Theorem pellfund14 43904

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 43866	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)
2		pellfundrp 43894	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
3	2	adantr 486	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℝ+)
4		pellfundne1 43895	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)
5	4	adantr 486	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 1)
6		reglogcl 43896	. . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ (PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
7	1, 3, 5, 6	syl3anc 1398	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
8	7	flcld 13938	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
9		pell14qrre 43863	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)
10	9	recnd 11337	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ)
11	3, 8	rpexpcld 14391	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
12	11	rpcnd 13166	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
13	8	znegcld 12805	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
14	3, 13	rpexpcld 14391	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
15	14	rpcnd 13166	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
16	14	rpne0d 13169	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ≠ 0)
17		simpl 488	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐷 ∈ (ℕ ∖ ◻NN))
18		pell1qrss14 43874	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19		pellfundex 43892	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
20	18, 19	sseldd 3932	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷))
21	20	adantr 486	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷))
22		pell14qrexpcl 43873	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ (Pell14QR‘𝐷))
24		pell14qrmulcl 43869	. . . . . 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 1491	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ (Pell14QR‘𝐷))
26		1rp 13124	. . . . . . . . . 10 ⊢ 1 ∈ ℝ+
27	26	a1i 11	. . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ∈ ℝ+)
28		modge0 14019	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
29	7, 27, 28	syl2anc 596	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
30	7	recnd 11337	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℂ)
31	8	zcnd 12804	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℂ)
32	30, 31	negsubd 11675	. . . . . . . . 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 14024	. . . . . . . . . 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 18	. . . . . . . . 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 2799	. . . . . . . 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 5133	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
37		reglog1 43902	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
38	3, 5, 37	syl2anc 596	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
39		reglogmul 43899	. . . . . . . . 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 1401	. . . . . . . 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 43903	. . . . . . . . . 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 850	. . . . . . . . 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 7436	. . . . . . . 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 2796	. . . . . . 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 5137	. . . . . 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 13180	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ ℝ+)
47		pellfundgt1 43889	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))
48	47	adantr 486	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 < (PellFund‘𝐷))
49		reglogleb 43898	. . . . . . 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 852	. . . . . 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 14020	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
53	7, 27, 52	syl2anc 596	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
54	35, 53	eqbrtrd 5127	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) < 1)
55		reglogbas 43901	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
56	3, 5, 55	syl2anc 596	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
57	54, 44, 56	3brtr4d 5137	. . . . . 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 43897	. . . . . . 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 852	. . . . . 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 43893	. . . . 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 1401	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) = 1)
63	31	negidd 11659	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) = 0)
64	63	oveq2d 7436	. . . . 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 13166	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℂ)
66	3	rpne0d 13169	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 0)
67		expaddz 14249	. . . . . 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 852	. . . . 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 14283	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑0) = 1)
70	64, 68, 69	3eqtr3rd 2805	. . . 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 2796	. . 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 11952	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
73		oveq2 7428	. . 3 ⊢ (𝑥 = (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) → ((PellFund‘𝐷)↑𝑥) = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
74	73	rspceeqv 3599	. 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 596	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   ∖ cdif 3896   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  ℂcc 11198  ℝcr 11199  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205   < clt 11343   ≤ cle 11344   − cmin 11541  -cneg 11542   / cdiv 11973  ℕcn 12335  ℤcz 12693  ℝ⁺crp 13120  ⌊cfl 13930   mod cmo 14009  ↑cexp 14204  logclog 26882  ◻_NNcsquarenn 43842  Pell1QRcpell1qr 43843  Pell14QRcpell14qr 43845  PellFundcpellfund 43846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278  ax-addf 11279

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-supp 8178  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-omul 8481  df-er 8717  df-map 8849  df-pm 8850  df-ixp 8926  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-fsupp 9354  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9504  df-card 10020  df-acn 10023  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-xnn0 12680  df-z 12694  df-dec 12815  df-uz 12966  df-q 13076  df-rp 13121  df-xneg 13241  df-xadd 13242  df-xmul 13243  df-ioo 13480  df-ioc 13481  df-ico 13482  df-icc 13483  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-fac 14418  df-bc 14447  df-hash 14475  df-shft 15220  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-limsup 15638  df-clim 15655  df-rlim 15656  df-sum 15854  df-ef 16233  df-sin 16235  df-cos 16236  df-pi 16238  df-dvds 16423  df-gcd 16665  df-numer 16911  df-denom 16912  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-starv 17443  df-sca 17444  df-vsca 17445  df-ip 17446  df-tset 17447  df-ple 17448  df-ds 17450  df-unif 17451  df-hom 17452  df-cco 17453  df-rest 17593  df-topn 17594  df-0g 17612  df-gsum 17613  df-topgen 17614  df-pt 17615  df-prds 17618  df-xrs 17674  df-qtop 17679  df-imas 17680  df-xps 17682  df-mre 17756  df-mrc 17757  df-acs 17759  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-mulg 19278  df-cntz 19531  df-cmn 19996  df-psmet 21670  df-xmet 21671  df-met 21672  df-bl 21673  df-mopn 21674  df-fbas 21675  df-fg 21676  df-cnfld 21679  df-top 23212  df-topon 23229  df-topsp 23251  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-lp 23454  df-perf 23455  df-cn 23545  df-cnp 23546  df-haus 23633  df-tx 23881  df-hmeo 24074  df-fil 24165  df-fm 24257  df-flim 24258  df-flf 24259  df-xms 24639  df-ms 24640  df-tms 24641  df-cncf 25199  df-limc 26186  df-dv 26187  df-log 26884  df-squarenn 43847  df-pell1qr 43848  df-pell14qr 43849  df-pell1234qr 43850  df-pellfund 43851

This theorem is used by:  pellfund14b  43905