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Theorem pell1234qrne0 35770
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 35768 . . 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 772 . . . . . . 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 9626 . . . . . . . . 9  |-  1  =/=  0
4 eldifi 3544 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  e.  ( NN  \NN )  ->  D  e.  NN )
54adantr 472 . . . . . . . . . . . . . . . . . . 19  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  NN )
65nncnd 10647 . . . . . . . . . . . . . . . . . 18  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  RR )  ->  D  e.  CC )
76ad3antrrr 744 . . . . . . . . . . . . . . . . 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 13576 . . . . . . . . . . . . . . . 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 10966 . . . . . . . . . . . . . . . . . 18  |-  ( b  e.  ZZ  ->  b  e.  CC )
109ad2antll 743 . . . . . . . . . . . . . . . . 17  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  b  e.  CC )
1110ad2antrr 740 . . . . . . . . . . . . . . . 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 12466 . . . . . . . . . . . . . . 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 13578 . . . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . . . 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 2506 . . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . 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 10966 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ZZ  ->  a  e.  CC )
1817ad2antrl 742 . . . . . . . . . . . . . . 15  |-  ( ( ( D  e.  ( NN  \NN )  /\  A  e.  RR )  /\  (
a  e.  ZZ  /\  b  e.  ZZ )
)  ->  a  e.  CC )
1918ad2antrr 740 . . . . . . . . . . . . . 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 9681 . . . . . . . . . . . . . 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 12420 . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . 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 2505 . . . . . . . . . . . 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 770 . . . . . . . . . . . 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 468 . . . . . . . . . . . . . 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 6323 . . . . . . . . . . . . 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 10005 . . . . . . . . . . . . . 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 9849 . . . . . . . . . . . . 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 2505 . . . . . . . . . . . 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 2513 . . . . . . . . . . 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 441 . . . . . . . . . 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 2664 . . . . . . . . 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 730 . . . . . . 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 2710 . . . . . 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 441 . . . . 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 2878 . . . 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 614 . . 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 223 . 2  |-  ( D  e.  ( NN  \NN )  -> 
( A  e.  (Pell1234QR `  D )  ->  A  =/=  0 ) )
4039imp 436 1  |-  ( ( D  e.  ( NN 
\NN )  /\  A  e.  (Pell1234QR `  D ) )  ->  A  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    \ cdif 3387   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880   NNcn 10631   2c2 10681   ZZcz 10961   ^cexp 12310   sqrcsqrt 13373  ◻NNcsquarenn 35751  Pell1234QRcpell1234qr 35753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-pell1234qr 35760
This theorem is referenced by:  pell1234qrreccl  35771  pell14qrne0  35775
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