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Theorem rmxyelqirr 29251
Description: The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Assertion
Ref Expression
rmxyelqirr  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  e.  { a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) } )
Distinct variable groups:    A, a,
c, d    N, a
Allowed substitution hints:    N( c, d)

Proof of Theorem rmxyelqirr
StepHypRef Expression
1 rmspecnonsq 29248 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  ( NN  \NN ) )
21adantr 465 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A ^ 2 )  -  1 )  e.  ( NN  \NN ) )
3 pell14qrval 29189 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
(Pell14QR `  ( ( A ^ 2 )  - 
1 ) )  =  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  (
a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )  /\  (
( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^
2 ) ) )  =  1 ) } )
42, 3syl 16 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (Pell14QR `  ( ( A ^
2 )  -  1 ) )  =  {
a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  (
a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )  /\  (
( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^
2 ) ) )  =  1 ) } )
5 simpl 457 . . . . . . . . . 10  |-  ( ( a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )  /\  (
( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^
2 ) ) )  =  1 )  -> 
a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) ) )
65reximi 2823 . . . . . . . . 9  |-  ( E. d  e.  ZZ  (
a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )  /\  (
( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^
2 ) ) )  =  1 )  ->  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) ) )
76reximi 2823 . . . . . . . 8  |-  ( E. c  e.  NN0  E. d  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^ 2 ) ) )  =  1 )  ->  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) )
87rgenw 2783 . . . . . . 7  |-  A. a  e.  RR  ( E. c  e.  NN0  E. d  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^ 2 ) ) )  =  1 )  ->  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) )
98a1i 11 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  A. a  e.  RR  ( E. c  e.  NN0  E. d  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^ 2 ) ) )  =  1 )  ->  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) ) )
10 ss2rab 3428 . . . . . 6  |-  ( { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  (
a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )  /\  (
( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^
2 ) ) )  =  1 ) } 
C_  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }  <->  A. a  e.  RR  ( E. c  e.  NN0  E. d  e.  ZZ  (
a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) )  /\  (
( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^
2 ) ) )  =  1 )  ->  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) ) )
119, 10sylibr 212 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^ 2 ) ) )  =  1 ) }  C_  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) } )
12 ssv 3376 . . . . . 6  |-  RR  C_  _V
13 rabss2 3435 . . . . . 6  |-  ( RR  C_  _V  ->  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }  C_  { a  e.  _V  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) } )
1412, 13ax-mp 5 . . . . 5  |-  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }  C_  { a  e.  _V  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }
1511, 14syl6ss 3368 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^ 2 ) ) )  =  1 ) }  C_  { a  e.  _V  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) } )
16 rabab 2990 . . . 4  |-  { a  e.  _V  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) }  =  { a  |  E. c  e. 
NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) ) }
1715, 16syl6sseq 3402 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  { a  e.  RR  |  E. c  e.  NN0  E. d  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) )  /\  ( ( c ^ 2 )  -  ( ( ( A ^ 2 )  -  1 )  x.  ( d ^ 2 ) ) )  =  1 ) }  C_  { a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^ 2 )  -  1 ) )  x.  d ) ) } )
184, 17eqsstrd 3390 . 2  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (Pell14QR `  ( ( A ^
2 )  -  1 ) )  C_  { a  |  E. c  e. 
NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  ( ( A ^
2 )  -  1 ) )  x.  d
) ) } )
19 simpr 461 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  N  e.  ZZ )
20 rmspecfund 29250 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  =  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
2120adantr 465 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (PellFund `  ( ( A ^
2 )  -  1 ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
2221eqcomd 2448 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) )  =  (PellFund `  ( ( A ^ 2 )  - 
1 ) ) )
2322oveq1d 6106 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ N ) )
24 oveq2 6099 . . . . . 6  |-  ( a  =  N  ->  (
(PellFund `  ( ( A ^ 2 )  - 
1 ) ) ^
a )  =  ( (PellFund `  ( ( A ^ 2 )  - 
1 ) ) ^ N ) )
2524eqeq2d 2454 . . . . 5  |-  ( a  =  N  ->  (
( ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ a )  <-> 
( ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ N ) ) )
2625rspcev 3073 . . . 4  |-  ( ( N  e.  ZZ  /\  ( ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ N ) )  ->  E. a  e.  ZZ  ( ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) ^ N
)  =  ( (PellFund `  ( ( A ^
2 )  -  1 ) ) ^ a
) )
2719, 23, 26syl2anc 661 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  E. a  e.  ZZ  ( ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) ^ N
)  =  ( (PellFund `  ( ( A ^
2 )  -  1 ) ) ^ a
) )
28 pellfund14b 29240 . . . 4  |-  ( ( ( A ^ 2 )  -  1 )  e.  ( NN  \NN )  -> 
( ( ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) ^ N
)  e.  (Pell14QR `  (
( A ^ 2 )  -  1 ) )  <->  E. a  e.  ZZ  ( ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ a ) ) )
292, 28syl 16 . . 3  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  e.  (Pell14QR `  (
( A ^ 2 )  -  1 ) )  <->  E. a  e.  ZZ  ( ( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ a ) ) )
3027, 29mpbird 232 . 2  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  e.  (Pell14QR `  (
( A ^ 2 )  -  1 ) ) )
3118, 30sseldd 3357 1  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  e.  { a  |  E. c  e.  NN0  E. d  e.  ZZ  a  =  ( c  +  ( ( sqr `  (
( A ^ 2 )  -  1 ) )  x.  d ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972    \ cdif 3325    C_ wss 3328   ` cfv 5418  (class class class)co 6091   RRcr 9281   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595   NNcn 10322   2c2 10371   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   ^cexp 11865   sqrcsqr 12722  ◻NNcsquarenn 29177  Pell14QRcpell14qr 29180  PellFundcpellfund 29181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-omul 6925  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-acn 8112  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-sum 13164  df-ef 13353  df-sin 13355  df-cos 13356  df-pi 13358  df-dvds 13536  df-gcd 13691  df-numer 13813  df-denom 13814  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-squarenn 29182  df-pell1qr 29183  df-pell14qr 29184  df-pell1234qr 29185  df-pellfund 29186
This theorem is referenced by:  rmxyelxp  29253  rmxyval  29256
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