Theorem elpell14qr 39790

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 ∈ ℕ0 ∃𝑤 ∈ 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.

Step	Hyp	Ref	Expression
1		pell14qrval 39789	. . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
2	1	eleq2d 2875	. 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ 𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}))
3		eqeq1 2802	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ↔ 𝐴 = (𝑧 + ((√‘𝐷) · 𝑤))))
4	3	anbi1d 632	. . . 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 3259	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
6	5	elrab 3628	. 2 ⊢ (𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
7	2, 6	syl6bb 290	1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110   ∖ cdif 3878  ‘cfv 6324  (class class class)co 7135  ℝcr 10525  1c1 10527   + caddc 10529   · cmul 10531   − cmin 10859  ℕcn 11625  2c2 11680  ℕ₀cn0 11885  ℤcz 11969  ↑cexp 13425  √csqrt 14584  ◻_NNcsquarenn 39777  Pell14QRcpell14qr 39780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-cnex 10582  ax-resscn 10583

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-pell14qr 39784

This theorem is referenced by:  pell14qrss1234  39797  pell14qrgt0  39800  pell1234qrdich  39802  pell1qrss14  39809  pell14qrdich  39810  rmxycomplete  39858