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Theorem pell1qrval 36925
 Description: Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Assertion
Ref Expression
pell1qrval (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
Distinct variable group:   𝑦,𝑧,𝑤,𝐷

Proof of Theorem pell1qrval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6153 . . . . . . . 8 (𝑎 = 𝐷 → (√‘𝑎) = (√‘𝐷))
21oveq1d 6625 . . . . . . 7 (𝑎 = 𝐷 → ((√‘𝑎) · 𝑤) = ((√‘𝐷) · 𝑤))
32oveq2d 6626 . . . . . 6 (𝑎 = 𝐷 → (𝑧 + ((√‘𝑎) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤)))
43eqeq2d 2631 . . . . 5 (𝑎 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤))))
5 oveq1 6617 . . . . . . 7 (𝑎 = 𝐷 → (𝑎 · (𝑤↑2)) = (𝐷 · (𝑤↑2)))
65oveq2d 6626 . . . . . 6 (𝑎 = 𝐷 → ((𝑧↑2) − (𝑎 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2))))
76eqeq1d 2623 . . . . 5 (𝑎 = 𝐷 → (((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))
84, 7anbi12d 746 . . . 4 (𝑎 = 𝐷 → ((𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1) ↔ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
982rexbidv 3051 . . 3 (𝑎 = 𝐷 → (∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
109rabbidv 3180 . 2 (𝑎 = 𝐷 → {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)} = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
11 df-pell1qr 36921 . 2 Pell1QR = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)})
12 reex 9979 . . 3 ℝ ∈ V
1312rabex 4778 . 2 {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V
1410, 11, 13fvmpt 6244 1 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  {crab 2911   ∖ cdif 3556  ‘cfv 5852  (class class class)co 6610  ℝcr 9887  1c1 9889   + caddc 9891   · cmul 9893   − cmin 10218  ℕcn 10972  2c2 11022  ℕ0cn0 11244  ↑cexp 12808  √csqrt 13915  ◻NNcsquarenn 36915  Pell1QRcpell1qr 36916 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-cnex 9944  ax-resscn 9945 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-pell1qr 36921 This theorem is referenced by:  elpell1qr  36926
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