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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfund14gap | Structured version Visualization version GIF version |
Description: There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) |
Ref | Expression |
---|---|
pellfund14gap | ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 𝐴 = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3r 1200 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 𝐴 < (PellFund‘𝐷)) | |
2 | pell14qrre 40702 | . . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ) | |
3 | 2 | 3adant3 1130 | . . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 𝐴 ∈ ℝ) |
4 | pellfundre 40726 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ) | |
5 | 4 | 3ad2ant1 1131 | . . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → (PellFund‘𝐷) ∈ ℝ) |
6 | 3, 5 | ltnled 11150 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → (𝐴 < (PellFund‘𝐷) ↔ ¬ (PellFund‘𝐷) ≤ 𝐴)) |
7 | 1, 6 | mpbid 231 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → ¬ (PellFund‘𝐷) ≤ 𝐴) |
8 | simpl1 1189 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) ∧ 1 < 𝐴) → 𝐷 ∈ (ℕ ∖ ◻NN)) | |
9 | simpl2 1190 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) ∧ 1 < 𝐴) → 𝐴 ∈ (Pell14QR‘𝐷)) | |
10 | simpr 484 | . . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) ∧ 1 < 𝐴) → 1 < 𝐴) | |
11 | pellfundlb 40729 | . . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴) | |
12 | 8, 9, 10, 11 | syl3anc 1369 | . . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴) |
13 | 7, 12 | mtand 812 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → ¬ 1 < 𝐴) |
14 | simp3l 1199 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 1 ≤ 𝐴) | |
15 | 1re 11003 | . . . . 5 ⊢ 1 ∈ ℝ | |
16 | leloe 11089 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (1 ≤ 𝐴 ↔ (1 < 𝐴 ∨ 1 = 𝐴))) | |
17 | 15, 3, 16 | sylancr 586 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → (1 ≤ 𝐴 ↔ (1 < 𝐴 ∨ 1 = 𝐴))) |
18 | 14, 17 | mpbid 231 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → (1 < 𝐴 ∨ 1 = 𝐴)) |
19 | orel1 885 | . . 3 ⊢ (¬ 1 < 𝐴 → ((1 < 𝐴 ∨ 1 = 𝐴) → 1 = 𝐴)) | |
20 | 13, 18, 19 | sylc 65 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 1 = 𝐴) |
21 | 20 | eqcomd 2739 | 1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴 ∧ 𝐴 < (PellFund‘𝐷))) → 𝐴 = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1537 ∈ wcel 2101 ∖ cdif 3886 class class class wbr 5077 ‘cfv 6447 ℝcr 10898 1c1 10900 < clt 11037 ≤ cle 11038 ℕcn 12001 ◻NNcsquarenn 40681 Pell14QRcpell14qr 40684 PellFundcpellfund 40685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-om 7733 df-1st 7851 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-oadd 8321 df-omul 8322 df-er 8518 df-map 8637 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-sup 9229 df-inf 9230 df-oi 9297 df-card 9725 df-acn 9728 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-n0 12262 df-xnn0 12334 df-z 12348 df-uz 12611 df-q 12717 df-rp 12759 df-ico 13113 df-fz 13268 df-fl 13540 df-mod 13618 df-seq 13750 df-exp 13811 df-hash 14073 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-dvds 15992 df-gcd 16230 df-numer 16467 df-denom 16468 df-squarenn 40686 df-pell1qr 40687 df-pell14qr 40688 df-pell1234qr 40689 df-pellfund 40690 |
This theorem is referenced by: pellfund14 40743 |
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