Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundgt1 | Structured version Visualization version GIF version |
Description: Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) |
Ref | Expression |
---|---|
pellfundgt1 | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 10817 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ∈ ℝ) | |
2 | eldifi 4031 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ) | |
3 | 2 | peano2nnd 11830 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ) |
4 | 3 | nnrpd 12609 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ+) |
5 | 4 | rpsqrtcld 14958 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ+) |
6 | 5 | rpred 12611 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ) |
7 | 2 | nnrpd 12609 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ+) |
8 | 7 | rpsqrtcld 14958 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ+) |
9 | 8 | rpred 12611 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ) |
10 | 6, 9 | readdcld 10845 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ∈ ℝ) |
11 | pellfundre 40358 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ) | |
12 | sqrt1 14818 | . . . . 5 ⊢ (√‘1) = 1 | |
13 | 12, 1 | eqeltrid 2838 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ∈ ℝ) |
14 | 13, 13 | readdcld 10845 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘1) + (√‘1)) ∈ ℝ) |
15 | 1lt2 11984 | . . . . 5 ⊢ 1 < 2 | |
16 | 12, 12 | oveq12i 7214 | . . . . . 6 ⊢ ((√‘1) + (√‘1)) = (1 + 1) |
17 | 1p1e2 11938 | . . . . . 6 ⊢ (1 + 1) = 2 | |
18 | 16, 17 | eqtri 2762 | . . . . 5 ⊢ ((√‘1) + (√‘1)) = 2 |
19 | 15, 18 | breqtrri 5070 | . . . 4 ⊢ 1 < ((√‘1) + (√‘1)) |
20 | 19 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < ((√‘1) + (√‘1))) |
21 | 3 | nnge1d 11861 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ≤ (𝐷 + 1)) |
22 | 0le1 11338 | . . . . . . 7 ⊢ 0 ≤ 1 | |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ 1) |
24 | 2 | nnred 11828 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ) |
25 | peano2re 10988 | . . . . . . 7 ⊢ (𝐷 ∈ ℝ → (𝐷 + 1) ∈ ℝ) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ) |
27 | 3 | nnnn0d 12133 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ0) |
28 | 27 | nn0ge0d 12136 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ (𝐷 + 1)) |
29 | 1, 23, 26, 28 | sqrtled 14973 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (1 ≤ (𝐷 + 1) ↔ (√‘1) ≤ (√‘(𝐷 + 1)))) |
30 | 21, 29 | mpbid 235 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ≤ (√‘(𝐷 + 1))) |
31 | 2 | nnge1d 11861 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ≤ 𝐷) |
32 | 2 | nnnn0d 12133 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ0) |
33 | 32 | nn0ge0d 12136 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ 𝐷) |
34 | 1, 23, 24, 33 | sqrtled 14973 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (1 ≤ 𝐷 ↔ (√‘1) ≤ (√‘𝐷))) |
35 | 31, 34 | mpbid 235 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ≤ (√‘𝐷)) |
36 | 13, 13, 6, 9, 30, 35 | le2addd 11434 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘1) + (√‘1)) ≤ ((√‘(𝐷 + 1)) + (√‘𝐷))) |
37 | 1, 14, 10, 20, 36 | ltletrd 10975 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < ((√‘(𝐷 + 1)) + (√‘𝐷))) |
38 | pellfundge 40359 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷)) | |
39 | 1, 10, 11, 37, 38 | ltletrd 10975 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∖ cdif 3854 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 < clt 10850 ≤ cle 10851 ℕcn 11813 2c2 11868 √csqrt 14779 ◻NNcsquarenn 40313 PellFundcpellfund 40317 |
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-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 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-int 4850 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-se 5499 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-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-oadd 8195 df-omul 8196 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-card 9538 df-acn 9541 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-xnn0 12146 df-z 12160 df-uz 12422 df-q 12528 df-rp 12570 df-ico 12924 df-fz 13079 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-hash 13880 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-dvds 15797 df-gcd 16035 df-numer 16272 df-denom 16273 df-squarenn 40318 df-pell1qr 40319 df-pell14qr 40320 df-pell1234qr 40321 df-pellfund 40322 |
This theorem is referenced by: pellfundex 40363 pellfundrp 40365 pellfundne1 40366 pellfund14 40375 |
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