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Mathbox for Stefan O'Rear |
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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 | β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β π΄ = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3r 1203 | . . . . 5 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β π΄ < (PellFundβπ·)) | |
2 | pell14qrre 41227 | . . . . . . 7 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·)) β π΄ β β) | |
3 | 2 | 3adant3 1133 | . . . . . 6 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β π΄ β β) |
4 | pellfundre 41251 | . . . . . . 7 β’ (π· β (β β β»NN) β (PellFundβπ·) β β) | |
5 | 4 | 3ad2ant1 1134 | . . . . . 6 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β (PellFundβπ·) β β) |
6 | 3, 5 | ltnled 11310 | . . . . 5 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β (π΄ < (PellFundβπ·) β Β¬ (PellFundβπ·) β€ π΄)) |
7 | 1, 6 | mpbid 231 | . . . 4 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β Β¬ (PellFundβπ·) β€ π΄) |
8 | simpl1 1192 | . . . . 5 β’ (((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β§ 1 < π΄) β π· β (β β β»NN)) | |
9 | simpl2 1193 | . . . . 5 β’ (((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β§ 1 < π΄) β π΄ β (Pell14QRβπ·)) | |
10 | simpr 486 | . . . . 5 β’ (((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β§ 1 < π΄) β 1 < π΄) | |
11 | pellfundlb 41254 | . . . . 5 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ 1 < π΄) β (PellFundβπ·) β€ π΄) | |
12 | 8, 9, 10, 11 | syl3anc 1372 | . . . 4 β’ (((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β§ 1 < π΄) β (PellFundβπ·) β€ π΄) |
13 | 7, 12 | mtand 815 | . . 3 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β Β¬ 1 < π΄) |
14 | simp3l 1202 | . . . 4 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β 1 β€ π΄) | |
15 | 1re 11163 | . . . . 5 β’ 1 β β | |
16 | leloe 11249 | . . . . 5 β’ ((1 β β β§ π΄ β β) β (1 β€ π΄ β (1 < π΄ β¨ 1 = π΄))) | |
17 | 15, 3, 16 | sylancr 588 | . . . 4 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β (1 β€ π΄ β (1 < π΄ β¨ 1 = π΄))) |
18 | 14, 17 | mpbid 231 | . . 3 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β (1 < π΄ β¨ 1 = π΄)) |
19 | orel1 888 | . . 3 β’ (Β¬ 1 < π΄ β ((1 < π΄ β¨ 1 = π΄) β 1 = π΄)) | |
20 | 13, 18, 19 | sylc 65 | . 2 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β 1 = π΄) |
21 | 20 | eqcomd 2739 | 1 β’ ((π· β (β β β»NN) β§ π΄ β (Pell14QRβπ·) β§ (1 β€ π΄ β§ π΄ < (PellFundβπ·))) β π΄ = 1) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 397 β¨ wo 846 β§ w3a 1088 = wceq 1542 β wcel 2107 β cdif 3911 class class class wbr 5109 βcfv 6500 βcr 11058 1c1 11060 < clt 11197 β€ cle 11198 βcn 12161 β»NNcsquarenn 41206 Pell14QRcpell14qr 41209 PellFundcpellfund 41210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-oadd 8420 df-omul 8421 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-acn 9886 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-xnn0 12494 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-ico 13279 df-fz 13434 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-dvds 16145 df-gcd 16383 df-numer 16618 df-denom 16619 df-squarenn 41211 df-pell1qr 41212 df-pell14qr 41213 df-pell1234qr 41214 df-pellfund 41215 |
This theorem is referenced by: pellfund14 41268 |
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