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Theorem pell14qrss1234 29039
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 10656 . . . . . . 7  |-  ( b  e.  NN0  ->  b  e.  ZZ )
21a1i 11 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( b  e.  NN0  ->  b  e.  ZZ ) )
32anim1d 559 . . . . 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 2815 . . . 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 560 . . 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 29032 . . 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 29034 . . 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 3350 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   E.wrex 2706    \ cdif 3313    C_ wss 3316   ` cfv 5406  (class class class)co 6080   RRcr 9268   1c1 9270    + caddc 9272    x. cmul 9274    - cmin 9582   NNcn 10309   2c2 10358   NN0cn0 10566   ZZcz 10633   ^cexp 11848   sqrcsqr 12705  ◻NNcsquarenn 29019  Pell1234QRcpell1234qr 29021  Pell14QRcpell14qr 29022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-nn 10310  df-n0 10567  df-z 10634  df-pell14qr 29026  df-pell1234qr 29027
This theorem is referenced by:  pell14qrre  29040  pell14qrne0  29041  elpell14qr2  29045
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