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Theorem pell14qrss1234 29337
Description: A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell14qrss1234  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )

Proof of Theorem pell14qrss1234
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0z 10772 . . . . . . 7  |-  ( b  e.  NN0  ->  b  e.  ZZ )
21a1i 11 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( b  e.  NN0  ->  b  e.  ZZ ) )
32anim1d 564 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( ( b  e. 
NN0  /\  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) )  -> 
( b  e.  ZZ  /\ 
E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D )  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  1 ) ) ) )
43reximdv2 2923 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( E. b  e. 
NN0  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D )  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  (
c ^ 2 ) ) )  =  1 )  ->  E. b  e.  ZZ  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) )
54anim2d 565 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( a  e.  RR  /\  E. b  e.  NN0  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) )  -> 
( a  e.  RR  /\ 
E. b  e.  ZZ  E. c  e.  ZZ  (
a  =  ( b  +  ( ( sqr `  D )  x.  c
) )  /\  (
( b ^ 2 )  -  ( D  x.  ( c ^
2 ) ) )  =  1 ) ) ) )
6 elpell14qr 29330 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  <->  ( a  e.  RR  /\  E. b  e.  NN0  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) ) )
7 elpell1234qr 29332 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell1234QR `  D )  <->  ( a  e.  RR  /\  E. b  e.  ZZ  E. c  e.  ZZ  ( a  =  ( b  +  ( ( sqr `  D
)  x.  c ) )  /\  ( ( b ^ 2 )  -  ( D  x.  ( c ^ 2 ) ) )  =  1 ) ) ) )
85, 6, 73imtr4d 268 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  ->  a  e.  (Pell1234QR `  D ) ) )
98ssrdv 3462 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796    \ cdif 3425    C_ wss 3428   ` cfv 5518  (class class class)co 6192   RRcr 9384   1c1 9386    + caddc 9388    x. cmul 9390    - cmin 9698   NNcn 10425   2c2 10474   NN0cn0 10682   ZZcz 10749   ^cexp 11968   sqrcsqr 12826  ◻NNcsquarenn 29317  Pell1234QRcpell1234qr 29319  Pell14QRcpell14qr 29320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-pell14qr 29324  df-pell1234qr 29325
This theorem is referenced by:  pell14qrre  29338  pell14qrne0  29339  elpell14qr2  29343
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