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Mathbox for Stefan O'Rear |
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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 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)}) |
Colors of variables: wff setvar class |
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 |
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