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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundge | Structured version Visualization version GIF version |
Description: Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
Ref | Expression |
---|---|
pellfundge | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3687 | . . . 4 ⊢ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷) | |
2 | pell14qrre 37421 | . . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑎 ∈ (Pell14QR‘𝐷)) → 𝑎 ∈ ℝ) | |
3 | 2 | ex 450 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑎 ∈ (Pell14QR‘𝐷) → 𝑎 ∈ ℝ)) |
4 | 3 | ssrdv 3609 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ) |
5 | 1, 4 | syl5ss 3614 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ) |
6 | pell1qrss14 37432 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷)) | |
7 | pellqrex 37443 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎) | |
8 | ssrexv 3667 | . . . . 5 ⊢ ((Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷) → (∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎 → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎)) | |
9 | 6, 7, 8 | sylc 65 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎) |
10 | rabn0 3958 | . . . 4 ⊢ ({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ↔ ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎) | |
11 | 9, 10 | sylibr 224 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅) |
12 | eldifi 3732 | . . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ) | |
13 | 12 | peano2nnd 11037 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ) |
14 | 13 | nnrpd 11870 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ+) |
15 | 14 | rpsqrtcld 14150 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ+) |
16 | 15 | rpred 11872 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ) |
17 | 12 | nnrpd 11870 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ+) |
18 | 17 | rpsqrtcld 14150 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ+) |
19 | 18 | rpred 11872 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ) |
20 | 16, 19 | readdcld 10069 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ∈ ℝ) |
21 | breq2 4657 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (1 < 𝑎 ↔ 1 < 𝑏)) | |
22 | 21 | elrab 3363 | . . . . 5 ⊢ (𝑏 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑏 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑏)) |
23 | pell14qrgap 37439 | . . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑏 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑏) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏) | |
24 | 23 | 3expib 1268 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((𝑏 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑏) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏)) |
25 | 22, 24 | syl5bi 232 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑏 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏)) |
26 | 25 | ralrimiv 2965 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∀𝑏 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏) |
27 | infmrgelbi 37442 | . . 3 ⊢ ((({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ ∧ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ∧ ((√‘(𝐷 + 1)) + (√‘𝐷)) ∈ ℝ) ∧ ∀𝑏 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) | |
28 | 5, 11, 20, 26, 27 | syl31anc 1329 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) |
29 | pellfundval 37444 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) | |
30 | 28, 29 | breqtrrd 4681 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 infcinf 8347 ℝcr 9935 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 ℕcn 11020 √csqrt 13973 ◻NNcsquarenn 37400 Pell1QRcpell1qr 37401 Pell14QRcpell14qr 37403 PellFundcpellfund 37404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-ico 12181 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-numer 15443 df-denom 15444 df-squarenn 37405 df-pell1qr 37406 df-pell14qr 37407 df-pell1234qr 37408 df-pellfund 37409 |
This theorem is referenced by: pellfundgt1 37447 rmspecfund 37474 |
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