Theorem pell1qrval 43109

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 ∈ ℕ0 ∃𝑤 ∈ 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.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . . . . . 8 ⊢ (𝑎 = 𝐷 → (√‘𝑎) = (√‘𝐷))
2	1	oveq1d 7373	. . . . . . 7 ⊢ (𝑎 = 𝐷 → ((√‘𝑎) · 𝑤) = ((√‘𝐷) · 𝑤))
3	2	oveq2d 7374	. . . . . 6 ⊢ (𝑎 = 𝐷 → (𝑧 + ((√‘𝑎) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤)))
4	3	eqeq2d 2747	. . . . 5 ⊢ (𝑎 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤))))
5		oveq1 7365	. . . . . . 7 ⊢ (𝑎 = 𝐷 → (𝑎 · (𝑤↑2)) = (𝐷 · (𝑤↑2)))
6	5	oveq2d 7374	. . . . . 6 ⊢ (𝑎 = 𝐷 → ((𝑧↑2) − (𝑎 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2))))
7	6	eqeq1d 2738	. . . . 5 ⊢ (𝑎 = 𝐷 → (((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))
8	4, 7	anbi12d 632	. . . 4 ⊢ (𝑎 = 𝐷 → ((𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1) ↔ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
9	8	2rexbidv 3201	. . 3 ⊢ (𝑎 = 𝐷 → (∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
10	9	rabbidv 3406	. 2 ⊢ (𝑎 = 𝐷 → {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)} = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
11		df-pell1qr 43105	. 2 ⊢ Pell1QR = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)})
12		reex 11119	. . 3 ⊢ ℝ ∈ V
13	12	rabex 5284	. 2 ⊢ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V
14	10, 11, 13	fvmpt 6941	1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  {crab 3399   ∖ cdif 3898  ‘cfv 6492  (class class class)co 7358  ℝcr 11027  1c1 11029   + caddc 11031   · cmul 11033   − cmin 11366  ℕcn 12147  2c2 12202  ℕ₀cn0 12403  ↑cexp 13986  √csqrt 15158  ◻_NNcsquarenn 43099  Pell1QRcpell1qr 43100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-cnex 11084  ax-resscn 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-pell1qr 43105

This theorem is referenced by:  elpell1qr  43110