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Theorem pell1qrss14 35165
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 10928 . . . . . . . 8  |-  ( b  e.  NN0  ->  b  e.  ZZ )
21a1i 11 . . . . . . 7  |-  ( D  e.  ( NN  \NN )  -> 
( b  e.  NN0  ->  b  e.  ZZ ) )
32anim1d 562 . . . . . 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 2875 . . . . 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 2878 . . . 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 563 . . 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 35144 . . 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 35146 . . 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 3448 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell1QR `  D )  C_  (Pell14QR `  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755    \ cdif 3411    C_ wss 3414   ` cfv 5569  (class class class)co 6278   RRcr 9521   1c1 9523    + caddc 9525    x. cmul 9527    - cmin 9841   NNcn 10576   2c2 10626   NN0cn0 10836   ZZcz 10905   ^cexp 12210   sqrcsqrt 13215  ◻NNcsquarenn 35133  Pell1QRcpell1qr 35134  Pell14QRcpell14qr 35136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-n0 10837  df-z 10906  df-pell1qr 35139  df-pell14qr 35140
This theorem is referenced by:  elpell1qr2  35169  pellfundre  35178  pellfundge  35179  pellfundglb  35182  pellfundex  35183  pellfund14  35195  pellfund14b  35196  rmspecfund  35206
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