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Mathbox for Stefan O'Rear |
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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 4073 | . . . 4 ⊢ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷) | |
2 | pell14qrre 41366 | . . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑎 ∈ (Pell14QR‘𝐷)) → 𝑎 ∈ ℝ) | |
3 | 2 | ex 413 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑎 ∈ (Pell14QR‘𝐷) → 𝑎 ∈ ℝ)) |
4 | 3 | ssrdv 3984 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ) |
5 | 1, 4 | sstrid 3989 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ) |
6 | pell1qrss14 41377 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷)) | |
7 | pellqrex 41388 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎) | |
8 | ssrexv 4047 | . . . . 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 4381 | . . . 4 ⊢ ({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ↔ ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎) | |
11 | 9, 10 | sylibr 233 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅) |
12 | eldifi 4122 | . . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℕ) | |
13 | 12 | peano2nnd 12211 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℕ) |
14 | 13 | nnrpd 12996 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝐷 + 1) ∈ ℝ+) |
15 | 14 | rpsqrtcld 15340 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ+) |
16 | 15 | rpred 12998 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘(𝐷 + 1)) ∈ ℝ) |
17 | 12 | nnrpd 12996 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → 𝐷 ∈ ℝ+) |
18 | 17 | rpsqrtcld 15340 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ+) |
19 | 18 | rpred 12998 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (√‘𝐷) ∈ ℝ) |
20 | 16, 19 | readdcld 11225 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ∈ ℝ) |
21 | breq2 5145 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (1 < 𝑎 ↔ 1 < 𝑏)) | |
22 | 21 | elrab 3679 | . . . . 5 ⊢ (𝑏 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑏 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑏)) |
23 | pell14qrgap 41384 | . . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑏 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑏) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏) | |
24 | 23 | 3expib 1122 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((𝑏 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑏) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏)) |
25 | 22, 24 | biimtrid 241 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑏 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏)) |
26 | 25 | ralrimiv 3144 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∀𝑏 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝑏) |
27 | infmrgelbi 41387 | . . 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 1373 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) |
29 | pellfundval 41389 | . 2 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) | |
30 | 28, 29 | breqtrrd 5169 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3431 ∖ cdif 3941 ⊆ wss 3944 ∅c0 4318 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 infcinf 9418 ℝcr 11091 1c1 11093 + caddc 11095 < clt 11230 ≤ cle 11231 ℕcn 12194 √csqrt 15162 ◻NNcsquarenn 41345 Pell1QRcpell1qr 41346 Pell14QRcpell14qr 41348 PellFundcpellfund 41349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-oadd 8452 df-omul 8453 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-acn 9919 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-xnn0 12527 df-z 12541 df-uz 12805 df-q 12915 df-rp 12957 df-ico 13312 df-fz 13467 df-fl 13739 df-mod 13817 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-dvds 16180 df-gcd 16418 df-numer 16653 df-denom 16654 df-squarenn 41350 df-pell1qr 41351 df-pell14qr 41352 df-pell1234qr 41353 df-pellfund 41354 |
This theorem is referenced by: pellfundgt1 41392 rmspecfund 41418 |
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