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Theorem elpell1234qr 35451
 Description: Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell1234qr NN Pell1234QR
Distinct variable groups:   ,,   ,,

Proof of Theorem elpell1234qr
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pell1234qrval 35450 . . 3 NN Pell1234QR
21eleq2d 2490 . 2 NN Pell1234QR
3 eqeq1 2424 . . . . 5
43anbi1d 709 . . . 4
542rexbidv 2944 . . 3
65elrab 3226 . 2
72, 6syl6bb 264 1 NN Pell1234QR
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wceq 1437   wcel 1867  wrex 2774  crab 2777   cdif 3430  cfv 5592  (class class class)co 6296  cr 9527  c1 9529   caddc 9531   cmul 9533   cmin 9849  cn 10598  c2 10648  cz 10926  cexp 12258  csqrt 13264  ◻NNcsquarenn 35434  Pell1234QRcpell1234qr 35436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-cnex 9584  ax-resscn 9585 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-pell1234qr 35443 This theorem is referenced by:  pell1234qrre  35452  pell1234qrne0  35453  pell1234qrreccl  35454  pell1234qrmulcl  35455  pell14qrss1234  35456  pell1234qrdich  35461
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