Theorem pellfundglb 37449

Description: If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)

Assertion

Ref	Expression
pellfundglb	|- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> E. x e. (Pell1QR ` D ) ( (PellFund ` D ) <_ x /\ x < A ) )

Distinct variable groups:   x, D   x, A

Proof of Theorem pellfundglb

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pellfundval 37444	. . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) = inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) )
2	1	3ad2ant1 1082	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> (PellFund ` D ) = inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) )
3		simp3 1063	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> (PellFund ` D ) < A )
4	2, 3	eqbrtrrd 4677	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) < A )
5		pellfundre 37445	. . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> (PellFund ` D ) e. RR )
6	5	3ad2ant1 1082	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> (PellFund ` D ) e. RR )
7	2, 6	eqeltrrd 2702	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) e. RR )
8		simp2 1062	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> A e. RR )
9	7, 8	ltnled 10184	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> (inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) < A <-> -. A <_ inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) ) )
10	4, 9	mpbid 222	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> -. A <_ inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) )
11		ssrab2 3687	. . . . . 6 |- { a e. (Pell14QR ` D ) | 1 < a } C_ (Pell14QR ` D )
12		pell14qrre 37421	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ a e. (Pell14QR ` D ) ) -> a e. RR )
13	12	ex 450	. . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> ( a e. (Pell14QR ` D ) -> a e. RR ) )
14	13	ssrdv 3609	. . . . . . 7 |- ( D e. ( NN \ ◻NN ) -> (Pell14QR ` D ) C_ RR )
15	14	3ad2ant1 1082	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> (Pell14QR ` D ) C_ RR )
16	11, 15	syl5ss 3614	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> { a e. (Pell14QR ` D ) | 1 < a } C_ RR )
17		pell1qrss14 37432	. . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> (Pell1QR ` D ) C_ (Pell14QR ` D ) )
18	17	3ad2ant1 1082	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> (Pell1QR ` D ) C_ (Pell14QR ` D ) )
19		pellqrex 37443	. . . . . . . 8 |- ( D e. ( NN \ ◻NN ) -> E. a e. (Pell1QR ` D ) 1 < a )
20	19	3ad2ant1 1082	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> E. a e. (Pell1QR ` D ) 1 < a )
21		ssrexv 3667	. . . . . . 7 |- ( (Pell1QR ` D ) C_ (Pell14QR ` D ) -> ( E. a e. (Pell1QR ` D ) 1 < a -> E. a e. (Pell14QR ` D ) 1 < a ) )
22	18, 20, 21	sylc 65	. . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> E. a e. (Pell14QR ` D ) 1 < a )
23		rabn0 3958	. . . . . 6 |- ( { a e. (Pell14QR ` D ) | 1 < a } =/= (/) <-> E. a e. (Pell14QR ` D ) 1 < a )
24	22, 23	sylibr 224	. . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> { a e. (Pell14QR ` D ) | 1 < a } =/= (/) )
25		infmrgelbi 37442	. . . . . 6 |- ( ( ( { a e. (Pell14QR ` D ) | 1 < a } C_ RR /\ { a e. (Pell14QR ` D ) | 1 < a } =/= (/) /\ A e. RR ) /\ A. x e. { a e. (Pell14QR ` D ) | 1 < a } A <_ x ) -> A <_ inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) )
26	25	ex 450	. . . . 5 |- ( ( { a e. (Pell14QR ` D ) | 1 < a } C_ RR /\ { a e. (Pell14QR ` D ) | 1 < a } =/= (/) /\ A e. RR ) -> ( A. x e. { a e. (Pell14QR ` D ) | 1 < a } A <_ x -> A <_ inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) ) )
27	16, 24, 8, 26	syl3anc 1326	. . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> ( A. x e. { a e. (Pell14QR ` D ) | 1 < a } A <_ x -> A <_ inf ( { a e. (Pell14QR ` D ) | 1 < a } , RR , < ) ) )
28	10, 27	mtod 189	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> -. A. x e. { a e. (Pell14QR ` D ) | 1 < a } A <_ x )
29		rexnal 2995	. . 3 |- ( E. x e. { a e. (Pell14QR ` D ) | 1 < a } -. A <_ x <-> -. A. x e. { a e. (Pell14QR ` D ) | 1 < a } A <_ x )
30	28, 29	sylibr 224	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> E. x e. { a e. (Pell14QR ` D ) | 1 < a } -. A <_ x )
31		breq2 4657	. . . . . . . 8 |- ( a = x -> ( 1 < a <-> 1 < x ) )
32	31	elrab 3363	. . . . . . 7 |- ( x e. { a e. (Pell14QR ` D ) | 1 < a } <-> ( x e. (Pell14QR ` D ) /\ 1 < x ) )
