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Theorem rmspecfund 26972
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 26970 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  ( NN  \NN ) )
2 eluzelz 10496 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
3 zsqcl 11452 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
42, 3syl 16 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  ZZ )
54zred 10375 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
6 1re 9090 . . . . . . . . . . . 12  |-  1  e.  RR
76a1i 11 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
85, 7resubcld 9465 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR )
9 sq1 11476 . . . . . . . . . . . . 13  |-  ( 1 ^ 2 )  =  1
109a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
11 eluz2b2 10548 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
1211simprbi 451 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
13 eluzelre 10497 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
14 0le1 9551 . . . . . . . . . . . . . . 15  |-  0  <_  1
1514a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  1 )
16 2nn0 10238 . . . . . . . . . . . . . . . 16  |-  2  e.  NN0
17 eluznn0 10546 . . . . . . . . . . . . . . . 16  |-  ( ( 2  e.  NN0  /\  A  e.  ( ZZ>= ` 
2 ) )  ->  A  e.  NN0 )
1816, 17mpan 652 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN0 )
1918nn0ge0d 10277 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  A )
207, 13, 15, 19lt2sqd 11557 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 1 ^ 2 )  < 
( A ^ 2 ) ) )
2112, 20mpbid 202 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  < 
( A ^ 2 ) )
2210, 21eqbrtrrd 4234 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A ^ 2 ) )
237, 5posdifd 9613 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  ( A ^
2 )  <->  0  <  ( ( A ^ 2 )  -  1 ) ) )
2422, 23mpbid 202 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  ( ( A ^ 2 )  -  1 ) )
258, 24elrpd 10646 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR+ )
2625rpsqrcld 12214 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR+ )
2726rpred 10648 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR )
2827recnd 9114 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
2928mulid1d 9105 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( A ^ 2 )  - 
1 ) )  x.  1 )  =  ( sqr `  ( ( A ^ 2 )  -  1 ) ) )
3029oveq2d 6097 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
31 pell1qrss14 26931 . . . . . 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 10237 . . . . . . 7  |-  1  e.  NN0
3433a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  NN0 )
359oveq2i 6092 . . . . . . . . 9  |-  ( ( ( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( ( A ^ 2 )  - 
1 )  x.  1 )
368recnd 9114 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
3736mulid1d 9105 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  1 )  =  ( ( A ^
2 )  -  1 ) )
3835, 37syl5eq 2480 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( A ^
2 )  -  1 ) )
3938oveq2d 6097 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( ( A ^ 2 )  -  1 ) ) )
405recnd 9114 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
41 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
4241a1i 11 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  CC )
4340, 42nncand 9416 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( A ^
2 )  -  1 ) )  =  1 )
4439, 43eqtrd 2468 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  1 )
45 pellqrexplicit 26940 . . . . . 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 1187 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell1QR `  ( ( A ^
2 )  -  1 ) ) )
4732, 46sseldd 3349 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
4830, 47eqeltrrd 2511 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
497, 27readdcld 9115 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
5013, 27readdcld 9115 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
517, 26ltaddrpd 10677 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( 1  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
527, 13, 27, 12ltadd1dd 9637 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
537, 49, 50, 51, 52lttrd 9231 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
54 pellfundlb 26947 . . 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 1184 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  <_  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
5640, 42npcand 9415 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  +  1 )  =  ( A ^ 2 ) )
5756fveq2d 5732 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  ( sqr `  ( A ^ 2 ) ) )
5813, 19sqrsqd 12222 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( A ^ 2 ) )  =  A )
5957, 58eqtrd 2468 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  A )
6059oveq1d 6096 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
61 pellfundge 26945 . . . 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 4234 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
64 pellfundre 26944 . . . 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 9215 . 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 889 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 1652    e. wcel 1725    \ cdif 3317    C_ wss 3320   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    - cmin 9291   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ^cexp 11382   sqrcsqr 12038  ◻NNcsquarenn 26899  Pell1QRcpell1qr 26900  Pell14QRcpell14qr 26902  PellFundcpellfund 26903
This theorem is referenced by:  rmxyelqirr  26973  rmxycomplete  26980  rmbaserp  26982
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-ico 10922  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-numer 13127  df-denom 13128  df-squarenn 26904  df-pell1qr 26905  df-pell14qr 26906  df-pell1234qr 26907  df-pellfund 26908
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