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Theorem pell14qrss1234 30396
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 10883 . . . . . . 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 2934 . . . 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 30389 . . 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 30391 . . 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 3510 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    \ cdif 3473    C_ wss 3476   ` cfv 5586  (class class class)co 6282   RRcr 9487   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   NNcn 10532   2c2 10581   NN0cn0 10791   ZZcz 10860   ^cexp 12130   sqrcsqrt 13025  ◻NNcsquarenn 30376  Pell1234QRcpell1234qr 30378  Pell14QRcpell14qr 30379
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-pell14qr 30383  df-pell1234qr 30384
This theorem is referenced by:  pell14qrre  30397  pell14qrne0  30398  elpell14qr2  30402
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