Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pell14qrss1234 Structured version   Unicode version

Theorem pell14qrss1234 26919
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 10304 . . . . . . 7  |-  ( b  e.  NN0  ->  b  e.  ZZ )
21a1i 11 . . . . . 6  |-  ( D  e.  ( NN  \NN )  -> 
( b  e.  NN0  ->  b  e.  ZZ ) )
32anim1d 548 . . . . 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 549 . . 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 26912 . . 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 26914 . . 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 260 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( a  e.  (Pell14QR `  D )  ->  a  e.  (Pell1234QR `  D ) ) )
98ssrdv 3354 1  |-  ( D  e.  ( NN  \NN )  -> 
(Pell14QR `  D )  C_  (Pell1234QR `
 D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    \ cdif 3317    C_ wss 3320   ` cfv 5454  (class class class)co 6081   RRcr 8989   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ^cexp 11382   sqrcsqr 12038  ◻NNcsquarenn 26899  Pell1234QRcpell1234qr 26901  Pell14QRcpell14qr 26902
This theorem is referenced by:  pell14qrre  26920  pell14qrne0  26921  elpell14qr2  26925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-pell14qr 26906  df-pell1234qr 26907
  Copyright terms: Public domain W3C validator