Theorem pell1234qrval 38932

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

Assertion

Ref	Expression
pell1234qrval	⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Distinct variable group:   𝑦,𝑧,𝑤,𝐷

Proof of Theorem pell1234qrval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6538	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (√‘𝑑) = (√‘𝐷))
2	1	oveq1d 7031	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ((√‘𝑑) · 𝑤) = ((√‘𝐷) · 𝑤))
3	2	oveq2d 7032	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑧 + ((√‘𝑑) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤)))
4	3	eqeq2d 2805	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤))))
5		oveq1 7023	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑑 · (𝑤↑2)) = (𝐷 · (𝑤↑2)))
6	5	oveq2d 7032	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑧↑2) − (𝑑 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2))))
7	6	eqeq1d 2797	. . . . 5 ⊢ (𝑑 = 𝐷 → (((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))
8	4, 7	anbi12d 630	. . . 4 ⊢ (𝑑 = 𝐷 → ((𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1) ↔ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
9	8	2rexbidv 3263	. . 3 ⊢ (𝑑 = 𝐷 → (∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)))
10	9	rabbidv 3425	. 2 ⊢ (𝑑 = 𝐷 → {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1)} = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})
11		df-pell1234qr 38926	. 2 ⊢ Pell1234QR = (𝑑 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1)})
12		reex 10474	. . 3 ⊢ ℝ ∈ V
13	12	rabex 5126	. 2 ⊢ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V
14	10, 11, 13	fvmpt 6635	1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∃wrex 3106  {crab 3109   ∖ cdif 3856  ‘cfv 6225  (class class class)co 7016  ℝcr 10382  1c1 10384   + caddc 10386   · cmul 10388   − cmin 10717  ℕcn 11486  2c2 11540  ℤcz 11829  ↑cexp 13279  √csqrt 14426  ◻_NNcsquarenn 38918  Pell1234QRcpell1234qr 38920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221  ax-cnex 10439  ax-resscn 10440

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-pell1234qr 38926

This theorem is referenced by:  elpell1234qr  38933