Theorem elpell1234qr 38189

Description: Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)

Assertion

Ref	Expression
elpell1234qr	⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1234QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Distinct variable groups:   𝑧,𝑤,𝐷   𝑧,𝐴,𝑤

Proof of Theorem elpell1234qr

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pell1234qrval 38188	. . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
2	1	eleq2d 2862	. 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1234QR‘𝐷) ↔ 𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}))
3		eqeq1 2801	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ↔ 𝐴 = (𝑧 + ((√‘𝐷) · 𝑤))))
4	3	anbi1d 624	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
5	4	2rexbidv 3236	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
6	5	elrab 3554	. 2 ⊢ (𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
7	2, 6	syl6bb 279	1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1234QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∃wrex 3088  {crab 3091   ∖ cdif 3764  ‘cfv 6099  (class class class)co 6876  ℝcr 10221  1c1 10223   + caddc 10225   · cmul 10227   − cmin 10554  ℕcn 11310  2c2 11364  ℤcz 11662  ↑cexp 13110  √csqrt 14311  ◻_NNcsquarenn 38174  Pell1234QRcpell1234qr 38176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-cnex 10278  ax-resscn 10279

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6879  df-pell1234qr 38182

This theorem is referenced by:  pell1234qrre  38190  pell1234qrne0  38191  pell1234qrreccl  38192  pell1234qrmulcl  38193  pell14qrss1234  38194  pell1234qrdich  38199