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Theorem elpell1qr 40241
Description: Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell1qr (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))
Distinct variable groups:   𝑧,𝑤,𝐷   𝑧,𝐴,𝑤

Proof of Theorem elpell1qr
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 pell1qrval 40240 . . 3 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
21eleq2d 2818 . 2 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ 𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}))
3 eqeq1 2742 . . . . 5 (𝑎 = 𝐴 → (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ↔ 𝐴 = (𝑧 + ((√‘𝐷) · 𝑤))))
43anbi1d 633 . . . 4 (𝑎 = 𝐴 → ((𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
542rexbidv 3210 . . 3 (𝑎 = 𝐴 → (∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
65elrab 3588 . 2 (𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
72, 6bitrdi 290 1 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wrex 3054  {crab 3057  cdif 3840  cfv 6339  (class class class)co 7170  cr 10614  1c1 10616   + caddc 10618   · cmul 10620  cmin 10948  cn 11716  2c2 11771  0cn0 11976  cexp 13521  csqrt 14682  NNcsquarenn 40230  Pell1QRcpell1qr 40231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-cnex 10671  ax-resscn 10672
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-pell1qr 40236
This theorem is referenced by:  pell1qrss14  40262  pell14qrdich  40263  pell1qrge1  40264  pell1qr1  40265  pell1qrgap  40268  pellqrexplicit  40271
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