Theorem pellfundex 37450

Description: The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete.

Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 37440. (Contributed by Stefan O'Rear, 18-Sep-2014.)

Assertion

Ref	Expression
pellfundex	|- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. (Pell1QR ` D ) )

Proof of Theorem pellfundex

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2re 11090	. . . 4 |- 2 e. RR
2		pellfundre 37445	. . . 4 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. RR )
3		remulcl 10021	. . . 4 |- ( ( 2 e. RR /\ (PellFund ` D ) e. RR ) -> ( 2 x. (PellFund ` D ) ) e. RR )
4	1, 2, 3	sylancr 695	. . 3 |- ( D e. ( NN \ ◻NN ) -> ( 2 x. (PellFund ` D ) ) e. RR )
5		0red 10041	. . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> 0 e. RR )
6		1red 10055	. . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> 1 e. RR )
7		0lt1 10550	. . . . . . . 8 |- 0 < 1
8	7	a1i 11	. . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> 0 < 1 )
9		pellfundgt1 37447	. . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> 1 < (PellFund ` D ) )
10	5, 6, 2, 8, 9	lttrd 10198	. . . . . 6 |- ( D e. ( NN \ ◻NN ) -> 0 < (PellFund ` D ) )
11	2, 10	elrpd 11869	. . . . 5 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. RR+ )
12	2, 11	ltaddrpd 11905	. . . 4 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) < ( (PellFund ` D ) + (PellFund ` D ) ) )
13	2	recnd 10068	. . . . 5 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. CC )
14	13	2timesd 11275	. . . 4 |- ( D e. ( NN \ ◻NN ) -> ( 2 x. (PellFund ` D ) ) = ( (PellFund ` D ) + (PellFund ` D ) ) )
15	12, 14	breqtrrd 4681	. . 3 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) < ( 2 x. (PellFund ` D ) ) )
16		pellfundglb 37449	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ ( 2 x. (PellFund ` D ) ) e. RR /\ (PellFund ` D ) < ( 2 x. (PellFund ` D ) ) ) -> E. a e. (Pell1QR ` D ) ( (PellFund ` D ) <_ a /\ a < ( 2 x. (PellFund ` D ) ) ) )
17	4, 15, 16	mpd3an23 1426	. 2 |- ( D e. ( NN \ ◻NN ) -> E. a e. (Pell1QR ` D ) ( (PellFund ` D ) <_ a /\ a < ( 2 x. (PellFund ` D ) ) ) )
18	2	adantr 481	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> (PellFund ` D ) e. RR )
19		pell1qrss14 37432	. . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> (Pell1QR ` D ) C_ (Pell14QR ` D ) )
20	19	sselda 3603	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> a e. (Pell14QR ` D ) )
21		pell14qrre 37421	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell14QR ` D ) ) -> a e. RR )
22	20, 21	syldan 487	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> a e. RR )
23	18, 22	leloed 10180	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> ( (PellFund ` D ) <_ a <-> ( (PellFund ` D ) < a \/ (PellFund ` D ) = a ) ) )
24		simp-4l 806	. . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> D e. ( NN \ ◻NN ) )
25		simp-4r 807	. . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> a e. (Pell1QR ` D ) )
26		simplr 792	. . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> b e. (Pell1QR ` D ) )
27		simprr 796	. . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> b < a )
28	22	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> a e. RR )
29	4	ad4antr 768	. . . . . . . . . 10 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> ( 2 x. (PellFund ` D ) ) e. RR )
30	19	ad4antr 768	. . . . . . . . . . . . 13 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> (Pell1QR ` D ) C_ (Pell14QR ` D ) )
31	30, 26	sseldd 3604	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> b e. (Pell14QR ` D ) )
32		pell14qrre 37421	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ b e. (Pell14QR ` D ) ) -> b e. RR )
33	24, 31, 32	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> b e. RR )
34		remulcl 10021	. . . . . . . . . . 11 |- ( ( 2 e. RR /\ b e. RR ) -> ( 2 x. b ) e. RR )
35	1, 33, 34	sylancr 695	. . . . . . . . . 10 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> ( 2 x. b ) e. RR )
36		simprr 796	. . . . . . . . . . 11 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) -> a < ( 2 x. (PellFund ` D ) ) )
37	36	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> a < ( 2 x. (PellFund ` D ) ) )
