Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pell1234qrne0 Structured version   Unicode version

Theorem pell1234qrne0 35150
Description: No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell1234qrne0  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )

Proof of Theorem pell1234qrne0
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpell1234qr 35148 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  <->  ( A  e.  RR  /\  E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) ) ) )
2 simprl 756 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  =  ( a  +  ( ( sqr `  D )  x.  b
) ) )
3 ax-1ne0 9591 . . . . . . . . 9  |-  1  =/=  0
4 eldifi 3565 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
54adantr 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  NN )
65nncnd 10592 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  CC )
76ad3antrrr 728 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  ->  D  e.  CC )
87sqrtcld 13417 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( sqr `  D
)  e.  CC )
9 zcn 10910 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  ZZ  ->  b  e.  CC )
109ad2antll 727 . . . . . . . . . . . . . . . . 17  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  b  e.  CC )
1110ad2antrr 724 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
b  e.  CC )
128, 11sqmuld 12366 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( ( sqr `  D )  x.  b
) ^ 2 )  =  ( ( ( sqr `  D ) ^ 2 )  x.  ( b ^ 2 ) ) )
137sqsqrtd 13419 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( sqr `  D
) ^ 2 )  =  D )
1413oveq1d 6293 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( ( sqr `  D ) ^ 2 )  x.  ( b ^ 2 ) )  =  ( D  x.  ( b ^ 2 ) ) )
1512, 14eqtr2d 2444 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( D  x.  (
b ^ 2 ) )  =  ( ( ( sqr `  D
)  x.  b ) ^ 2 ) )
1615oveq2d 6294 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  ( ( a ^ 2 )  -  ( ( ( sqr `  D )  x.  b ) ^
2 ) ) )
17 zcn 10910 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ZZ  ->  a  e.  CC )
1817ad2antrl 726 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  a  e.  CC )
1918ad2antrr 724 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
a  e.  CC )
208, 11mulcld 9646 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( sqr `  D
)  x.  b )  e.  CC )
21 subsq 12320 . . . . . . . . . . . . . 14  |-  ( ( a  e.  CC  /\  ( ( sqr `  D
)  x.  b )  e.  CC )  -> 
( ( a ^
2 )  -  (
( ( sqr `  D
)  x.  b ) ^ 2 ) )  =  ( ( a  +  ( ( sqr `  D )  x.  b
) )  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) ) )
2219, 20, 21syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( a ^
2 )  -  (
( ( sqr `  D
)  x.  b ) ^ 2 ) )  =  ( ( a  +  ( ( sqr `  D )  x.  b
) )  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) ) )
2316, 22eqtrd 2443 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  ( ( a  +  ( ( sqr `  D )  x.  b ) )  x.  ( a  -  ( ( sqr `  D
)  x.  b ) ) ) )
24 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( a ^
2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )
25 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( a  +  ( ( sqr `  D
)  x.  b ) )  =  0 )
2625oveq1d 6293 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( a  +  ( ( sqr `  D
)  x.  b ) )  x.  ( a  -  ( ( sqr `  D )  x.  b
) ) )  =  ( 0  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) ) )
2719, 20subcld 9967 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( a  -  (
( sqr `  D
)  x.  b ) )  e.  CC )
2827mul02d 9812 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( 0  x.  (
a  -  ( ( sqr `  D )  x.  b ) ) )  =  0 )
2926, 28eqtrd 2443 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
( ( a  +  ( ( sqr `  D
)  x.  b ) )  x.  ( a  -  ( ( sqr `  D )  x.  b
) ) )  =  0 )
3023, 24, 293eqtr3d 2451 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  ( a  e.  ZZ  /\  b  e.  ZZ ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  /\  (
a  +  ( ( sqr `  D )  x.  b ) )  =  0 )  -> 
1  =  0 )
3130ex 432 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  /\  ( (
a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  (
( a  +  ( ( sqr `  D
)  x.  b ) )  =  0  -> 
1  =  0 ) )
3231necon3d 2627 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  /\  ( (
a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  (
1  =/=  0  -> 
( a  +  ( ( sqr `  D
)  x.  b ) )  =/=  0 ) )
333, 32mpi 20 . . . . . . . 8  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  /\  ( (
a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  (
a  +  ( ( sqr `  D )  x.  b ) )  =/=  0 )
3433adantrl 714 . . . . . . 7  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  -> 
( a  +  ( ( sqr `  D
)  x.  b ) )  =/=  0 )
352, 34eqnetrd 2696 . . . . . 6  |-  ( ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  /\  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  =/=  0 )
3635ex 432 . . . . 5  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  ( ( A  =  ( a  +  ( ( sqr `  D )  x.  b
) )  /\  (
( a ^ 2 )  -  ( D  x.  ( b ^
2 ) ) )  =  1 )  ->  A  =/=  0 ) )
3736rexlimdvva 2903 . . . 4  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  ( E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 )  ->  A  =/=  0 ) )
3837expimpd 601 . . 3  |-  ( D  e.  ( NN  \NN )  -> 
( ( A  e.  RR  /\  E. a  e.  ZZ  E. b  e.  ZZ  ( A  =  ( a  +  ( ( sqr `  D
)  x.  b ) )  /\  ( ( a ^ 2 )  -  ( D  x.  ( b ^ 2 ) ) )  =  1 ) )  ->  A  =/=  0 ) )
391, 38sylbid 215 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  ->  A  =/=  0 ) )
4039imp 427 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755    \ cdif 3411   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    - cmin 9841   NNcn 10576   2c2 10626   ZZcz 10905   ^cexp 12210   sqrcsqrt 13215  ◻NNcsquarenn 35133  Pell1234QRcpell1234qr 35135
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  ax-pre-sup 9600
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-rmo 2762  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-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-pell1234qr 35141
This theorem is referenced by:  pell1234qrreccl  35151  pell14qrne0  35155
  Copyright terms: Public domain W3C validator