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Theorem pellfund14 42214
Description: Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
Assertion
Ref Expression
pellfund14 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))
Distinct variable groups:   𝑥,𝐷   𝑥,𝐴

Proof of Theorem pellfund14
StepHypRef Expression
1 pell14qrrp 42176 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)
2 pellfundrp 42204 . . . . 5 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
32adantr 480 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℝ+)
4 pellfundne1 42205 . . . . 5 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)
54adantr 480 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 1)
6 reglogcl 42206 . . . 4 ((𝐴 ∈ ℝ+ ∧ (PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
71, 3, 5, 6syl3anc 1368 . . 3 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
87flcld 13769 . 2 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
9 pell14qrre 42173 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)
109recnd 11246 . . 3 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ)
113, 8rpexpcld 14215 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
1211rpcnd 13024 . . 3 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
138znegcld 12672 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
143, 13rpexpcld 14215 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
1514rpcnd 13024 . . 3 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
1614rpne0d 13027 . . 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 42184 . . . . . . . . 9 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19 pellfundex 42202 . . . . . . . . 9 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
2018, 19sseldd 3978 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷))
2120adantr 480 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷))
22 pell14qrexpcl 42183 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ (PellFund‘𝐷) ∈ (Pell14QR‘𝐷) ∧ -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ (Pell14QR‘𝐷))
2317, 21, 13, 22syl3anc 1368 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ (Pell14QR‘𝐷))
24 pell14qrmulcl 42179 . . . . . 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))
2523, 24mpd3an3 1458 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ (Pell14QR‘𝐷))
26 1rp 12984 . . . . . . . . . 10 1 ∈ ℝ+
2726a1i 11 . . . . . . . . 9 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ∈ ℝ+)
28 modge0 13850 . . . . . . . . 9 ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
297, 27, 28syl2anc 583 . . . . . . . 8 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
307recnd 11246 . . . . . . . . . 10 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℂ)
318zcnd 12671 . . . . . . . . . 10 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℂ)
3230, 31negsubd 11581 . . . . . . . . 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 13855 . . . . . . . . . 10 (((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) = (((log‘𝐴) / (log‘(PellFund‘𝐷))) − (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
347, 33syl 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))))))
3532, 34eqtr4d 2769 . . . . . . . 8 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) = (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
3629, 35breqtrrd 5169 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
37 reglog1 42212 . . . . . . . 8 (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
383, 5, 37syl2anc 583 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
39 reglogmul 42209 . . . . . . . . 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)))))
401, 14, 3, 5, 39syl112anc 1371 . . . . . . . 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 42213 . . . . . . . . . 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)))))
4213, 3, 5, 41syl12anc 834 . . . . . . . . 9 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) / (log‘(PellFund‘𝐷))) = -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))
4342oveq2d 7421 . . . . . . . 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))))))
4440, 43eqtrd 2766 . . . . . . 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))))))
4536, 38, 443brtr4d 5173 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) / (log‘(PellFund‘𝐷))) ≤ ((log‘(𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))) / (log‘(PellFund‘𝐷))))
461, 14rpmulcld 13038 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ ℝ+)
47 pellfundgt1 42199 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))
4847adantr 480 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 < (PellFund‘𝐷))
49 reglogleb 42208 . . . . . . 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)))))
5027, 46, 3, 48, 49syl22anc 836 . . . . . 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)))))
5145, 50mpbird 257 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ≤ (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))))
52 modlt 13851 . . . . . . . . 9 ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
537, 27, 52syl2anc 583 . . . . . . . 8 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
5435, 53eqbrtrd 5163 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) < 1)
55 reglogbas 42211 . . . . . . . 8 (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
563, 5, 55syl2anc 583 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
5754, 44, 563brtr4d 5173 . . . . . 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 42207 . . . . . . 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)))))
5946, 3, 3, 48, 58syl22anc 836 . . . . . 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)))))
6057, 59mpbird 257 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) < (PellFund‘𝐷))
61 pellfund14gap 42203 . . . . 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)
6217, 25, 51, 60, 61syl112anc 1371 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) = 1)
6331negidd 11565 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) = 0)
6463oveq2d 7421 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) = ((PellFund‘𝐷)↑0))
653rpcnd 13024 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℂ)
663rpne0d 13027 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 0)
67 expaddz 14077 . . . . . 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)))))))
6865, 66, 8, 13, 67syl22anc 836 . . . . 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)))))))
6965exp0d 14110 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑0) = 1)
7064, 68, 693eqtr3rd 2775 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 = (((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))))
7162, 70eqtrd 2766 . . 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)))))))
7210, 12, 15, 16, 71mulcan2ad 11854 . 2 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
73 oveq2 7413 . . 3 (𝑥 = (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) → ((PellFund‘𝐷)↑𝑥) = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
7473rspceeqv 3628 . 2 (((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ ∧ 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))
758, 72, 74syl2anc 583 1 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wne 2934  wrex 3064  cdif 3940   class class class wbr 5141  cfv 6537  (class class class)co 7405  cc 11110  cr 11111  0cc0 11112  1c1 11113   + caddc 11115   · cmul 11117   < clt 11252  cle 11253  cmin 11448  -cneg 11449   / cdiv 11875  cn 12216  cz 12562  +crp 12980  cfl 13761   mod cmo 13840  cexp 14032  logclog 26443  NNcsquarenn 42152  Pell1QRcpell1qr 42153  Pell14QRcpell14qr 42155  PellFundcpellfund 42156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-oadd 8471  df-omul 8472  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-acn 9939  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ioo 13334  df-ioc 13335  df-ico 13336  df-icc 13337  df-fz 13491  df-fzo 13634  df-fl 13763  df-mod 13841  df-seq 13973  df-exp 14033  df-fac 14239  df-bc 14268  df-hash 14296  df-shft 15020  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15421  df-clim 15438  df-rlim 15439  df-sum 15639  df-ef 16017  df-sin 16019  df-cos 16020  df-pi 16022  df-dvds 16205  df-gcd 16443  df-numer 16680  df-denom 16681  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-starv 17221  df-sca 17222  df-vsca 17223  df-ip 17224  df-tset 17225  df-ple 17226  df-ds 17228  df-unif 17229  df-hom 17230  df-cco 17231  df-rest 17377  df-topn 17378  df-0g 17396  df-gsum 17397  df-topgen 17398  df-pt 17399  df-prds 17402  df-xrs 17457  df-qtop 17462  df-imas 17463  df-xps 17465  df-mre 17539  df-mrc 17540  df-acs 17542  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-mulg 18996  df-cntz 19233  df-cmn 19702  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-mopn 21236  df-fbas 21237  df-fg 21238  df-cnfld 21241  df-top 22751  df-topon 22768  df-topsp 22790  df-bases 22804  df-cld 22878  df-ntr 22879  df-cls 22880  df-nei 22957  df-lp 22995  df-perf 22996  df-cn 23086  df-cnp 23087  df-haus 23174  df-tx 23421  df-hmeo 23614  df-fil 23705  df-fm 23797  df-flim 23798  df-flf 23799  df-xms 24181  df-ms 24182  df-tms 24183  df-cncf 24753  df-limc 25750  df-dv 25751  df-log 26445  df-squarenn 42157  df-pell1qr 42158  df-pell14qr 42159  df-pell1234qr 42160  df-pellfund 42161
This theorem is referenced by:  pellfund14b  42215
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