Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pell1qrgap | Structured version Visualization version GIF version |
Description: First-quadrant Pell solutions are bounded away from 1. (This particular bound allows us to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.) |
Ref | Expression |
---|---|
pell1qrgap | ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpell1qr 40324 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)))) | |
2 | 1 | adantr 484 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)))) |
3 | eldifi 4031 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ) | |
4 | 3 | ad4antr 732 | . . . . . . . . . 10 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → 𝐷 ∈ ℕ) |
5 | simplr 769 | . . . . . . . . . 10 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) | |
6 | simp-4r 784 | . . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → 1 < 𝐴) | |
7 | simprl 771 | . . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → 𝐴 = (𝑎 + ((√‘𝐷) · 𝑏))) | |
8 | 6, 7 | breqtrd 5069 | . . . . . . . . . 10 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → 1 < (𝑎 + ((√‘𝐷) · 𝑏))) |
9 | simprr 773 | . . . . . . . . . 10 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1) | |
10 | pell1qrgaplem 40350 | . . . . . . . . . 10 ⊢ (((𝐷 ∈ ℕ ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (1 < (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (𝑎 + ((√‘𝐷) · 𝑏))) | |
11 | 4, 5, 8, 9, 10 | syl22anc 839 | . . . . . . . . 9 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (𝑎 + ((√‘𝐷) · 𝑏))) |
12 | 11, 7 | breqtrrd 5071 | . . . . . . . 8 ⊢ (((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴) |
13 | 12 | ex 416 | . . . . . . 7 ⊢ ((((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → ((𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)) |
14 | 13 | rexlimdvva 3206 | . . . . . 6 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) ∧ 𝐴 ∈ ℝ) → (∃𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)) |
15 | 14 | expimpd 457 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) → ((𝐴 ∈ ℝ ∧ ∃𝑎 ∈ ℕ0 ∃𝑏 ∈ ℕ0 (𝐴 = (𝑎 + ((√‘𝐷) · 𝑏)) ∧ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)) |
16 | 2, 15 | sylbid 243 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 1 < 𝐴) → (𝐴 ∈ (Pell1QR‘𝐷) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)) |
17 | 16 | ex 416 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (1 < 𝐴 → (𝐴 ∈ (Pell1QR‘𝐷) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴))) |
18 | 17 | com23 86 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) → (1 < 𝐴 → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴))) |
19 | 18 | 3imp 1113 | 1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 ∖ cdif 3854 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 1c1 10713 + caddc 10715 · cmul 10717 < clt 10850 ≤ cle 10851 − cmin 11045 ℕcn 11813 2c2 11868 ℕ0cn0 12073 ↑cexp 13618 √csqrt 14779 ◻NNcsquarenn 40313 Pell1QRcpell1qr 40314 |
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-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 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-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-pell1qr 40319 |
This theorem is referenced by: pell14qrgap 40352 |
Copyright terms: Public domain | W3C validator |