Theorem rmspecfund 37474

Description: The base of exponent used to define the X and Y sequences is the fundamental solution of the corresponding Pell equation. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Assertion

Ref	Expression
rmspecfund	|- ( A e. ( ZZ>= ` 2 ) -> (PellFund ` ( ( A ^ 2 ) - 1 ) ) = ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )

Proof of Theorem rmspecfund

Step	Hyp	Ref	Expression
1		rmspecnonsq 37472	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. ( NN \ ◻NN ) )
2		eluzelz 11697	. . . . . . . . . . . . 13 |- ( A e. ( ZZ>= ` 2 ) -> A e. ZZ )
3		zsqcl 12934	. . . . . . . . . . . . 13 |- ( A e. ZZ -> ( A ^ 2 ) e. ZZ )
4	2, 3	syl 17	. . . . . . . . . . . 12 |- ( A e. ( ZZ>= ` 2 ) -> ( A ^ 2 ) e. ZZ )
5	4	zred 11482	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) -> ( A ^ 2 ) e. RR )
6		1red 10055	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) -> 1 e. RR )
7	5, 6	resubcld 10458	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. RR )
8		sq1 12958	. . . . . . . . . . . . 13 |- ( 1 ^ 2 ) = 1
9	8	a1i 11	. . . . . . . . . . . 12 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 ^ 2 ) = 1 )
10		eluz2b2 11761	. . . . . . . . . . . . . 14 |- ( A e. ( ZZ>= ` 2 ) <-> ( A e. NN /\ 1 < A ) )
11	10	simprbi 480	. . . . . . . . . . . . 13 |- ( A e. ( ZZ>= ` 2 ) -> 1 < A )
12		eluzelre 11698	. . . . . . . . . . . . . 14 |- ( A e. ( ZZ>= ` 2 ) -> A e. RR )
13		0le1 10551	. . . . . . . . . . . . . . 15 |- 0 <_ 1
14	13	a1i 11	. . . . . . . . . . . . . 14 |- ( A e. ( ZZ>= ` 2 ) -> 0 <_ 1 )
15		eluzge2nn0 11727	. . . . . . . . . . . . . . 15 |- ( A e. ( ZZ>= ` 2 ) -> A e. NN0 )
16	15	nn0ge0d 11354	. . . . . . . . . . . . . 14 |- ( A e. ( ZZ>= ` 2 ) -> 0 <_ A )
17	6, 12, 14, 16	lt2sqd 13043	. . . . . . . . . . . . 13 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 < A <-> ( 1 ^ 2 ) < ( A ^ 2 ) ) )
18	11, 17	mpbid 222	. . . . . . . . . . . 12 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 ^ 2 ) < ( A ^ 2 ) )
19	9, 18	eqbrtrrd 4677	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) -> 1 < ( A ^ 2 ) )
20	6, 5	posdifd 10614	. . . . . . . . . . 11 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 < ( A ^ 2 ) <-> 0 < ( ( A ^ 2 ) - 1 ) ) )
21	19, 20	mpbid 222	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> 0 < ( ( A ^ 2 ) - 1 ) )
22	7, 21	elrpd 11869	. . . . . . . . 9 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. RR+ )
23	22	rpsqrtcld 14150	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. RR+ )
24	23	rpred 11872	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. RR )
25	24	recnd 10068	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( A ^ 2 ) - 1 ) ) e. CC )
26	25	mulid1d 10057	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. 1 ) = ( sqr ` ( ( A ^ 2 ) - 1 ) ) )
27	26	oveq2d 6666	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( A + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. 1 ) ) = ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
28		pell1qrss14 37432	. . . . . 6 |- ( ( ( A ^ 2 ) - 1 ) e. ( NN \ ◻NN ) -> (Pell1QR ` ( ( A ^ 2 ) - 1 ) ) C_ (Pell14QR ` ( ( A ^ 2 ) - 1 ) ) )
29	1, 28	syl 17	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> (Pell1QR ` ( ( A ^ 2 ) - 1 ) ) C_ (Pell14QR ` ( ( A ^ 2 ) - 1 ) ) )
30		1nn0 11308	. . . . . . 7 |- 1 e. NN0
31	30	a1i 11	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> 1 e. NN0 )
32	8	oveq2i 6661	. . . . . . . . 9 |- ( ( ( A ^ 2 ) - 1 ) x. ( 1 ^ 2 ) ) = ( ( ( A ^ 2 ) - 1 ) x. 1 )
33	7	recnd 10068	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - 1 ) e. CC )
34	33	mulid1d 10057	. . . . . . . . 9 |- ( A e. ( ZZ>= ` 2 ) -> ( ( ( A ^ 2 ) - 1 ) x. 1 ) = ( ( A ^ 2 ) - 1 ) )
35	32, 34	syl5eq 2668	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( ( ( A ^ 2 ) - 1 ) x. ( 1 ^ 2 ) ) = ( ( A ^ 2 ) - 1 ) )
36	35	oveq2d 6666	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( 1 ^ 2 ) ) ) = ( ( A ^ 2 ) - ( ( A ^ 2 ) - 1 ) ) )
37	5	recnd 10068	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> ( A ^ 2 ) e. CC )
38		1cnd 10056	. . . . . . . 8 |- ( A e. ( ZZ>= ` 2 ) -> 1 e. CC )
39	37, 38	nncand 10397	. . . . . . 7 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - ( ( A ^ 2 ) - 1 ) ) = 1 )
