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Theorem pell14qrval 38870
Description: Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell14qrval (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
Distinct variable group:   𝑦,𝑧,𝑤,𝐷

Proof of Theorem pell14qrval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6496 . . . . . . . 8 (𝑎 = 𝐷 → (√‘𝑎) = (√‘𝐷))
21oveq1d 6989 . . . . . . 7 (𝑎 = 𝐷 → ((√‘𝑎) · 𝑤) = ((√‘𝐷) · 𝑤))
32oveq2d 6990 . . . . . 6 (𝑎 = 𝐷 → (𝑧 + ((√‘𝑎) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤)))
43eqeq2d 2782 . . . . 5 (𝑎 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤))))
5 oveq1 6981 . . . . . . 7 (𝑎 = 𝐷 → (𝑎 · (𝑤↑2)) = (𝐷 · (𝑤↑2)))
65oveq2d 6990 . . . . . 6 (𝑎 = 𝐷 → ((𝑧↑2) − (𝑎 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2))))
76eqeq1d 2774 . . . . 5 (𝑎 = 𝐷 → (((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))
84, 7anbi12d 621 . . . 4 (𝑎 = 𝐷 → ((𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1) ↔ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
982rexbidv 3239 . . 3 (𝑎 = 𝐷 → (∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
109rabbidv 3397 . 2 (𝑎 = 𝐷 → {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)} = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
11 df-pell14qr 38865 . 2 Pell14QR = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)})
12 reex 10424 . . 3 ℝ ∈ V
1312rabex 5087 . 2 {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V
1410, 11, 13fvmpt 6593 1 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2050  wrex 3083  {crab 3086  cdif 3820  cfv 6185  (class class class)co 6974  cr 10332  1c1 10334   + caddc 10336   · cmul 10338  cmin 10668  cn 11437  2c2 11493  0cn0 11705  cz 11791  cexp 13242  csqrt 14451  NNcsquarenn 38858  Pell14QRcpell14qr 38861
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 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-cnex 10389  ax-resscn 10390
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 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-ov 6977  df-pell14qr 38865
This theorem is referenced by:  elpell14qr  38871  rmxyelqirr  38932
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