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Theorem pell1234qrne0 29194
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 29192 . . 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 755 . . . . . . 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 9351 . . . . . . . . 9  |-  1  =/=  0
4 eldifi 3478 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
54adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  NN )
65nncnd 10338 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  CC )
76ad3antrrr 729 . . . . . . . . . . . . . . . . 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 12923 . . . . . . . . . . . . . . . 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 10651 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  ZZ  ->  b  e.  CC )
109ad2antll 728 . . . . . . . . . . . . . . . . 17  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  b  e.  CC )
1110ad2antrr 725 . . . . . . . . . . . . . . . 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 12020 . . . . . . . . . . . . . . 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 12925 . . . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . 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 10651 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ZZ  ->  a  e.  CC )
1817ad2antrl 727 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  a  e.  CC )
1918ad2antrr 725 . . . . . . . . . . . . . 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 9406 . . . . . . . . . . . . . 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 11973 . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 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 461 . . . . . . . . . . . . . 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 6106 . . . . . . . . . . . . 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 9719 . . . . . . . . . . . . . 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 9567 . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 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 2483 . . . . . . . . . . 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 434 . . . . . . . . . 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 2646 . . . . . . . . 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 715 . . . . . . 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 2626 . . . . . 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 434 . . . . 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 2848 . . . 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 603 . . 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 429 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716    \ cdif 3325   ` cfv 5418  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595   NNcn 10322   2c2 10371   ZZcz 10646   ^cexp 11865   sqrcsqr 12722  ◻NNcsquarenn 29177  Pell1234QRcpell1234qr 29179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-pell1234qr 29185
This theorem is referenced by:  pell1234qrreccl  29195  pell14qrne0  29199
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