33		simprl 794	. . . . . . . . 9 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> x e. (Pell14QR ` D ) )
34		1red 10055	. . . . . . . . . 10 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> 1 e. RR )
35		simpl1 1064	. . . . . . . . . . 11 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> D e. ( NN \ ◻NN ) )
36		pell14qrre 37421	. . . . . . . . . . 11 |- ( ( D e. ( NN \ ◻NN ) /\ x e. (Pell14QR ` D ) ) -> x e. RR )
37	35, 33, 36	syl2anc 693	. . . . . . . . . 10 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> x e. RR )
38		simprr 796	. . . . . . . . . 10 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> 1 < x )
39	34, 37, 38	ltled 10185	. . . . . . . . 9 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> 1 <_ x )
40	33, 39	jca 554	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> ( x e. (Pell14QR ` D ) /\ 1 <_ x ) )
41		elpell1qr2 37436	. . . . . . . . 9 |- ( D e. ( NN \ ◻NN ) -> ( x e. (Pell1QR ` D ) <-> ( x e. (Pell14QR ` D ) /\ 1 <_ x ) ) )
42	35, 41	syl 17	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> ( x e. (Pell1QR ` D ) <-> ( x e. (Pell14QR ` D ) /\ 1 <_ x ) ) )
43	40, 42	mpbird 247	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. (Pell14QR ` D ) /\ 1 < x ) ) -> x e. (Pell1QR ` D ) )
44	32, 43	sylan2b 492	. . . . . 6 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ x e. { a e. (Pell14QR ` D ) | 1 < a } ) -> x e. (Pell1QR ` D ) )
45	44	adantrr 753	. . . . 5 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> x e. (Pell1QR ` D ) )
46		simpl1 1064	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> D e. ( NN \ ◻NN ) )
47		simprl 794	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> x e. { a e. (Pell14QR ` D ) | 1 < a } )
48	11, 47	sseldi 3601	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> x e. (Pell14QR ` D ) )
49		simpr 477	. . . . . . . . . . 11 |- ( ( x e. (Pell14QR ` D ) /\ 1 < x ) -> 1 < x )
50	49	a1i 11	. . . . . . . . . 10 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> ( ( x e. (Pell14QR ` D ) /\ 1 < x ) -> 1 < x ) )
51	32, 50	syl5bi 232	. . . . . . . . 9 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> ( x e. { a e. (Pell14QR ` D ) | 1 < a } -> 1 < x ) )
52	51	imp 445	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ x e. { a e. (Pell14QR ` D ) | 1 < a } ) -> 1 < x )
53	52	adantrr 753	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> 1 < x )
54		pellfundlb 37448	. . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ x e. (Pell14QR ` D ) /\ 1 < x ) -> (PellFund ` D ) <_ x )
55	46, 48, 53, 54	syl3anc 1326	. . . . . 6 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> (PellFund ` D ) <_ x )
56		simprr 796	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> -. A <_ x )
57	15	adantr 481	. . . . . . . . 9 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> (Pell14QR ` D ) C_ RR )
58	57, 48	sseldd 3604	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> x e. RR )
59		simpl2 1065	. . . . . . . 8 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> A e. RR )
60	58, 59	ltnled 10184	. . . . . . 7 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> ( x < A <-> -. A <_ x ) )
61	56, 60	mpbird 247	. . . . . 6 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> x < A )
62	55, 61	jca 554	. . . . 5 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> ( (PellFund ` D ) <_ x /\ x < A ) )
63	45, 62	jca 554	. . . 4 |- ( ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) /\ ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) ) -> ( x e. (Pell1QR ` D ) /\ ( (PellFund ` D ) <_ x /\ x < A ) ) )
64	63	ex 450	. . 3 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> ( ( x e. { a e. (Pell14QR ` D ) | 1 < a } /\ -. A <_ x ) -> ( x e. (Pell1QR ` D ) /\ ( (PellFund ` D ) <_ x /\ x < A ) ) ) )
65	64	reximdv2 3014	. 2 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> ( E. x e. { a e. (Pell14QR ` D ) | 1 < a } -. A <_ x -> E. x e. (Pell1QR ` D ) ( (PellFund ` D ) <_ x /\ x < A ) ) )
66	30, 65	mpd 15	1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. RR /\ (PellFund ` D ) < A ) -> E. x e. (Pell1QR ` D ) ( (PellFund ` D ) <_ x /\ x < A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   class class class wbr 4653   ` cfv 5888  infcinf 8347   RRcr 9935   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020  ◻_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:  pellfundex  37450