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Mirrors > Home > MPE Home > Th. List > Mathboxes > pell14qrval | Structured version Visualization version GIF version |
Description: Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
Ref | Expression |
---|---|
pell14qrval | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6706 | . . . . . . . 8 ⊢ (𝑎 = 𝐷 → (√‘𝑎) = (√‘𝐷)) | |
2 | 1 | oveq1d 7217 | . . . . . . 7 ⊢ (𝑎 = 𝐷 → ((√‘𝑎) · 𝑤) = ((√‘𝐷) · 𝑤)) |
3 | 2 | oveq2d 7218 | . . . . . 6 ⊢ (𝑎 = 𝐷 → (𝑧 + ((√‘𝑎) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤))) |
4 | 3 | eqeq2d 2745 | . . . . 5 ⊢ (𝑎 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)))) |
5 | oveq1 7209 | . . . . . . 7 ⊢ (𝑎 = 𝐷 → (𝑎 · (𝑤↑2)) = (𝐷 · (𝑤↑2))) | |
6 | 5 | oveq2d 7218 | . . . . . 6 ⊢ (𝑎 = 𝐷 → ((𝑧↑2) − (𝑎 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2)))) |
7 | 6 | eqeq1d 2736 | . . . . 5 ⊢ (𝑎 = 𝐷 → (((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)) |
8 | 4, 7 | anbi12d 634 | . . . 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 3212 | . . 3 ⊢ (𝑎 = 𝐷 → (∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1) ↔ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))) |
10 | 9 | rabbidv 3383 | . 2 ⊢ (𝑎 = 𝐷 → {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)} = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) |
11 | df-pell14qr 40320 | . 2 ⊢ Pell14QR = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)}) | |
12 | reex 10803 | . . 3 ⊢ ℝ ∈ V | |
13 | 12 | rabex 5214 | . 2 ⊢ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V |
14 | 10, 11, 13 | fvmpt 6807 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 {crab 3058 ∖ cdif 3854 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 1c1 10713 + caddc 10715 · cmul 10717 − cmin 11045 ℕcn 11813 2c2 11868 ℕ0cn0 12073 ℤcz 12159 ↑cexp 13618 √csqrt 14779 ◻NNcsquarenn 40313 Pell14QRcpell14qr 40316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-cnex 10768 ax-resscn 10769 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-iota 6327 df-fun 6371 df-fv 6377 df-ov 7205 df-pell14qr 40320 |
This theorem is referenced by: elpell14qr 40326 rmxyelqirr 40387 |
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