤ (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))))) |
| 52 | | modlt 13920 |
. . . . . . . . 9
⊢
((((log‘𝐴) /
(log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈
ℝ+) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) <
1) |
| 53 | 7, 27, 52 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) <
1) |
| 54 | 35, 53 | eqbrtrd 5165 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) +
-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) < 1) |
| 55 | | reglogbas 42906 |
. . . . . . . 8
⊢
(((PellFund‘𝐷)
∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) →
((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1) |
| 56 | 3, 5, 55 | syl2anc 584 |
. . . . . . 7
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘(PellFund‘𝐷)) /
(log‘(PellFund‘𝐷))) = 1) |
| 57 | 54, 44, 56 | 3brtr4d 5175 |
. . . . . 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 42902 |
. . . . . . 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 257 |
. . . . 5
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) < (PellFund‘𝐷)) |
| 61 | | pellfund14gap 42898 |
. . . . 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 11610 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷))))) = 0) |
| 64 | 63 | oveq2d 7447 |
. . . . 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 13079 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℂ) |
| 66 | 3 | rpne0d 13082 |
. . . . . 6
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 0) |
| 67 | | expaddz 14147 |
. . . . . 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 14180 |
. . . . 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 11899 |
. 2
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) |
| 73 | | oveq2 7439 |
. . 3
⊢ (𝑥 =
(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) →
((PellFund‘𝐷)↑𝑥) = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) /
(log‘(PellFund‘𝐷)))))) |
| 74 | 73 | rspceeqv 3645 |
. 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 584 |
1
⊢ ((𝐷 ∈ (ℕ ∖
◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)) |