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Theorem pell1234qrne0 26916
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 26914 . . 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 733 . . . . . . 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 9059 . . . . . . . . 9  |-  1  =/=  0
4 eldifi 3469 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
54adantr 452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  NN )
65nncnd 10016 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  CC )
76ad3antrrr 711 . . . . . . . . . . . . . . . . 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 )
87sqrcld 12239 . . . . . . . . . . . . . . . 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 10287 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  ZZ  ->  b  e.  CC )
109ad2antll 710 . . . . . . . . . . . . . . . . 17  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  b  e.  CC )
1110ad2antrr 707 . . . . . . . . . . . . . . . 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 11535 . . . . . . . . . . . . . . 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 ) ) )
137sqsqrd 12241 . . . . . . . . . . . . . . . 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 6096 . . . . . . . . . . . . . . 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 2469 . . . . . . . . . . . . . 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 6097 . . . . . . . . . . . . 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 10287 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ZZ  ->  a  e.  CC )
1817ad2antrl 709 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  a  e.  CC )
1918ad2antrr 707 . . . . . . . . . . . . . 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 9108 . . . . . . . . . . . . . 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 11488 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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 2468 . . . . . . . . . . . 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 732 . . . . . . . . . . . 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 448 . . . . . . . . . . . . . 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 6096 . . . . . . . . . . . . 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 9411 . . . . . . . . . . . . . 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 9264 . . . . . . . . . . . . 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 2468 . . . . . . . . . . . 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 2476 . . . . . . . . . . 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 424 . . . . . . . . . 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 2639 . . . . . . . . 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 17 . . . . . . . 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 697 . . . . . . 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 2619 . . . . . 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 424 . . . . 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 2837 . . . 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 587 . . 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 207 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  ->  A  =/=  0 ) )
4039imp 419 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    \ cdif 3317   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   NNcn 10000   2c2 10049   ZZcz 10282   ^cexp 11382   sqrcsqr 12038  ◻NNcsquarenn 26899  Pell1234QRcpell1234qr 26901
This theorem is referenced by:  pell1234qrreccl  26917  pell14qrne0  26921
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  ax-pre-sup 9068
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-rmo 2713  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-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-pell1234qr 26907
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