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Theorem pell1234qrval 37233
Description: Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell1234qrval (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
Distinct variable group:   𝑦,𝑧,𝑤,𝐷

Proof of Theorem pell1234qrval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6178 . . . . . . . 8 (𝑑 = 𝐷 → (√‘𝑑) = (√‘𝐷))
21oveq1d 6650 . . . . . . 7 (𝑑 = 𝐷 → ((√‘𝑑) · 𝑤) = ((√‘𝐷) · 𝑤))
32oveq2d 6651 . . . . . 6 (𝑑 = 𝐷 → (𝑧 + ((√‘𝑑) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤)))
43eqeq2d 2630 . . . . 5 (𝑑 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤))))
5 oveq1 6642 . . . . . . 7 (𝑑 = 𝐷 → (𝑑 · (𝑤↑2)) = (𝐷 · (𝑤↑2)))
65oveq2d 6651 . . . . . 6 (𝑑 = 𝐷 → ((𝑧↑2) − (𝑑 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2))))
76eqeq1d 2622 . . . . 5 (𝑑 = 𝐷 → (((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))
84, 7anbi12d 746 . . . 4 (𝑑 = 𝐷 → ((𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1) ↔ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
982rexbidv 3053 . . 3 (𝑑 = 𝐷 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
109rabbidv 3184 . 2 (𝑑 = 𝐷 → {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1)} = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
11 df-pell1234qr 37227 . 2 Pell1234QR = (𝑑 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1)})
12 reex 10012 . . 3 ℝ ∈ V
1312rabex 4804 . 2 {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V
1410, 11, 13fvmpt 6269 1 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wrex 2910  {crab 2913  cdif 3564  cfv 5876  (class class class)co 6635  cr 9920  1c1 9922   + caddc 9924   · cmul 9926  cmin 10251  cn 11005  2c2 11055  cz 11362  cexp 12843  csqrt 13954  NNcsquarenn 37219  Pell1234QRcpell1234qr 37221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-cnex 9977  ax-resscn 9978
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-pell1234qr 37227
This theorem is referenced by:  elpell1234qr  37234
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