Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pell1234qrval Structured version   Visualization version   GIF version

Theorem pell1234qrval 39325
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 6663 . . . . . . . 8 (𝑑 = 𝐷 → (√‘𝑑) = (√‘𝐷))
21oveq1d 7160 . . . . . . 7 (𝑑 = 𝐷 → ((√‘𝑑) · 𝑤) = ((√‘𝐷) · 𝑤))
32oveq2d 7161 . . . . . 6 (𝑑 = 𝐷 → (𝑧 + ((√‘𝑑) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤)))
43eqeq2d 2829 . . . . 5 (𝑑 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤))))
5 oveq1 7152 . . . . . . 7 (𝑑 = 𝐷 → (𝑑 · (𝑤↑2)) = (𝐷 · (𝑤↑2)))
65oveq2d 7161 . . . . . 6 (𝑑 = 𝐷 → ((𝑧↑2) − (𝑑 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2))))
76eqeq1d 2820 . . . . 5 (𝑑 = 𝐷 → (((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))
84, 7anbi12d 630 . . . 4 (𝑑 = 𝐷 → ((𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1) ↔ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
982rexbidv 3297 . . 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 3478 . 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 39319 . 2 Pell1234QR = (𝑑 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1)})
12 reex 10616 . . 3 ℝ ∈ V
1312rabex 5226 . 2 {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V
1410, 11, 13fvmpt 6761 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 396   = wceq 1528  wcel 2105  wrex 3136  {crab 3139  cdif 3930  cfv 6348  (class class class)co 7145  cr 10524  1c1 10526   + caddc 10528   · cmul 10530  cmin 10858  cn 11626  2c2 11680  cz 11969  cexp 13417  csqrt 14580  NNcsquarenn 39311  Pell1234QRcpell1234qr 39313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-pell1234qr 39319
This theorem is referenced by:  elpell1234qr  39326
  Copyright terms: Public domain W3C validator