Theorem elpell1qr 44045

Description: Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.)

Assertion

Ref	Expression
elpell1qr	⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

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

Proof of Theorem elpell1qr

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pell1qrval 44044	. . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
2	1	eleq2d 2850	. 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ 𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}))
3		eqeq1 2767	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ↔ 𝐴 = (𝑧 + ((√‘𝐷) · 𝑤))))
4	3	anbi1d 643	. . . 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 3231	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
6	5	elrab 3648	. 2 ⊢ (𝐴 ∈ {𝑎 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑎 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
7	2, 6	bitrdi 290	1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416   ∖ cdif 4048  ‘cfv 6555  (class class class)co 7440  ℝcr 11223  1c1 11225   + caddc 11227   · cmul 11229   − cmin 11565  ℕcn 12359  2c2 12421  ℕ₀cn0 12630  ↑cexp 14228  √csqrt 15426  ◻_NNcsquarenn 44034  Pell1QRcpell1qr 44035

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-cnex 11280  ax-resscn 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563  df-ov 7443  df-pell1qr 44040

This theorem is used by:  pell1qrss14  44066  pell14qrdich  44067  pell1qrge1  44068  pell1qr1  44069  pell1qrgap  44072  pellqrexplicit  44075