Theorem pellfund14 37462

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 e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> E. x e. ZZ A = ( (PellFund ` D ) ^ x ) )

Distinct variable groups:   x, D   x, A

Proof of Theorem pellfund14

Step	Hyp	Ref	Expression
1		pell14qrrp 37424	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A e. RR+ )
2		pellfundrp 37452	. . . . 5 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. RR+ )
3	2	adantr 481	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> (PellFund ` D ) e. RR+ )
4		pellfundne1 37453	. . . . 5 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) =/= 1 )
5	4	adantr 481	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> (PellFund ` D ) =/= 1 )
6		reglogcl 37454	. . . 4 |- ( ( A e. RR+ /\ (PellFund ` D ) e. RR+ /\ (PellFund ` D ) =/= 1 ) -> ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) e. RR )
7	1, 3, 5, 6	syl3anc 1326	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) e. RR )
8	7	flcld 12599	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. ZZ )
9		pell14qrre 37421	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A e. RR )
10	9	recnd 10068	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A e. CC )
11	3, 8	rpexpcld 13032	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. RR+ )
12	11	rpcnd 11874	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. CC )
13	8	znegcld 11484	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. ZZ )
14	3, 13	rpexpcld 13032	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. RR+ )
15	14	rpcnd 11874	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. CC )
16	14	rpne0d 11877	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) =/= 0 )
17		simpl 473	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> D e. ( NN \ ◻NN ) )
18		pell1qrss14 37432	. . . . . . . . 9 |- ( D e. ( NN \ ◻NN ) -> (Pell1QR ` D ) C_ (Pell14QR ` D ) )
19		pellfundex 37450	. . . . . . . . 9 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. (Pell1QR ` D ) )
20	18, 19	sseldd 3604	. . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. (Pell14QR ` D ) )
21	20	adantr 481	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> (PellFund ` D ) e. (Pell14QR ` D ) )
22		pell14qrexpcl 37431	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ (PellFund ` D ) e. (Pell14QR ` D ) /\ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. ZZ ) -> ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. (Pell14QR ` D ) )
23	17, 21, 13, 22	syl3anc 1326	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. (Pell14QR ` D ) )
24		pell14qrmulcl 37427	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. (Pell14QR ` D ) ) -> ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) e. (Pell14QR ` D ) )
25	23, 24	mpd3an3 1425	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) e. (Pell14QR ` D ) )
26		1rp 11836	. . . . . . . . . 10 |- 1 e. RR+
27	26	a1i 11	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> 1 e. RR+ )
28		modge0 12678	. . . . . . . . 9 |- ( ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) e. RR /\ 1 e. RR+ ) -> 0 <_ ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) mod 1 ) )
29	7, 27, 28	syl2anc 693	. . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> 0 <_ ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) mod 1 ) )
30	7	recnd 10068	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) e. CC )
31	8	zcnd 11483	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. CC )
32	30, 31	negsubd 10398	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) - ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
33		modfrac 12683	. . . . . . . . . 10 |- ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) e. RR -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) mod 1 ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) - ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
34	7, 33	syl 17	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) mod 1 ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) - ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
35	32, 34	eqtr4d 2659	. . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) mod 1 ) )
36	29, 35	breqtrrd 4681	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> 0 <_ ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
37		reglog1 37460	. . . . . . . 8 |- ( ( (PellFund ` D ) e. RR+ /\ (PellFund ` D ) =/= 1 ) -> ( ( log ` 1 ) / ( log ` (PellFund ` D ) ) ) = 0 )
38	3, 5, 37	syl2anc 693	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` 1 ) / ( log ` (PellFund ` D ) ) ) = 0 )
39		reglogmul 37457	. . . . . . . . 9 |- ( ( A e. RR+ /\ ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) e. RR+ /\ ( (PellFund ` D ) e. RR+ /\ (PellFund ` D ) =/= 1 ) ) -> ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + ( ( log ` ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) ) )
40	1, 14, 3, 5, 39	syl112anc 1330	. . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + ( ( log ` ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) ) )
41		reglogexpbas 37461	. . . . . . . . . 10 |- ( ( -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. ZZ /\ ( (PellFund ` D ) e. RR+ /\ (PellFund ` D ) =/= 1 ) ) -> ( ( log ` ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) = -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) )
42	13, 3, 5, 41	syl12anc 1324	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) = -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) )
43	42	oveq2d 6666	. . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + ( ( log ` ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
44	40, 43	eqtrd 2656	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) = ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
45	36, 38, 44	3brtr4d 4685	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` 1 ) / ( log ` (PellFund ` D ) ) ) <_ ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) )
