Theorem pellfund14 36480

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

Step	Hyp	Ref	Expression
1		pell14qrrp 36442	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)
2		pellfundrp 36470	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
3	2	adantr 480	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℝ+)
4		pellfundne1 36471	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)
5	4	adantr 480	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 1)
6		reglogcl 36472	. . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ (PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
7	1, 3, 5, 6	syl3anc 1318	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
8	7	flcld 12461	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
9		pell14qrre 36439	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)
10	9	recnd 9947	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ)
11	3, 8	rpexpcld 12894	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
12	11	rpcnd 11750	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
13	8	znegcld 11360	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
14	3, 13	rpexpcld 12894	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
15	14	rpcnd 11750	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
16	14	rpne0d 11753	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ≠ 0)
17		simpl 472	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐷 ∈ (ℕ ∖ ◻NN))
18		pell1qrss14 36450	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19		pellfundex 36468	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
20	18, 19	sseldd 3569	. . . . . . . 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 36449	. . . . . . 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 1318	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ (Pell14QR‘𝐷))
24		pell14qrmulcl 36445	. . . . . 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 1417	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ (Pell14QR‘𝐷))
26		1rp 11712	. . . . . . . . . 10 ⊢ 1 ∈ ℝ+
27	26	a1i 11	. . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ∈ ℝ+)
28		modge0 12540	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
29	7, 27, 28	syl2anc 691	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
30	7	recnd 9947	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℂ)
31	8	zcnd 11359	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℂ)
32	30, 31	negsubd 10277	. . . . . . . . 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 12545	. . . . . . . . . 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 2647	. . . . . . . 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 4611	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
37		reglog1 36478	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
38	3, 5, 37	syl2anc 691	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
39		reglogmul 36475	. . . . . . . . 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 1322	. . . . . . . 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 36479	. . . . . . . . . 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 1316	. . . . . . . . 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 6565	. . . . . . . 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 2644	. . . . . . 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 4615	. . . . . 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 11764	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ ℝ+)
47		pellfundgt1 36465	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))
48	47	adantr 480	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 < (PellFund‘𝐷))
49		reglogleb 36474	. . . . . . 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 1319	. . . . . 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 246	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ≤ (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))))
52		modlt 12541	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
53	7, 27, 52	syl2anc 691	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
54	35, 53	eqbrtrd 4605	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) < 1)
55		reglogbas 36477	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
56	3, 5, 55	syl2anc 691	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
57	54, 44, 56	3brtr4d 4615	. . . . . 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 36473	. . . . . . 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 1319	. . . . . 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 246	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) < (PellFund‘𝐷))
61		pellfund14gap 36469	. . . . 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 1322	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) = 1)
63	31	negidd 10261	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) = 0)
64	63	oveq2d 6565	. . . . 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 11750	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℂ)
66	3	rpne0d 11753	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 0)
67		expaddz 12766	. . . . . 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 1319	. . . . 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 12864	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑0) = 1)
70	64, 68, 69	3eqtr3rd 2653	. . . 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 2644	. . 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 10542	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
73		oveq2 6557	. . . 4 ⊢ (𝑥 = (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) → ((PellFund‘𝐷)↑𝑥) = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
74	73	eqeq2d 2620	. . 3 ⊢ (𝑥 = (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) → (𝐴 = ((PellFund‘𝐷)↑𝑥) ↔ 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))))
75	74	rspcev 3282	. 2 ⊢ (((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ ∧ 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))
76	8, 72, 75	syl2anc 691	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145  -cneg 10146   / cdiv 10563  ℕcn 10897  ℤcz 11254  ℝ⁺crp 11708  ⌊cfl 12453   mod cmo 12530  ↑cexp 12722  logclog 24105  ◻_NNcsquarenn 36418  Pell1QRcpell1qr 36419  Pell14QRcpell14qr 36421  PellFundcpellfund 36422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-dvds 14822  df-gcd 15055  df-numer 15281  df-denom 15282  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-squarenn 36423  df-pell1qr 36424  df-pell14qr 36425  df-pell1234qr 36426  df-pellfund 36427

This theorem is referenced by:  pellfund14b  36481