Theorem pellfund14 40636

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 40598	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)
2		pellfundrp 40626	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
3	2	adantr 480	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℝ+)
4		pellfundne1 40627	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)
5	4	adantr 480	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 1)
6		reglogcl 40628	. . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ (PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
7	1, 3, 5, 6	syl3anc 1369	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
8	7	flcld 13446	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
9		pell14qrre 40595	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)
10	9	recnd 10934	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ)
11	3, 8	rpexpcld 13890	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
12	11	rpcnd 12703	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
13	8	znegcld 12357	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
14	3, 13	rpexpcld 13890	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
15	14	rpcnd 12703	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
16	14	rpne0d 12706	. . 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 40606	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19		pellfundex 40624	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
20	18, 19	sseldd 3918	. . . . . . . 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 40605	. . . . . . 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 1369	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ (Pell14QR‘𝐷))
24		pell14qrmulcl 40601	. . . . . 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 1460	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ (Pell14QR‘𝐷))
26		1rp 12663	. . . . . . . . . 10 ⊢ 1 ∈ ℝ+
27	26	a1i 11	. . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ∈ ℝ+)
28		modge0 13527	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
29	7, 27, 28	syl2anc 583	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
30	7	recnd 10934	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℂ)
31	8	zcnd 12356	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℂ)
32	30, 31	negsubd 11268	. . . . . . . . 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 13532	. . . . . . . . . 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 2781	. . . . . . . 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 5098	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
37		reglog1 40634	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
38	3, 5, 37	syl2anc 583	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
39		reglogmul 40631	. . . . . . . . 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 1372	. . . . . . . 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 40635	. . . . . . . . . 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 833	. . . . . . . . 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 7271	. . . . . . . 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 2778	. . . . . . 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 5102	. . . . . 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 12717	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ ℝ+)
47		pellfundgt1 40621	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))
48	47	adantr 480	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 < (PellFund‘𝐷))
49		reglogleb 40630	. . . . . . 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 835	. . . . . 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 256	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ≤ (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))))
52		modlt 13528	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
53	7, 27, 52	syl2anc 583	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
54	35, 53	eqbrtrd 5092	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) < 1)
55		reglogbas 40633	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
56	3, 5, 55	syl2anc 583	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
57	54, 44, 56	3brtr4d 5102	. . . . . 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 40629	. . . . . . 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 835	. . . . . 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 256	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) < (PellFund‘𝐷))
61		pellfund14gap 40625	. . . . 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 1372	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) = 1)
63	31	negidd 11252	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) = 0)
64	63	oveq2d 7271	. . . . 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 12703	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℂ)
66	3	rpne0d 12706	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 0)
67		expaddz 13755	. . . . . 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 835	. . . . 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 13786	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑0) = 1)
70	64, 68, 69	3eqtr3rd 2787	. . . 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 2778	. . 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 11541	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
73		oveq2 7263	. . 3 ⊢ (𝑥 = (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) → ((PellFund‘𝐷)↑𝑥) = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
74	73	rspceeqv 3567	. 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 583	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∖ cdif 3880   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807   < clt 10940   ≤ cle 10941   − cmin 11135  -cneg 11136   / cdiv 11562  ℕcn 11903  ℤcz 12249  ℝ⁺crp 12659  ⌊cfl 13438   mod cmo 13517  ↑cexp 13710  logclog 25615  ◻_NNcsquarenn 40574  Pell1QRcpell1qr 40575  Pell14QRcpell14qr 40577  PellFundcpellfund 40578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-addf 10881  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-acn 9631  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ioc 13013  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-mod 13518  df-seq 13650  df-exp 13711  df-fac 13916  df-bc 13945  df-hash 13973  df-shft 14706  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-limsup 15108  df-clim 15125  df-rlim 15126  df-sum 15326  df-ef 15705  df-sin 15707  df-cos 15708  df-pi 15710  df-dvds 15892  df-gcd 16130  df-numer 16367  df-denom 16368  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-fbas 20507  df-fg 20508  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-ntr 22079  df-cls 22080  df-nei 22157  df-lp 22195  df-perf 22196  df-cn 22286  df-cnp 22287  df-haus 22374  df-tx 22621  df-hmeo 22814  df-fil 22905  df-fm 22997  df-flim 22998  df-flf 22999  df-xms 23381  df-ms 23382  df-tms 23383  df-cncf 23947  df-limc 24935  df-dv 24936  df-log 25617  df-squarenn 40579  df-pell1qr 40580  df-pell14qr 40581  df-pell1234qr 40582  df-pellfund 40583

This theorem is referenced by:  pellfund14b  40637