Theorem pellfund14 43413

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 43375	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)
2		pellfundrp 43403	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)
3	2	adantr 483	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℝ+)
4		pellfundne1 43404	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)
5	4	adantr 483	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 1)
6		reglogcl 43405	. . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ (PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
7	1, 3, 5, 6	syl3anc 1382	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ)
8	7	flcld 13794	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
9		pell14qrre 43372	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)
10	9	recnd 11196	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℂ)
11	3, 8	rpexpcld 14246	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
12	11	rpcnd 13025	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
13	8	znegcld 12665	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℤ)
14	3, 13	rpexpcld 14246	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℝ+)
15	14	rpcnd 13025	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ ℂ)
16	14	rpne0d 13028	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ≠ 0)
17		simpl 485	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐷 ∈ (ℕ ∖ ◻NN))
18		pell1qrss14 43383	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19		pellfundex 43401	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))
20	18, 19	sseldd 3928	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷))
21	20	adantr 483	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ (Pell14QR‘𝐷))
22		pell14qrexpcl 43382	. . . . . . 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 1382	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) ∈ (Pell14QR‘𝐷))
24		pell14qrmulcl 43378	. . . . . 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 1473	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ (Pell14QR‘𝐷))
26		1rp 12983	. . . . . . . . . 10 ⊢ 1 ∈ ℝ+
27	26	a1i 11	. . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ∈ ℝ+)
28		modge0 13875	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
29	7, 27, 28	syl2anc 592	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1))
30	7	recnd 11196	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℂ)
31	8	zcnd 12664	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) ∈ ℂ)
32	30, 31	negsubd 11534	. . . . . . . . 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 13880	. . . . . . . . . 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 2790	. . . . . . . 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 5118	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 ≤ (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
37		reglog1 43411	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
38	3, 5, 37	syl2anc 592	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘1) / (log‘(PellFund‘𝐷))) = 0)
39		reglogmul 43408	. . . . . . . . 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 1385	. . . . . . . 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 43412	. . . . . . . . . 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 845	. . . . . . . . 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 7397	. . . . . . . 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 2787	. . . . . . 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 5122	. . . . . 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 13039	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) ∈ ℝ+)
47		pellfundgt1 43398	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))
48	47	adantr 483	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 < (PellFund‘𝐷))
49		reglogleb 43407	. . . . . . 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 847	. . . . . 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 259	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 1 ≤ (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))))
52		modlt 13876	. . . . . . . . 9 ⊢ ((((log‘𝐴) / (log‘(PellFund‘𝐷))) ∈ ℝ ∧ 1 ∈ ℝ+) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
53	7, 27, 52	syl2anc 592	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) mod 1) < 1)
54	35, 53	eqbrtrd 5112	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (((log‘𝐴) / (log‘(PellFund‘𝐷))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) < 1)
55		reglogbas 43410	. . . . . . . 8 ⊢ (((PellFund‘𝐷) ∈ ℝ+ ∧ (PellFund‘𝐷) ≠ 1) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
56	3, 5, 55	syl2anc 592	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((log‘(PellFund‘𝐷)) / (log‘(PellFund‘𝐷))) = 1)
57	54, 44, 56	3brtr4d 5122	. . . . . 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 43406	. . . . . . 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 847	. . . . . 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 259	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) < (PellFund‘𝐷))
61		pellfund14gap 43402	. . . . 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 1385	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 · ((PellFund‘𝐷)↑-(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))))) = 1)
63	31	negidd 11518	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) + -(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))) = 0)
64	63	oveq2d 7397	. . . . 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 13025	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ∈ ℂ)
66	3	rpne0d 13028	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (PellFund‘𝐷) ≠ 0)
67		expaddz 14105	. . . . . 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 847	. . . . 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 14139	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ((PellFund‘𝐷)↑0) = 1)
70	64, 68, 69	3eqtr3rd 2796	. . . 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 2787	. . 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 11809	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
73		oveq2 7389	. . 3 ⊢ (𝑥 = (⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷)))) → ((PellFund‘𝐷)↑𝑥) = ((PellFund‘𝐷)↑(⌊‘((log‘𝐴) / (log‘(PellFund‘𝐷))))))
74	73	rspceeqv 3595	. 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 592	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ∃wrex 3076   ∖ cdif 3892   class class class wbr 5090  ‘cfv 6506  (class class class)co 7381  ℂcc 11057  ℝcr 11058  0cc0 11059  1c1 11060   + caddc 11062   · cmul 11064   < clt 11202   ≤ cle 11203   − cmin 11400  -cneg 11401   / cdiv 11830  ℕcn 12196  ℤcz 12554  ℝ⁺crp 12979  ⌊cfl 13786   mod cmo 13865  ↑cexp 14060  logclog 26585  ◻_NNcsquarenn 43351  Pell1QRcpell1qr 43352  Pell14QRcpell14qr 43354  PellFundcpellfund 43355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-inf2 9582  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-addf 11138