40	36, 39	eqtrd 2656	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> ( ( A ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( 1 ^ 2 ) ) ) = 1 )
41		pellqrexplicit 37441	. . . . . 6 |- ( ( ( ( ( A ^ 2 ) - 1 ) e. ( NN \ ◻NN ) /\ A e. NN0 /\ 1 e. NN0 ) /\ ( ( A ^ 2 ) - ( ( ( A ^ 2 ) - 1 ) x. ( 1 ^ 2 ) ) ) = 1 ) -> ( A + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. 1 ) ) e. (Pell1QR ` ( ( A ^ 2 ) - 1 ) ) )
42	1, 15, 31, 40, 41	syl31anc 1329	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( A + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. 1 ) ) e. (Pell1QR ` ( ( A ^ 2 ) - 1 ) ) )
43	29, 42	sseldd 3604	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( A + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. 1 ) ) e. (Pell14QR ` ( ( A ^ 2 ) - 1 ) ) )
44	27, 43	eqeltrrd 2702	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) e. (Pell14QR ` ( ( A ^ 2 ) - 1 ) ) )
45	6, 24	readdcld 10069	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) e. RR )
46	12, 24	readdcld 10069	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) e. RR )
47	6, 23	ltaddrpd 11905	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> 1 < ( 1 + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
48	6, 12, 24, 11	ltadd1dd 10638	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( 1 + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) < ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
49	6, 45, 46, 47, 48	lttrd 10198	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> 1 < ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
50		pellfundlb 37448	. . 3 |- ( ( ( ( A ^ 2 ) - 1 ) e. ( NN \ ◻NN ) /\ ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) e. (Pell14QR ` ( ( A ^ 2 ) - 1 ) ) /\ 1 < ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ) -> (PellFund ` ( ( A ^ 2 ) - 1 ) ) <_ ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
51	1, 44, 49, 50	syl3anc 1326	. 2 |- ( A e. ( ZZ>= ` 2 ) -> (PellFund ` ( ( A ^ 2 ) - 1 ) ) <_ ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
52	37, 38	npcand 10396	. . . . . 6 |- ( A e. ( ZZ>= ` 2 ) -> ( ( ( A ^ 2 ) - 1 ) + 1 ) = ( A ^ 2 ) )
53	52	fveq2d 6195	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( ( A ^ 2 ) - 1 ) + 1 ) ) = ( sqr ` ( A ^ 2 ) ) )
54	12, 16	sqrtsqd 14158	. . . . 5 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( A ^ 2 ) ) = A )
55	53, 54	eqtrd 2656	. . . 4 |- ( A e. ( ZZ>= ` 2 ) -> ( sqr ` ( ( ( A ^ 2 ) - 1 ) + 1 ) ) = A )
56	55	oveq1d 6665	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( sqr ` ( ( ( A ^ 2 ) - 1 ) + 1 ) ) + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) = ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
57		pellfundge 37446	. . . 4 |- ( ( ( A ^ 2 ) - 1 ) e. ( NN \ ◻NN ) -> ( ( sqr ` ( ( ( A ^ 2 ) - 1 ) + 1 ) ) + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) <_ (PellFund ` ( ( A ^ 2 ) - 1 ) ) )
58	1, 57	syl 17	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> ( ( sqr ` ( ( ( A ^ 2 ) - 1 ) + 1 ) ) + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) <_ (PellFund ` ( ( A ^ 2 ) - 1 ) ) )
59	56, 58	eqbrtrrd 4677	. 2 |- ( A e. ( ZZ>= ` 2 ) -> ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) <_ (PellFund ` ( ( A ^ 2 ) - 1 ) ) )
60		pellfundre 37445	. . . 4 |- ( ( ( A ^ 2 ) - 1 ) e. ( NN \ ◻NN ) -> (PellFund ` ( ( A ^ 2 ) - 1 ) ) e. RR )
61	1, 60	syl 17	. . 3 |- ( A e. ( ZZ>= ` 2 ) -> (PellFund ` ( ( A ^ 2 ) - 1 ) ) e. RR )
62	61, 46	letri3d 10179	. 2 |- ( A e. ( ZZ>= ` 2 ) -> ( (PellFund ` ( ( A ^ 2 ) - 1 ) ) = ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) <-> ( (PellFund ` ( ( A ^ 2 ) - 1 ) ) <_ ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) /\ ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) <_ (PellFund ` ( ( A ^ 2 ) - 1 ) ) ) ) )
63	51, 59, 62	mpbir2and 957	1 |- ( A e. ( ZZ>= ` 2 ) -> (PellFund ` ( ( A ^ 2 ) - 1 ) ) = ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ 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   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ^cexp 12860   sqrcsqrt 13973  ◻_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:  rmxyelqirr  37475  rmxycomplete  37482  rmbaserp  37484