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Theorem rmspecfund 27097
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 27095 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  ( NN  \NN ) )
2 eluzelz 10254 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
3 zsqcl 11190 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
42, 3syl 15 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  ZZ )
54zred 10133 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
6 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
76a1i 10 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
85, 7resubcld 9227 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR )
9 sq1 11214 . . . . . . . . . . . . 13  |-  ( 1 ^ 2 )  =  1
109a1i 10 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
11 eluz2b2 10306 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
1211simprbi 450 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
13 eluzelre 10255 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
14 0le1 9313 . . . . . . . . . . . . . . 15  |-  0  <_  1
1514a1i 10 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  1 )
16 2nn0 9998 . . . . . . . . . . . . . . . 16  |-  2  e.  NN0
17 eluznn0 10304 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  NN0  /\  A  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  NN0 )
1816, 17mpan 651 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN0 )
1918nn0ge0d 10037 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  A )
207, 13, 15, 19lt2sqd 11295 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 1 ^ 2 )  < 
( A ^ 2 ) ) )
2112, 20mpbid 201 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  < 
( A ^ 2 ) )
2210, 21eqbrtrrd 4061 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A ^ 2 ) )
237, 5posdifd 9375 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  ( A ^
2 )  <->  0  <  ( ( A ^ 2 )  -  1 ) ) )
2422, 23mpbid 201 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  ( ( A ^ 2 )  -  1 ) )
258, 24elrpd 10404 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR+ )
2625rpsqrcld 11910 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR+ )
2726rpred 10406 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR )
2827recnd 8877 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
2928mulid1d 8868 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( A ^ 2 )  - 
1 ) )  x.  1 )  =  ( sqr `  ( ( A ^ 2 )  -  1 ) ) )
3029oveq2d 5890 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
31 pell1qrss14 27056 . . . . . 6  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(Pell1QR `  ( ( A ^ 2 )  - 
1 ) )  C_  (Pell14QR `  ( ( A ^ 2 )  - 
1 ) ) )
321, 31syl 15 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  (Pell1QR `  (
( A ^ 2 )  -  1 ) )  C_  (Pell14QR `  (
( A ^ 2 )  -  1 ) ) )
33 1nn0 9997 . . . . . . 7  |-  1  e.  NN0
3433a1i 10 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  NN0 )
359oveq2i 5885 . . . . . . . . 9  |-  ( ( ( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( ( A ^ 2 )  - 
1 )  x.  1 )
368recnd 8877 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
3736mulid1d 8868 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  1 )  =  ( ( A ^
2 )  -  1 ) )
3835, 37syl5eq 2340 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( A ^
2 )  -  1 ) )
3938oveq2d 5890 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( ( A ^ 2 )  -  1 ) ) )
405recnd 8877 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
41 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
4241a1i 10 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  CC )
4340, 42nncand 9178 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( A ^
2 )  -  1 ) )  =  1 )
4439, 43eqtrd 2328 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  1 )
45 pellqrexplicit 27065 . . . . . 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 1185 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell1QR `  ( ( A ^
2 )  -  1 ) ) )
4732, 46sseldd 3194 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
4830, 47eqeltrrd 2371 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
497, 27readdcld 8878 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
5013, 27readdcld 8878 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
517, 26ltaddrpd 10435 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( 1  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
527, 13, 27, 12ltadd1dd 9399 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
537, 49, 50, 51, 52lttrd 8993 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
54 pellfundlb 27072 . . 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 1182 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  <_  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
5640, 42npcand 9177 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  +  1 )  =  ( A ^ 2 ) )
5756fveq2d 5545 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  ( sqr `  ( A ^ 2 ) ) )
5813, 19sqrsqd 11918 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( A ^ 2 ) )  =  A )
5957, 58eqtrd 2328 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  A )
6059oveq1d 5889 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
61 pellfundge 27070 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
( ( sqr `  (
( ( A ^
2 )  -  1 )  +  1 ) )  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) )  <_ 
(PellFund `  ( ( A ^ 2 )  - 
1 ) ) )
621, 61syl 15 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
6360, 62eqbrtrrd 4061 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
64 pellfundre 27069 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(PellFund `  ( ( A ^ 2 )  - 
1 ) )  e.  RR )
651, 64syl 15 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  e.  RR )
6665, 50letri3d 8977 . 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 888 1  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  =  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    \ cdif 3162    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ^cexp 11120   sqrcsqr 11734  ◻NNcsquarenn 27024  Pell1QRcpell1qr 27025  Pell14QRcpell14qr 27027  PellFundcpellfund 27028
This theorem is referenced by:  rmxyelqirr  27098  rmxycomplete  27105  rmbaserp  27107
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-ico 10678  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-numer 12822  df-denom 12823  df-squarenn 27029  df-pell1qr 27030  df-pell14qr 27031  df-pell1234qr 27032  df-pellfund 27033
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