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 10642 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ∈ ℝ) | |
2 | eldifi 4103 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ) | |
3 | 2 | peano2nnd 11655 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ) |
4 | 3 | nnrpd 12430 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ+) |
5 | 4 | rpsqrtcld 14771 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ+) |
6 | 5 | rpred 12432 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ) |
7 | 2 | nnrpd 12430 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ+) |
8 | 7 | rpsqrtcld 14771 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ+) |
9 | 8 | rpred 12432 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ) |
10 | 6, 9 | readdcld 10670 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ∈ ℝ) |
11 | pellfundre 39498 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ) | |
12 | sqrt1 14631 | . . . . 5 ⊢ (√‘1) = 1 | |
13 | 12, 1 | eqeltrid 2917 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ∈ ℝ) |
14 | 13, 13 | readdcld 10670 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘1) + (√‘1)) ∈ ℝ) |
15 | 1lt2 11809 | . . . . 5 ⊢ 1 < 2 | |
16 | 12, 12 | oveq12i 7168 | . . . . . 6 ⊢ ((√‘1) + (√‘1)) = (1 + 1) |
17 | 1p1e2 11763 | . . . . . 6 ⊢ (1 + 1) = 2 | |
18 | 16, 17 | eqtri 2844 | . . . . 5 ⊢ ((√‘1) + (√‘1)) = 2 |
19 | 15, 18 | breqtrri 5093 | . . . 4 ⊢ 1 < ((√‘1) + (√‘1)) |
20 | 19 | a1i 11 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < ((√‘1) + (√‘1))) |
21 | 3 | nnge1d 11686 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ≤ (𝐷 + 1)) |
22 | 0le1 11163 | . . . . . . 7 ⊢ 0 ≤ 1 | |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ 1) |
24 | 2 | nnred 11653 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ) |
25 | peano2re 10813 | . . . . . . 7 ⊢ (𝐷 ∈ ℝ → (𝐷 + 1) ∈ ℝ) | |
26 | 24, 25 | syl 17 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ) |
27 | 3 | nnnn0d 11956 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ0) |
28 | 27 | nn0ge0d 11959 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ (𝐷 + 1)) |
29 | 1, 23, 26, 28 | sqrtled 14786 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (1 ≤ (𝐷 + 1) ↔ (√‘1) ≤ (√‘(𝐷 + 1)))) |
30 | 21, 29 | mpbid 234 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ≤ (√‘(𝐷 + 1))) |
31 | 2 | nnge1d 11686 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 ≤ 𝐷) |
32 | 2 | nnnn0d 11956 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ0) |
33 | 32 | nn0ge0d 11959 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 0 ≤ 𝐷) |
34 | 1, 23, 24, 33 | sqrtled 14786 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (1 ≤ 𝐷 ↔ (√‘1) ≤ (√‘𝐷))) |
35 | 31, 34 | mpbid 234 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘1) ≤ (√‘𝐷)) |
36 | 13, 13, 6, 9, 30, 35 | le2addd 11259 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘1) + (√‘1)) ≤ ((√‘(𝐷 + 1)) + (√‘𝐷))) |
37 | 1, 14, 10, 20, 36 | ltletrd 10800 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < ((√‘(𝐷 + 1)) + (√‘𝐷))) |
38 | pellfundge 39499 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷)) | |
39 | 1, 10, 11, 37, 38 | ltletrd 10800 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∖ cdif 3933 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 ℕcn 11638 2c2 11693 √csqrt 14592 ◻NNcsquarenn 39453 PellFundcpellfund 39457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-ico 12745 df-fz 12894 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-gcd 15844 df-numer 16075 df-denom 16076 df-squarenn 39458 df-pell1qr 39459 df-pell14qr 39460 df-pell1234qr 39461 df-pellfund 39462 |
This theorem is referenced by: pellfundex 39503 pellfundrp 39505 pellfundne1 39506 pellfund14 39515 |
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