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Theorem rmxyelqirr 30780
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 30777 . . . . 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 30718 . . . 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 2935 . . . . . . . . 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 2935 . . . . . . . 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 2828 . . . . . . 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 3581 . . . . . 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 3529 . . . . . 6  |-  RR  C_  _V
13 rabss2 3588 . . . . . 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 3521 . . . 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 3136 . . . 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 3555 . . 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 3543 . 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 30779 . . . . . . 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 2475 . . . . 5  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  ( A  +  ( sqr `  ( ( A ^
2 )  -  1 ) ) )  =  (PellFund `  ( ( A ^ 2 )  - 
1 ) ) )
2322oveq1d 6310 . . . 4  |-  ( ( A  e.  ( ZZ>= ` 
2 )  /\  N  e.  ZZ )  ->  (
( A  +  ( sqr `  ( ( A ^ 2 )  -  1 ) ) ) ^ N )  =  ( (PellFund `  (
( A ^ 2 )  -  1 ) ) ^ N ) )
24 oveq2 6303 . . . . . 6  |-  ( a  =  N  ->  (
(PellFund `  ( ( A ^ 2 )  - 
1 ) ) ^
a )  =  ( (PellFund `  ( ( A ^ 2 )  - 
1 ) ) ^ N ) )
2524eqeq2d 2481 . . . . 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 3219 . . . 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 30769 . . . 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 3510 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 1379    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118    \ cdif 3478    C_ wss 3481   ` cfv 5594  (class class class)co 6295   RRcr 9503   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817   NNcn 10548   2c2 10597   NN0cn0 10807   ZZcz 10876   ZZ>=cuz 11094   ^cexp 12146   sqrcsqrt 13045  ◻NNcsquarenn 30706  Pell14QRcpell14qr 30709  PellFundcpellfund 30710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ioc 11546  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-fac 12334  df-bc 12361  df-hash 12386  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686  df-dvds 13864  df-gcd 14020  df-numer 14143  df-denom 14144  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839  df-mulg 15931  df-cntz 16226  df-cmn 16671  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-limc 22136  df-dv 22137  df-log 22808  df-squarenn 30711  df-pell1qr 30712  df-pell14qr 30713  df-pell1234qr 30714  df-pellfund 30715
This theorem is referenced by:  rmxyelxp  30782  rmxyval  30785
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