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Mirrors > Home > MPE Home > Th. List > Mathboxes > pell1234qrval | Structured version Visualization version GIF version |
Description: Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
Ref | Expression |
---|---|
pell1234qrval | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑑 = 𝐷 → (√‘𝑑) = (√‘𝐷)) | |
2 | 1 | oveq1d 7165 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → ((√‘𝑑) · 𝑤) = ((√‘𝐷) · 𝑤)) |
3 | 2 | oveq2d 7166 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑧 + ((√‘𝑑) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤))) |
4 | 3 | eqeq2d 2832 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)))) |
5 | oveq1 7157 | . . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑑 · (𝑤↑2)) = (𝐷 · (𝑤↑2))) | |
6 | 5 | oveq2d 7166 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑧↑2) − (𝑑 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2)))) |
7 | 6 | eqeq1d 2823 | . . . . 5 ⊢ (𝑑 = 𝐷 → (((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)) |
8 | 4, 7 | anbi12d 632 | . . . 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 3300 | . . 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 3480 | . 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 39434 | . 2 ⊢ Pell1234QR = (𝑑 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑑) · 𝑤)) ∧ ((𝑧↑2) − (𝑑 · (𝑤↑2))) = 1)}) | |
12 | reex 10622 | . . 3 ⊢ ℝ ∈ V | |
13 | 12 | rabex 5227 | . 2 ⊢ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V |
14 | 10, 11, 13 | fvmpt 6762 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {crab 3142 ∖ cdif 3932 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 ℕcn 11632 2c2 11686 ℤcz 11975 ↑cexp 13423 √csqrt 14586 ◻NNcsquarenn 39426 Pell1234QRcpell1234qr 39428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-pell1234qr 39434 |
This theorem is referenced by: elpell1234qr 39441 |
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