38		simprl 794	. . . . . . . . . . 11 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> (PellFund ` D ) <_ b )
39	2	ad4antr 768	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> (PellFund ` D ) e. RR )
40	1	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> 2 e. RR )
41		2pos 11112	. . . . . . . . . . . . 13 |- 0 < 2
42	41	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> 0 < 2 )
43		lemul2 10876	. . . . . . . . . . . 12 |- ( ( (PellFund ` D ) e. RR /\ b e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( (PellFund ` D ) <_ b <-> ( 2 x. (PellFund ` D ) ) <_ ( 2 x. b ) ) )
44	39, 33, 40, 42, 43	syl112anc 1330	. . . . . . . . . . 11 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> ( (PellFund ` D ) <_ b <-> ( 2 x. (PellFund ` D ) ) <_ ( 2 x. b ) ) )
45	38, 44	mpbid 222	. . . . . . . . . 10 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> ( 2 x. (PellFund ` D ) ) <_ ( 2 x. b ) )
46	28, 29, 35, 37, 45	ltletrd 10197	. . . . . . . . 9 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> a < ( 2 x. b ) )
47		simp1 1061	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> D e. ( NN \ ◻NN ) )
48	19	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> (Pell1QR ` D ) C_ (Pell14QR ` D ) )
49		simp2l 1087	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> a e. (Pell1QR ` D ) )
50	48, 49	sseldd 3604	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> a e. (Pell14QR ` D ) )
51		simp2r 1088	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> b e. (Pell1QR ` D ) )
52	48, 51	sseldd 3604	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> b e. (Pell14QR ` D ) )
53		pell14qrdivcl 37429	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell14QR ` D ) /\ b e. (Pell14QR ` D ) ) -> ( a / b ) e. (Pell14QR ` D ) )
54	47, 50, 52, 53	syl3anc 1326	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> ( a / b ) e. (Pell14QR ` D ) )
55	47, 52, 32	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> b e. RR )
56	55	recnd 10068	. . . . . . . . . . . . 13 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> b e. CC )
57	56	mulid2d 10058	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> ( 1 x. b ) = b )
58		simp3l 1089	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> b < a )
59	57, 58	eqbrtrd 4675	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> ( 1 x. b ) < a )
60		1red 10055	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> 1 e. RR )
61	47, 50, 21	syl2anc 693	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> a e. RR )
62		pell14qrgt0 37423	. . . . . . . . . . . . 13 |- ( ( D e. ( NN \ ◻NN ) /\ b e. (Pell14QR ` D ) ) -> 0 < b )
63	47, 52, 62	syl2anc 693	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> 0 < b )
64		ltmuldiv 10896	. . . . . . . . . . . 12 |- ( ( 1 e. RR /\ a e. RR /\ ( b e. RR /\ 0 < b ) ) -> ( ( 1 x. b ) < a <-> 1 < ( a / b ) ) )
65	60, 61, 55, 63, 64	syl112anc 1330	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> ( ( 1 x. b ) < a <-> 1 < ( a / b ) ) )
66	59, 65	mpbid 222	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> 1 < ( a / b ) )
67		simp3r 1090	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> a < ( 2 x. b ) )
68	1	a1i 11	. . . . . . . . . . . 12 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> 2 e. RR )
69		ltdivmul2 10900	. . . . . . . . . . . 12 |- ( ( a e. RR /\ 2 e. RR /\ ( b e. RR /\ 0 < b ) ) -> ( ( a / b ) < 2 <-> a < ( 2 x. b ) ) )
70	61, 68, 55, 63, 69	syl112anc 1330	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> ( ( a / b ) < 2 <-> a < ( 2 x. b ) ) )
71	67, 70	mpbird 247	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> ( a / b ) < 2 )
72		simprr 796	. . . . . . . . . . 11 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> ( a / b ) < 2 )
73		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> D e. ( NN \ ◻NN ) )
74		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> ( a / b ) e. (Pell14QR ` D ) )
75		simprl 794	. . . . . . . . . . . . 13 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> 1 < ( a / b ) )
76		pell14qrgapw 37440	. . . . . . . . . . . . 13 |- ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) /\ 1 < ( a / b ) ) -> 2 < ( a / b ) )
77	73, 74, 75, 76	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> 2 < ( a / b ) )
