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Theorem rmspecfund 31049
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 31047 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  ( NN  \NN ) )
2 eluzelz 11115 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  ZZ )
3 zsqcl 12241 . . . . . . . . . . . . 13  |-  ( A  e.  ZZ  ->  ( A ^ 2 )  e.  ZZ )
42, 3syl 16 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  ZZ )
54zred 10990 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  RR )
6 1red 9628 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  RR )
75, 6resubcld 10008 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR )
8 sq1 12265 . . . . . . . . . . . . 13  |-  ( 1 ^ 2 )  =  1
98a1i 11 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  =  1 )
10 eluz2b2 11179 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  <->  ( A  e.  NN  /\  1  < 
A ) )
1110simprbi 464 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  A )
12 eluzelre 11116 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  RR )
13 0le1 10097 . . . . . . . . . . . . . . 15  |-  0  <_  1
1413a1i 11 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  1 )
15 eluzge2nn0 11145 . . . . . . . . . . . . . . 15  |-  ( A  e.  ( ZZ>= `  2
)  ->  A  e.  NN0 )
1615nn0ge0d 10876 . . . . . . . . . . . . . 14  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <_  A )
176, 12, 14, 16lt2sqd 12347 . . . . . . . . . . . . 13  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  A  <->  ( 1 ^ 2 )  < 
( A ^ 2 ) ) )
1811, 17mpbid 210 . . . . . . . . . . . 12  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1 ^ 2 )  < 
( A ^ 2 ) )
199, 18eqbrtrrd 4478 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A ^ 2 ) )
206, 5posdifd 10160 . . . . . . . . . . 11  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  <  ( A ^
2 )  <->  0  <  ( ( A ^ 2 )  -  1 ) ) )
2119, 20mpbid 210 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  0  <  ( ( A ^ 2 )  -  1 ) )
227, 21elrpd 11279 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  RR+ )
2322rpsqrtcld 13255 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR+ )
2423rpred 11281 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  RR )
2524recnd 9639 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( A ^
2 )  -  1 ) )  e.  CC )
2625mulid1d 9630 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( A ^ 2 )  - 
1 ) )  x.  1 )  =  ( sqr `  ( ( A ^ 2 )  -  1 ) ) )
2726oveq2d 6312 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
28 pell1qrss14 31008 . . . . . 6  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(Pell1QR `  ( ( A ^ 2 )  - 
1 ) )  C_  (Pell14QR `  ( ( A ^ 2 )  - 
1 ) ) )
291, 28syl 16 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  (Pell1QR `  (
( A ^ 2 )  -  1 ) )  C_  (Pell14QR `  (
( A ^ 2 )  -  1 ) ) )
30 1nn0 10832 . . . . . . 7  |-  1  e.  NN0
3130a1i 11 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  NN0 )
328oveq2i 6307 . . . . . . . . 9  |-  ( ( ( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( ( A ^ 2 )  - 
1 )  x.  1 )
337recnd 9639 . . . . . . . . . 10  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  CC )
3433mulid1d 9630 . . . . . . . . 9  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  1 )  =  ( ( A ^
2 )  -  1 ) )
3532, 34syl5eq 2510 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  x.  ( 1 ^ 2 ) )  =  ( ( A ^
2 )  -  1 ) )
3635oveq2d 6312 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  ( ( A ^ 2 )  -  ( ( A ^ 2 )  -  1 ) ) )
375recnd 9639 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A ^ 2 )  e.  CC )
38 1cnd 9629 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  e.  CC )
3937, 38nncand 9955 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( A ^
2 )  -  1 ) )  =  1 )
4036, 39eqtrd 2498 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  -  ( ( ( A ^ 2 )  - 
1 )  x.  (
1 ^ 2 ) ) )  =  1 )
41 pellqrexplicit 31017 . . . . . 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 1231 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell1QR `  ( ( A ^
2 )  -  1 ) ) )
4329, 42sseldd 3500 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  1 ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
4427, 43eqeltrrd 2546 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  (Pell14QR `  ( ( A ^
2 )  -  1 ) ) )
456, 24readdcld 9640 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
4612, 24readdcld 9640 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  e.  RR )
476, 23ltaddrpd 11310 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( 1  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
486, 12, 24, 11ltadd1dd 10184 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( 1  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
496, 45, 46, 47, 48lttrd 9760 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  1  <  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
50 pellfundlb 31024 . . 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 1228 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  <_  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
5237, 38npcand 9954 . . . . . 6  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( (
( A ^ 2 )  -  1 )  +  1 )  =  ( A ^ 2 ) )
5352fveq2d 5876 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  ( sqr `  ( A ^ 2 ) ) )
5412, 16sqrtsqd 13263 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( A ^ 2 ) )  =  A )
5553, 54eqtrd 2498 . . . 4  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( sqr `  ( ( ( A ^ 2 )  - 
1 )  +  1 ) )  =  A )
5655oveq1d 6311 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
57 pellfundge 31022 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
( ( sqr `  (
( ( A ^
2 )  -  1 )  +  1 ) )  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) )  <_ 
(PellFund `  ( ( A ^ 2 )  - 
1 ) ) )
581, 57syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( sqr `  ( ( ( A ^ 2 )  -  1 )  +  1 ) )  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
5956, 58eqbrtrrd 4478 . 2  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) )  <_  (PellFund `  ( ( A ^
2 )  -  1 ) ) )
60 pellfundre 31021 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(PellFund `  ( ( A ^ 2 )  - 
1 ) )  e.  RR )
611, 60syl 16 . . 3  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  e.  RR )
6261, 46letri3d 9744 . 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 922 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 1395    e. wcel 1819    \ cdif 3468    C_ wss 3471   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    <_ cle 9646    - cmin 9824   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ^cexp 12169   sqrcsqrt 13078  ◻NNcsquarenn 30976  Pell1QRcpell1qr 30977  Pell14QRcpell14qr 30979  PellFundcpellfund 30980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-omul 7153  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-ico 11560  df-fz 11698  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-numer 14280  df-denom 14281  df-squarenn 30981  df-pell1qr 30982  df-pell14qr 30983  df-pell1234qr 30984  df-pellfund 30985
This theorem is referenced by:  rmxyelqirr  31050  rmxycomplete  31057  rmbaserp  31059
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