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Theorem pell1qrss14 29380
Description: First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell1qrss14  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )

Proof of Theorem pell1qrss14
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0z 10784 . . . . . . . 8  |-  ( b  e.  NN0  ->  b  e.  ZZ )
21a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
( b  e.  NN0  ->  b  e.  ZZ ) )
32anim1d 564 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( ( b  e. 
NN0  /\  ( a  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( b  e.  ZZ  /\  ( a  =  ( c  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 ) ) ) )
43reximdv2 2931 . . . . 5  |-  ( D  e.  ( NN  \NN )  -> 
( E. b  e. 
NN0  ( a  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  E. b  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) )
54reximdv 2933 . . . 4  |-  ( D  e.  ( NN  \NN )  -> 
( E. c  e. 
NN0  E. b  e.  NN0  ( a  =  ( c  +  ( ( sqr `  D )  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  (
b ^ 2 ) ) )  =  1 )  ->  E. c  e.  NN0  E. b  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) )
65anim2d 565 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( a  e.  RR  /\  E. c  e.  NN0  E. b  e. 
NN0  ( a  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( a  e.  RR  /\ 
E. c  e.  NN0  E. b  e.  ZZ  (
a  =  ( c  +  ( ( sqr `  D )  x.  b
) )  /\  (
( c ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 ) ) ) )
7 elpell1qr 29359 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell1QR `  D )  <->  ( a  e.  RR  /\  E. c  e.  NN0  E. b  e. 
NN0  ( a  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
8 elpell14qr 29361 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  <->  ( a  e.  RR  /\  E. c  e.  NN0  E. b  e.  ZZ  ( a  =  ( c  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( c ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
96, 7, 83imtr4d 268 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell1QR `  D )  ->  a  e.  (Pell14QR `  D )
) )
109ssrdv 3473 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800    \ cdif 3436    C_ wss 3439   ` cfv 5529  (class class class)co 6203   RRcr 9396   1c1 9398    + caddc 9400    x. cmul 9402    - cmin 9710   NNcn 10437   2c2 10486   NN0cn0 10694   ZZcz 10761   ^cexp 11986   sqrcsqr 12844  ◻NNcsquarenn 29348  Pell1QRcpell1qr 29349  Pell14QRcpell14qr 29351
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-n0 10695  df-z 10762  df-pell1qr 29354  df-pell14qr 29355
This theorem is referenced by:  elpell1qr2  29384  pellfundre  29393  pellfundge  29394  pellfundglb  29397  pellfundex  29398  pellfund14  29410  pellfund14b  29411  rmspecfund  29421
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