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Theorem rmspecfund 29175
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 29173 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  ( NN  \NN ) )
2 eluzelz 10866 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
3 zsqcl 11932 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
42, 3syl 16 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  ZZ )
54zred 10743 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
6 1re 9381 . . . . . . . . . . . 12  |-  1  e.  RR
76a1i 11 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
85, 7resubcld 9772 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR )
9 sq1 11956 . . . . . . . . . . . . 13  |-  ( 1 ^ 2 )  =  1
109a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
11 eluz2b2 10923 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
1211simprbi 461 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
13 eluzelre 10867 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
14 0le1 9859 . . . . . . . . . . . . . . 15  |-  0  <_  1
1514a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  1 )
16 2nn0 10592 . . . . . . . . . . . . . . . 16  |-  2  e.  NN0
17 eluznn0 10920 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  NN0  /\  A  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  NN0 )
1816, 17mpan 665 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN0 )
1918nn0ge0d 10635 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  A )
207, 13, 15, 19lt2sqd 12038 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 1 ^ 2 )  < 
( A ^ 2 ) ) )
2112, 20mpbid 210 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  < 
( A ^ 2 ) )
2210, 21eqbrtrrd 4311 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A ^ 2 ) )
237, 5posdifd 9922 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  ( A ^
2 )  <->  0  <  ( ( A ^ 2 )  -  1 ) ) )
2422, 23mpbid 210 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  ( ( A ^ 2 )  -  1 ) )
258, 24elrpd 11021 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR+ )
2625rpsqrcld 12894 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR+ )
2726rpred 11023 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR )
2827recnd 9408 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
2928mulid1d 9399 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( A ^ 2 )  - 
1 ) )  x.  1 )  =  ( sqr `  ( ( A ^ 2 )  -  1 ) ) )
3029oveq2d 6106 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
31 pell1qrss14 29134 . . . . . 6  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(Pell1QR `  ( ( A ^ 2 )  - 
1 ) )  C_  (Pell14QR `  ( ( A ^ 2 )  - 
1 ) ) )
321, 31syl 16 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  (Pell1QR `  (
( A ^ 2 )  -  1 ) )  C_  (Pell14QR `  (
( A ^ 2 )  -  1 ) ) )
33 1nn0 10591 . . . . . . 7  |-  1  e.  NN0
3433a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  NN0 )
359oveq2i 6101 . . . . . . . . 9  |-  ( ( ( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( ( A ^ 2 )  - 
1 )  x.  1 )
368recnd 9408 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
3736mulid1d 9399 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  1 )  =  ( ( A ^
2 )  -  1 ) )
3835, 37syl5eq 2485 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( A ^
2 )  -  1 ) )
3938oveq2d 6106 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( ( A ^ 2 )  -  1 ) ) )
405recnd 9408 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
41 ax-1cn 9336 . . . . . . . . 9  |-  1  e.  CC
4241a1i 11 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  CC )
4340, 42nncand 9720 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( A ^
2 )  -  1 ) )  =  1 )
4439, 43eqtrd 2473 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  1 )
45 pellqrexplicit 29143 . . . . . 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 ) ) )
461, 18, 34, 44, 45syl31anc 1216 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell1QR `  ( ( A ^
2 )  -  1 ) ) )
4732, 46sseldd 3354 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
4830, 47eqeltrrd 2516 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
497, 27readdcld 9409 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
5013, 27readdcld 9409 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
517, 26ltaddrpd 11052 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( 1  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
527, 13, 27, 12ltadd1dd 9946 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
537, 49, 50, 51, 52lttrd 9528 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
54 pellfundlb 29150 . . 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 ) ) ) )
551, 48, 53, 54syl3anc 1213 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  <_  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
5640, 42npcand 9719 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  +  1 )  =  ( A ^ 2 ) )
5756fveq2d 5692 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  ( sqr `  ( A ^ 2 ) ) )
5813, 19sqrsqd 12902 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( A ^ 2 ) )  =  A )
5957, 58eqtrd 2473 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  A )
6059oveq1d 6105 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
61 pellfundge 29148 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
( ( sqr `  (
( ( A ^
2 )  -  1 )  +  1 ) )  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) )  <_ 
(PellFund `  ( ( A ^ 2 )  - 
1 ) ) )
621, 61syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
6360, 62eqbrtrrd 4311 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
64 pellfundre 29147 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(PellFund `  ( ( A ^ 2 )  - 
1 ) )  e.  RR )
651, 64syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  e.  RR )
6665, 50letri3d 9512 . 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 ) ) ) ) )
6755, 63, 66mpbir2and 908 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 1364    e. wcel 1761    \ cdif 3322    C_ wss 3325   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    <_ cle 9415    - cmin 9591   NNcn 10318   2c2 10367   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ^cexp 11861   sqrcsqr 12718  ◻NNcsquarenn 29102  Pell1QRcpell1qr 29103  Pell14QRcpell14qr 29105  PellFundcpellfund 29106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-omul 6921  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-acn 8108  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-ico 11302  df-fz 11434  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-numer 13809  df-denom 13810  df-squarenn 29107  df-pell1qr 29108  df-pell14qr 29109  df-pell1234qr 29110  df-pellfund 29111
This theorem is referenced by:  rmxyelqirr  29176  rmxycomplete  29183  rmbaserp  29185
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