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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmxyelqirr | Structured version Visualization version GIF version |
Description: The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by SN, 23-Dec-2024.) |
Ref | Expression |
---|---|
rmxyelqirr | β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ + (ββ((π΄β2) β 1)))βπ) β {π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmspecnonsq 41277 | . . . . 5 β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β (β β β»NN)) | |
2 | 1 | adantr 482 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄β2) β 1) β (β β β»NN)) |
3 | pell14qrval 41218 | . . . 4 β’ (((π΄β2) β 1) β (β β β»NN) β (Pell14QRβ((π΄β2) β 1)) = {π β β β£ βπ β β0 βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1)}) | |
4 | 2, 3 | syl 17 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (Pell14QRβ((π΄β2) β 1)) = {π β β β£ βπ β β0 βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1)}) |
5 | rabssab 4047 | . . . 4 β’ {π β β β£ βπ β β0 βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1)} β {π β£ βπ β β0 βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1)} | |
6 | simpl 484 | . . . . . . 7 β’ ((π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1) β π = (π + ((ββ((π΄β2) β 1)) Β· π))) | |
7 | 6 | reximi 3084 | . . . . . 6 β’ (βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1) β βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))) |
8 | 7 | reximi 3084 | . . . . 5 β’ (βπ β β0 βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1) β βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))) |
9 | 8 | ss2abi 4027 | . . . 4 β’ {π β£ βπ β β0 βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1)} β {π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))} |
10 | 5, 9 | sstri 3957 | . . 3 β’ {π β β β£ βπ β β0 βπ β β€ (π = (π + ((ββ((π΄β2) β 1)) Β· π)) β§ ((πβ2) β (((π΄β2) β 1) Β· (πβ2))) = 1)} β {π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))} |
11 | 4, 10 | eqsstrdi 4002 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (Pell14QRβ((π΄β2) β 1)) β {π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))}) |
12 | simpr 486 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β π β β€) | |
13 | rmspecfund 41279 | . . . . . . 7 β’ (π΄ β (β€β₯β2) β (PellFundβ((π΄β2) β 1)) = (π΄ + (ββ((π΄β2) β 1)))) | |
14 | 13 | adantr 482 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (PellFundβ((π΄β2) β 1)) = (π΄ + (ββ((π΄β2) β 1)))) |
15 | 14 | eqcomd 2739 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ + (ββ((π΄β2) β 1))) = (PellFundβ((π΄β2) β 1))) |
16 | 15 | oveq1d 7376 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ + (ββ((π΄β2) β 1)))βπ) = ((PellFundβ((π΄β2) β 1))βπ)) |
17 | oveq2 7369 | . . . . 5 β’ (π = π β ((PellFundβ((π΄β2) β 1))βπ) = ((PellFundβ((π΄β2) β 1))βπ)) | |
18 | 17 | rspceeqv 3599 | . . . 4 β’ ((π β β€ β§ ((π΄ + (ββ((π΄β2) β 1)))βπ) = ((PellFundβ((π΄β2) β 1))βπ)) β βπ β β€ ((π΄ + (ββ((π΄β2) β 1)))βπ) = ((PellFundβ((π΄β2) β 1))βπ)) |
19 | 12, 16, 18 | syl2anc 585 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β βπ β β€ ((π΄ + (ββ((π΄β2) β 1)))βπ) = ((PellFundβ((π΄β2) β 1))βπ)) |
20 | pellfund14b 41269 | . . . 4 β’ (((π΄β2) β 1) β (β β β»NN) β (((π΄ + (ββ((π΄β2) β 1)))βπ) β (Pell14QRβ((π΄β2) β 1)) β βπ β β€ ((π΄ + (ββ((π΄β2) β 1)))βπ) = ((PellFundβ((π΄β2) β 1))βπ))) | |
21 | 2, 20 | syl 17 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ + (ββ((π΄β2) β 1)))βπ) β (Pell14QRβ((π΄β2) β 1)) β βπ β β€ ((π΄ + (ββ((π΄β2) β 1)))βπ) = ((PellFundβ((π΄β2) β 1))βπ))) |
22 | 19, 21 | mpbird 257 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ + (ββ((π΄β2) β 1)))βπ) β (Pell14QRβ((π΄β2) β 1))) |
23 | 11, 22 | sseldd 3949 | 1 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ + (ββ((π΄β2) β 1)))βπ) β {π β£ βπ β β0 βπ β β€ π = (π + ((ββ((π΄β2) β 1)) Β· π))}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 {cab 2710 βwrex 3070 {crab 3406 β cdif 3911 βcfv 6500 (class class class)co 7361 βcr 11058 1c1 11060 + caddc 11062 Β· cmul 11064 β cmin 11393 βcn 12161 2c2 12216 β0cn0 12421 β€cz 12507 β€β₯cuz 12771 βcexp 13976 βcsqrt 15127 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 ax-addf 11138 ax-mulf 11139 |
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-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 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-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-oadd 8420 df-omul 8421 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 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-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-xnn0 12494 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-fac 14183 df-bc 14212 df-hash 14240 df-shft 14961 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15362 df-clim 15379 df-rlim 15380 df-sum 15580 df-ef 15958 df-sin 15960 df-cos 15961 df-pi 15963 df-dvds 16145 df-gcd 16383 df-numer 16618 df-denom 16619 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-fbas 20816 df-fg 20817 df-cnfld 20820 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-nei 22472 df-lp 22510 df-perf 22511 df-cn 22601 df-cnp 22602 df-haus 22689 df-tx 22936 df-hmeo 23129 df-fil 23220 df-fm 23312 df-flim 23313 df-flf 23314 df-xms 23696 df-ms 23697 df-tms 23698 df-cncf 24264 df-limc 25253 df-dv 25254 df-log 25935 df-squarenn 41211 df-pell1qr 41212 df-pell14qr 41213 df-pell1234qr 41214 df-pellfund 41215 |
This theorem is referenced by: rmxyelxp 41283 rmxyval 41286 |
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