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-uni 4856  df-int 4896  df-iun 4941  df-iin 4942  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-isom 6515  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-of 7645  df-om 7832  df-1st 7955  df-2nd 7956  df-supp 8125  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-oadd 8425  df-omul 8426  df-er 8662  df-map 8794  df-pm 8795  df-ixp 8865  df-en 8913  df-dom 8914  df-sdom 8915  df-fin 8916  df-fsupp 9294  df-fi 9343  df-sup 9374  df-inf 9375  df-oi 9444  df-card 9883  df-acn 9886  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-div 11831  df-nn 12197  df-2 12266  df-3 12267  df-4 12268  df-5 12269  df-6 12270  df-7 12271  df-8 12272  df-9 12273  df-n0 12468  df-xnn0 12541  df-z 12555  df-dec 12675  df-uz 12826  df-q 12936  df-rp 12980  df-xneg 13100  df-xadd 13101  df-xmul 13102  df-ioo 13339  df-ioc 13340  df-ico 13341  df-icc 13342  df-fz 13499  df-fzo 13646  df-fl 13788  df-mod 13866  df-seq 14001  df-exp 14061  df-fac 14273  df-bc 14302  df-hash 14330  df-shft 15066  df-cj 15098  df-re 15099  df-im 15100  df-sqrt 15234  df-abs 15235  df-limsup 15470  df-clim 15487  df-rlim 15488  df-sum 15686  df-ef 16069  df-sin 16071  df-cos 16072  df-pi 16074  df-dvds 16259  df-gcd 16501  df-numer 16742  df-denom 16743  df-struct 17155  df-sets 17172  df-slot 17190  df-ndx 17202  df-base 17218  df-ress 17239  df-plusg 17271  df-mulr 17272  df-starv 17273  df-sca 17274  df-vsca 17275  df-ip 17276  df-tset 17277  df-ple 17278  df-ds 17280  df-unif 17281  df-hom 17282  df-cco 17283  df-rest 17423  df-topn 17424  df-0g 17442  df-gsum 17443  df-topgen 17444  df-pt 17445  df-prds 17448  df-xrs 17504  df-qtop 17509  df-imas 17510  df-xps 17512  df-mre 17586  df-mrc 17587  df-acs 17589  df-mgm 18646  df-sgrp 18725  df-mnd 18741  df-submnd 18790  df-mulg 19082  df-cntz 19329  df-cmn 19794  df-psmet 21385  df-xmet 21386  df-met 21387  df-bl 21388  df-mopn 21389  df-fbas 21390  df-fg 21391  df-cnfld 21394  df-top 22923  df-topon 22940  df-topsp 22962  df-bases 22975  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-lp 23165  df-perf 23166  df-cn 23256  df-cnp 23257  df-haus 23344  df-tx 23591  df-hmeo 23784  df-fil 23875  df-fm 23967  df-flim 23968  df-flf 23969  df-xms 24349  df-ms 24350  df-tms 24351  df-cncf 24909  df-limc 25897  df-dv 25898  df-log 26587  df-squarenn 43356  df-pell1qr 43357  df-pell14qr 43358  df-pell1234qr 43359  df-pellfund 43360

This theorem is referenced by:  pellfund14b  43414