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Theorem rmspecfund 35757
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
StepHypRef Expression
1 rmspecnonsq 35755 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  ( NN  \NN ) )
2 eluzelz 11168 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
3 zsqcl 12345 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
42, 3syl 17 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  ZZ )
54zred 11040 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
6 1red 9658 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
75, 6resubcld 10047 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR )
8 sq1 12369 . . . . . . . . . . . . 13  |-  ( 1 ^ 2 )  =  1
98a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
10 eluz2b2 11231 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
1110simprbi 466 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
12 eluzelre 11169 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
13 0le1 10137 . . . . . . . . . . . . . . 15  |-  0  <_  1
1413a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  1 )
15 eluzge2nn0 11198 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN0 )
1615nn0ge0d 10928 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  A )
176, 12, 14, 16lt2sqd 12450 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 1 ^ 2 )  < 
( A ^ 2 ) ) )
1811, 17mpbid 214 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  < 
( A ^ 2 ) )
199, 18eqbrtrrd 4425 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A ^ 2 ) )
206, 5posdifd 10200 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  ( A ^
2 )  <->  0  <  ( ( A ^ 2 )  -  1 ) ) )
2119, 20mpbid 214 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  ( ( A ^ 2 )  -  1 ) )
227, 21elrpd 11338 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR+ )
2322rpsqrtcld 13473 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR+ )
2423rpred 11341 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR )
2524recnd 9669 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
2625mulid1d 9660 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( A ^ 2 )  - 
1 ) )  x.  1 )  =  ( sqr `  ( ( A ^ 2 )  -  1 ) ) )
2726oveq2d 6306 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
28 pell1qrss14 35714 . . . . . 6  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(Pell1QR `  ( ( A ^ 2 )  - 
1 ) )  C_  (Pell14QR `  ( ( A ^ 2 )  - 
1 ) ) )
291, 28syl 17 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  (Pell1QR `  (
( A ^ 2 )  -  1 ) )  C_  (Pell14QR `  (
( A ^ 2 )  -  1 ) ) )
30 1nn0 10885 . . . . . . 7  |-  1  e.  NN0
3130a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  NN0 )
328oveq2i 6301 . . . . . . . . 9  |-  ( ( ( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( ( A ^ 2 )  - 
1 )  x.  1 )
337recnd 9669 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
3433mulid1d 9660 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  1 )  =  ( ( A ^
2 )  -  1 ) )
3532, 34syl5eq 2497 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( A ^
2 )  -  1 ) )
3635oveq2d 6306 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( ( A ^ 2 )  -  1 ) ) )
375recnd 9669 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
38 1cnd 9659 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  CC )
3937, 38nncand 9991 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( A ^
2 )  -  1 ) )  =  1 )
4036, 39eqtrd 2485 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  1 )
41 pellqrexplicit 35723 . . . . . 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 ) ) )
421, 15, 31, 40, 41syl31anc 1271 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell1QR `  ( ( A ^
2 )  -  1 ) ) )
4329, 42sseldd 3433 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
4427, 43eqeltrrd 2530 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
456, 24readdcld 9670 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
4612, 24readdcld 9670 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
476, 23ltaddrpd 11371 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( 1  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
486, 12, 24, 11ltadd1dd 10224 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
496, 45, 46, 47, 48lttrd 9796 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
50 pellfundlb 35732 . . 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 ) ) ) )
511, 44, 49, 50syl3anc 1268 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  <_  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
5237, 38npcand 9990 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  +  1 )  =  ( A ^ 2 ) )
5352fveq2d 5869 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  ( sqr `  ( A ^ 2 ) ) )
5412, 16sqrtsqd 13481 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( A ^ 2 ) )  =  A )
5553, 54eqtrd 2485 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  A )
5655oveq1d 6305 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
57 pellfundge 35730 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
( ( sqr `  (
( ( A ^
2 )  -  1 )  +  1 ) )  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) )  <_ 
(PellFund `  ( ( A ^ 2 )  - 
1 ) ) )
581, 57syl 17 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
5956, 58eqbrtrrd 4425 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
60 pellfundre 35729 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(PellFund `  ( ( A ^ 2 )  - 
1 ) )  e.  RR )
611, 60syl 17 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  e.  RR )
6261, 46letri3d 9777 . 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 ) ) ) ) )
6351, 59, 62mpbir2and 933 1  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  =  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887    \ cdif 3401    C_ wss 3404   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ^cexp 12272   sqrcsqrt 13296  ◻NNcsquarenn 35680  Pell1QRcpell1qr 35681  Pell14QRcpell14qr 35683  PellFundcpellfund 35684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-ico 11641  df-fz 11785  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-gcd 14469  df-numer 14684  df-denom 14685  df-squarenn 35686  df-pell1qr 35687  df-pell14qr 35688  df-pell1234qr 35689  df-pellfund 35690
This theorem is referenced by:  rmxyelqirr  35758  rmxycomplete  35765  rmbaserp  35767
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