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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundlb | Structured version Visualization version GIF version |
Description: A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.) |
Ref | Expression |
---|---|
pellfundlb | ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pellfundval 37444 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) | |
2 | 1 | 3ad2ant1 1082 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) |
3 | ssrab2 3687 | . . . . 5 ⊢ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷) | |
4 | pell14qrre 37421 | . . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑑 ∈ (Pell14QR‘𝐷)) → 𝑑 ∈ ℝ) | |
5 | 4 | ex 450 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑑 ∈ (Pell14QR‘𝐷) → 𝑑 ∈ ℝ)) |
6 | 5 | ssrdv 3609 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ) |
7 | 3, 6 | syl5ss 3614 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ) |
8 | 7 | 3ad2ant1 1082 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ) |
9 | 1re 10039 | . . . 4 ⊢ 1 ∈ ℝ | |
10 | breq2 4657 | . . . . . . . 8 ⊢ (𝑎 = 𝑐 → (1 < 𝑎 ↔ 1 < 𝑐)) | |
11 | 10 | elrab 3363 | . . . . . . 7 ⊢ (𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑐 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑐)) |
12 | pell14qrre 37421 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑐 ∈ (Pell14QR‘𝐷)) → 𝑐 ∈ ℝ) | |
13 | ltle 10126 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (1 < 𝑐 → 1 ≤ 𝑐)) | |
14 | 9, 12, 13 | sylancr 695 | . . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑐 ∈ (Pell14QR‘𝐷)) → (1 < 𝑐 → 1 ≤ 𝑐)) |
15 | 14 | expimpd 629 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((𝑐 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑐) → 1 ≤ 𝑐)) |
16 | 11, 15 | syl5bi 232 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} → 1 ≤ 𝑐)) |
17 | 16 | ralrimiv 2965 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐) |
18 | 17 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐) |
19 | breq1 4656 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏 ≤ 𝑐 ↔ 1 ≤ 𝑐)) | |
20 | 19 | ralbidv 2986 | . . . . 5 ⊢ (𝑏 = 1 → (∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐 ↔ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐)) |
21 | 20 | rspcev 3309 | . . . 4 ⊢ ((1 ∈ ℝ ∧ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐) → ∃𝑏 ∈ ℝ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐) |
22 | 9, 18, 21 | sylancr 695 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ∃𝑏 ∈ ℝ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐) |
23 | simp2 1062 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 𝐴 ∈ (Pell14QR‘𝐷)) | |
24 | simp3 1063 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 1 < 𝐴) | |
25 | breq2 4657 | . . . . 5 ⊢ (𝑎 = 𝐴 → (1 < 𝑎 ↔ 1 < 𝐴)) | |
26 | 25 | elrab 3363 | . . . 4 ⊢ (𝐴 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴)) |
27 | 23, 24, 26 | sylanbrc 698 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 𝐴 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) |
28 | infrelb 11008 | . . 3 ⊢ (({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ ∧ ∃𝑏 ∈ ℝ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐 ∧ 𝐴 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) ≤ 𝐴) | |
29 | 8, 22, 27, 28 | syl3anc 1326 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) ≤ 𝐴) |
30 | 2, 29 | eqbrtrd 4675 | 1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 ∖ cdif 3571 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 infcinf 8347 ℝcr 9935 1c1 9937 < clt 10074 ≤ cle 10075 ℕcn 11020 ◻NNcsquarenn 37400 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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 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-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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-pell14qr 37407 df-pell1234qr 37408 df-pellfund 37409 |
This theorem is referenced by: pellfundglb 37449 pellfund14gap 37451 rmspecfund 37474 |
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