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Theorem pell1234qrre 42840
Description: General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell1234qrre ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)

Proof of Theorem pell1234qrre
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpell1234qr 42839 . 2 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1234QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℤ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1))))
21simprbda 498 1 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  cdif 3960  cfv 6563  (class class class)co 7431  cr 11152  1c1 11154   + caddc 11156   · cmul 11158  cmin 11490  cn 12264  2c2 12319  cz 12611  cexp 14099  csqrt 15269  NNcsquarenn 42824  Pell1234QRcpell1234qr 42826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-pell1234qr 42832
This theorem is referenced by:  pell1234qrreccl  42842  pell14qrre  42845  elpell14qr2  42850  pell14qrmulcl  42851  pell14qrreccl  42852
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