46	1, 14	rpmulcld 11888	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) e. RR+ )
47		pellfundgt1 37447	. . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> 1 < (PellFund ` D ) )
48	47	adantr 481	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> 1 < (PellFund ` D ) )
49		reglogleb 37456	. . . . . . 7 |- ( ( ( 1 e. RR+ /\ ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) e. RR+ ) /\ ( (PellFund ` D ) e. RR+ /\ 1 < (PellFund ` D ) ) ) -> ( 1 <_ ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) <-> ( ( log ` 1 ) / ( log ` (PellFund ` D ) ) ) <_ ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) ) )
50	27, 46, 3, 48, 49	syl22anc 1327	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( 1 <_ ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) <-> ( ( log ` 1 ) / ( log ` (PellFund ` D ) ) ) <_ ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) ) )
51	45, 50	mpbird 247	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> 1 <_ ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) )
52		modlt 12679	. . . . . . . . 9 |- ( ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) e. RR /\ 1 e. RR+ ) -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) mod 1 ) < 1 )
53	7, 27, 52	syl2anc 693	. . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) mod 1 ) < 1 )
54	35, 53	eqbrtrd 4675	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) < 1 )
55		reglogbas 37459	. . . . . . . 8 |- ( ( (PellFund ` D ) e. RR+ /\ (PellFund ` D ) =/= 1 ) -> ( ( log ` (PellFund ` D ) ) / ( log ` (PellFund ` D ) ) ) = 1 )
56	3, 5, 55	syl2anc 693	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` (PellFund ` D ) ) / ( log ` (PellFund ` D ) ) ) = 1 )
57	54, 44, 56	3brtr4d 4685	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) < ( ( log ` (PellFund ` D ) ) / ( log ` (PellFund ` D ) ) ) )
58		reglogltb 37455	. . . . . . 7 |- ( ( ( ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) e. RR+ /\ (PellFund ` D ) e. RR+ ) /\ ( (PellFund ` D ) e. RR+ /\ 1 < (PellFund ` D ) ) ) -> ( ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) < (PellFund ` D ) <-> ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) < ( ( log ` (PellFund ` D ) ) / ( log ` (PellFund ` D ) ) ) ) )
59	46, 3, 3, 48, 58	syl22anc 1327	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) < (PellFund ` D ) <-> ( ( log ` ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) ) / ( log ` (PellFund ` D ) ) ) < ( ( log ` (PellFund ` D ) ) / ( log ` (PellFund ` D ) ) ) ) )
60	57, 59	mpbird 247	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) < (PellFund ` D ) )
61		pellfund14gap 37451	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) e. (Pell14QR ` D ) /\ ( 1 <_ ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) /\ ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) < (PellFund ` D ) ) ) -> ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) = 1 )
62	17, 25, 51, 60, 61	syl112anc 1330	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) = 1 )
63	31	negidd 10382	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) = 0 )
64	63	oveq2d 6666	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ ( ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) = ( (PellFund ` D ) ^ 0 ) )
65	3	rpcnd 11874	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> (PellFund ` D ) e. CC )
66	3	rpne0d 11877	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> (PellFund ` D ) =/= 0 )
67		expaddz 12904	. . . . . 6 |- ( ( ( (PellFund ` D ) e. CC /\ (PellFund ` D ) =/= 0 ) /\ ( ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. ZZ /\ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. ZZ ) ) -> ( (PellFund ` D ) ^ ( ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) = ( ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) )
68	65, 66, 8, 13, 67	syl22anc 1327	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ ( ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) + -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) = ( ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) )
69	65	exp0d 13002	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( (PellFund ` D ) ^ 0 ) = 1 )
70	64, 68, 69	3eqtr3rd 2665	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> 1 = ( ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) )
71	62, 70	eqtrd 2656	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) = ( ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) x. ( (PellFund ` D ) ^ -u ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) )
72	10, 12, 15, 16, 71	mulcan2ad 10663	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A = ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
73		oveq2 6658	. . . 4 |- ( x = ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) -> ( (PellFund ` D ) ^ x ) = ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) )
74	73	eqeq2d 2632	. . 3 |- ( x = ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) -> ( A = ( (PellFund ` D ) ^ x ) <-> A = ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) )
75	74	rspcev 3309	. 2 |- ( ( ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) e. ZZ /\ A = ( (PellFund ` D ) ^ ( |_ ` ( ( log ` A ) / ( log ` (PellFund ` D ) ) ) ) ) ) -> E. x e. ZZ A = ( (PellFund ` D ) ^ x ) )
76	8, 72, 75	syl2anc 693	1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> E. x e. ZZ A = ( (PellFund ` D ) ^ x ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   ZZcz 11377   RR+crp 11832   |_cfl 12591   mod cmo 12668   ^cexp 12860   logclog 24301  ◻_NNcsquarenn 37400  Pell1QRcpell1qr 37401  Pell14QRcpell14qr 37403  PellFundcpellfund 37404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-squarenn 37405  df-pell1qr 37406  df-pell14qr 37407  df-pell1234qr 37408  df-pellfund 37409

This theorem is referenced by:  pellfund14b  37463