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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pell1qrval | Structured version Visualization version GIF version |
Description: Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.) |
Ref | Expression |
---|---|
pell1qrval | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6437 | . . . . . . . 8 ⊢ (𝑎 = 𝐷 → (√‘𝑎) = (√‘𝐷)) | |
2 | 1 | oveq1d 6925 | . . . . . . 7 ⊢ (𝑎 = 𝐷 → ((√‘𝑎) · 𝑤) = ((√‘𝐷) · 𝑤)) |
3 | 2 | oveq2d 6926 | . . . . . 6 ⊢ (𝑎 = 𝐷 → (𝑧 + ((√‘𝑎) · 𝑤)) = (𝑧 + ((√‘𝐷) · 𝑤))) |
4 | 3 | eqeq2d 2835 | . . . . 5 ⊢ (𝑎 = 𝐷 → (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ↔ 𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)))) |
5 | oveq1 6917 | . . . . . . 7 ⊢ (𝑎 = 𝐷 → (𝑎 · (𝑤↑2)) = (𝐷 · (𝑤↑2))) | |
6 | 5 | oveq2d 6926 | . . . . . 6 ⊢ (𝑎 = 𝐷 → ((𝑧↑2) − (𝑎 · (𝑤↑2))) = ((𝑧↑2) − (𝐷 · (𝑤↑2)))) |
7 | 6 | eqeq1d 2827 | . . . . 5 ⊢ (𝑎 = 𝐷 → (((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1 ↔ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)) |
8 | 4, 7 | anbi12d 624 | . . . 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 3267 | . . 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 3402 | . 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 38245 | . 2 ⊢ Pell1QR = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑎) · 𝑤)) ∧ ((𝑧↑2) − (𝑎 · (𝑤↑2))) = 1)}) | |
12 | reex 10350 | . . 3 ⊢ ℝ ∈ V | |
13 | 12 | rabex 5039 | . 2 ⊢ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)} ∈ V |
14 | 10, 11, 13 | fvmpt 6533 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0 ∃𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∃wrex 3118 {crab 3121 ∖ cdif 3795 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 1c1 10260 + caddc 10262 · cmul 10264 − cmin 10592 ℕcn 11357 2c2 11413 ℕ0cn0 11625 ↑cexp 13161 √csqrt 14357 ◻NNcsquarenn 38239 Pell1QRcpell1qr 38240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-cnex 10315 ax-resscn 10316 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-ov 6913 df-pell1qr 38245 |
This theorem is referenced by: elpell1qr 38250 |
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