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Theorem elpell14qr 38787
Description: Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
elpell14qr (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))
Distinct variable groups:   𝑧,𝑤,𝐷   𝑧,𝐴,𝑤

Proof of Theorem elpell14qr
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 pell14qrval 38786 . . 3 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
21eleq2d 2845 . 2 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ 𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}))
3 eqeq1 2776 . . . . 5 (𝑎 = 𝐴 → (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ↔ 𝐴 = (𝑧 + ((√‘𝐷) · 𝑤))))
43anbi1d 620 . . . 4 (𝑎 = 𝐴 → ((𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
542rexbidv 3239 . . 3 (𝑎 = 𝐴 → (∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
65elrab 3589 . 2 (𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
72, 6syl6bb 279 1 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wrex 3083  {crab 3086  cdif 3822  cfv 6182  (class class class)co 6970  cr 10326  1c1 10328   + caddc 10330   · cmul 10332  cmin 10662  cn 11431  2c2 11488  0cn0 11700  cz 11786  cexp 13237  csqrt 14443  NNcsquarenn 38774  Pell14QRcpell14qr 38777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-cnex 10383  ax-resscn 10384
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-iota 6146  df-fun 6184  df-fv 6190  df-ov 6973  df-pell14qr 38781
This theorem is referenced by:  pell14qrss1234  38794  pell14qrgt0  38797  pell1234qrdich  38799  pell1qrss14  38806  pell14qrdich  38807  rmxycomplete  38855
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