78		pell14qrre 37421	. . . . . . . . . . . . . 14 |- ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) -> ( a / b ) e. RR )
79	78	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> ( a / b ) e. RR )
80		ltnsym 10135	. . . . . . . . . . . . 13 |- ( ( 2 e. RR /\ ( a / b ) e. RR ) -> ( 2 < ( a / b ) -> -. ( a / b ) < 2 ) )
81	1, 79, 80	sylancr 695	. . . . . . . . . . . 12 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> ( 2 < ( a / b ) -> -. ( a / b ) < 2 ) )
82	77, 81	mpd 15	. . . . . . . . . . 11 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> -. ( a / b ) < 2 )
83	72, 82	pm2.21dd 186	. . . . . . . . . 10 |- ( ( ( D e. ( NN \ ◻NN ) /\ ( a / b ) e. (Pell14QR ` D ) ) /\ ( 1 < ( a / b ) /\ ( a / b ) < 2 ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) )
84	47, 54, 66, 71, 83	syl22anc 1327	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ ( a e. (Pell1QR ` D ) /\ b e. (Pell1QR ` D ) ) /\ ( b < a /\ a < ( 2 x. b ) ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) )
85	24, 25, 26, 27, 46, 84	syl122anc 1335	. . . . . . . 8 |- ( ( ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) /\ b e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) <_ b /\ b < a ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) )
86		simpll 790	. . . . . . . . 9 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) -> D e. ( NN \ ◻NN ) )
87	22	adantr 481	. . . . . . . . 9 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) -> a e. RR )
88		simprl 794	. . . . . . . . 9 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) -> (PellFund ` D ) < a )
89		pellfundglb 37449	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ a e. RR /\ (PellFund ` D ) < a ) -> E. b e. (Pell1QR ` D ) ( (PellFund ` D ) <_ b /\ b < a ) )
90	86, 87, 88, 89	syl3anc 1326	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) -> E. b e. (Pell1QR ` D ) ( (PellFund ` D ) <_ b /\ b < a ) )
91	85, 90	r19.29a 3078	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ ( (PellFund ` D ) < a /\ a < ( 2 x. (PellFund ` D ) ) ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) )
92	91	exp32 631	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> ( (PellFund ` D ) < a -> ( a < ( 2 x. (PellFund ` D ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) ) ) )
93		simp2 1062	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ (PellFund ` D ) = a /\ a < ( 2 x. (PellFund ` D ) ) ) -> (PellFund ` D ) = a )
94		simp1r 1086	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ (PellFund ` D ) = a /\ a < ( 2 x. (PellFund ` D ) ) ) -> a e. (Pell1QR ` D ) )
95	93, 94	eqeltrd 2701	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) /\ (PellFund ` D ) = a /\ a < ( 2 x. (PellFund ` D ) ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) )
96	95	3exp 1264	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> ( (PellFund ` D ) = a -> ( a < ( 2 x. (PellFund ` D ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) ) ) )
97	92, 96	jaod 395	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> ( ( (PellFund ` D ) < a \/ (PellFund ` D ) = a ) -> ( a < ( 2 x. (PellFund ` D ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) ) ) )
98	23, 97	sylbid 230	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> ( (PellFund ` D ) <_ a -> ( a < ( 2 x. (PellFund ` D ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) ) ) )
99	98	impd 447	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell1QR ` D ) ) -> ( ( (PellFund ` D ) <_ a /\ a < ( 2 x. (PellFund ` D ) ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) ) )
100	99	rexlimdva 3031	. 2 |- ( D e. ( NN \ ◻NN ) -> ( E. a e. (Pell1QR ` D ) ( (PellFund ` D ) <_ a /\ a < ( 2 x. (PellFund ` D ) ) ) -> (PellFund ` D ) e. (Pell1QR ` D ) ) )
101	17, 100	mpd 15	1 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. (Pell1QR ` D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070  ◻_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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ico 12181  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444  df-squarenn 37405  df-pell1qr 37406  df-pell14qr 37407  df-pell1234qr 37408  df-pellfund 37409

This theorem is referenced by:  pellfund14  37462  pellfund14b  37463