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Theorem rmxyelqirr 26995
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 26992 . . . . 5  |-  ( A  e.  ( ZZ>= `  2
)  ->  ( ( A ^ 2 )  - 
1 )  e.  ( NN  \NN ) )
21adantr 451 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A ^ 2 )  -  1 )  e.  ( NN  \NN ) )
3 pell14qrval 26933 . . . 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 15 . . 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 443 . . . . . . . . . 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 2650 . . . . . . . . 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 2650 . . . . . . . 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 2610 . . . . . . 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 10 . . . . . 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 3249 . . . . . 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 203 . . . . 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 3198 . . . . . 6  |-  RR  C_  _V
13 rabss2 3256 . . . . . 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 8 . . . . 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 3191 . . . 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 2805 . . . 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 3224 . . 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 3212 . 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 447 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  N  e.  ZZ )
20 rmspecfund 26994 . . . . . . 7  |-  ( A  e.  ( ZZ>= `  2
)  ->  (PellFund `  (
( A ^ 2 )  -  1 ) )  =  ( A  +  ( sqr `  (
( A ^ 2 )  -  1 ) ) ) )
2120adantr 451 . . . . . 6  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (PellFund `  ( ( A ^
2 )  -  1 ) )  =  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) ) )
2221eqcomd 2288 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) )  =  (PellFund `  ( ( A ^ 2 )  - 
1 ) ) )
2322oveq1d 5873 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ N ) )
24 oveq2 5866 . . . . . 6  |-  ( a  =  N  ->  (
(PellFund `  ( ( A ^ 2 )  - 
1 ) ) ^
a )  =  ( (PellFund `  ( ( A ^ 2 )  - 
1 ) ) ^ N ) )
2524eqeq2d 2294 . . . . 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 2884 . . . 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 642 . . 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 26984 . . . 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 15 . . 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 223 . 2  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  e.  (Pell14QR `  (
( A ^ 2 )  -  1 ) ) )
3118, 30sseldd 3181 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104   sqrcsqr 11718  ◻NNcsquarenn 26921  Pell14QRcpell14qr 26924  PellFundcpellfund 26925
This theorem is referenced by:  rmxyelxp  26997  rmxyval  27000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-numer 12806  df-denom 12807  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-squarenn 26926  df-pell1qr 26927  df-pell14qr 26928  df-pell1234qr 26929  df-pellfund 26930
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