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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > jm2.19lem1 | Structured version Visualization version GIF version |
Description: Lemma for jm2.19 42478. X and Y values are coprime. (Contributed by Stefan O'Rear, 23-Sep-2014.) |
Ref | Expression |
---|---|
jm2.19lem1 | β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) gcd (π΄ Yrm π)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmx 42398 | . . . . . . 7 β’ Xrm :((β€β₯β2) Γ β€)βΆβ0 | |
2 | 1 | fovcl 7545 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) β β0) |
3 | 2 | nn0cnd 12562 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) β β) |
4 | 3 | sqcld 14138 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π)β2) β β) |
5 | rmspecnonsq 42391 | . . . . . . . 8 β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β (β β β»NN)) | |
6 | 5 | eldifad 3952 | . . . . . . 7 β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β β) |
7 | 6 | adantr 479 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄β2) β 1) β β) |
8 | 7 | nncnd 12256 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄β2) β 1) β β) |
9 | frmy 42399 | . . . . . . . 8 β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | |
10 | 9 | fovcl 7545 | . . . . . . 7 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm π) β β€) |
11 | 10 | zcnd 12695 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm π) β β) |
12 | 11 | sqcld 14138 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Yrm π)β2) β β) |
13 | 8, 12 | mulcld 11262 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄β2) β 1) Β· ((π΄ Yrm π)β2)) β β) |
14 | 4, 13 | negsubd 11605 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Xrm π)β2) + -(((π΄β2) β 1) Β· ((π΄ Yrm π)β2))) = (((π΄ Xrm π)β2) β (((π΄β2) β 1) Β· ((π΄ Yrm π)β2)))) |
15 | 3 | sqvald 14137 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π)β2) = ((π΄ Xrm π) Β· (π΄ Xrm π))) |
16 | 11 | sqvald 14137 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Yrm π)β2) = ((π΄ Yrm π) Β· (π΄ Yrm π))) |
17 | 16 | oveq2d 7431 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (-((π΄β2) β 1) Β· ((π΄ Yrm π)β2)) = (-((π΄β2) β 1) Β· ((π΄ Yrm π) Β· (π΄ Yrm π)))) |
18 | 8, 12 | mulneg1d 11695 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (-((π΄β2) β 1) Β· ((π΄ Yrm π)β2)) = -(((π΄β2) β 1) Β· ((π΄ Yrm π)β2))) |
19 | nnnegz 12589 | . . . . . . . 8 β’ (((π΄β2) β 1) β β β -((π΄β2) β 1) β β€) | |
20 | 7, 19 | syl 17 | . . . . . . 7 β’ ((π΄ β (β€β₯β2) β§ π β β€) β -((π΄β2) β 1) β β€) |
21 | 20 | zcnd 12695 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β -((π΄β2) β 1) β β) |
22 | 21, 11, 11 | mul12d 11451 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (-((π΄β2) β 1) Β· ((π΄ Yrm π) Β· (π΄ Yrm π))) = ((π΄ Yrm π) Β· (-((π΄β2) β 1) Β· (π΄ Yrm π)))) |
23 | 17, 18, 22 | 3eqtr3d 2773 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β -(((π΄β2) β 1) Β· ((π΄ Yrm π)β2)) = ((π΄ Yrm π) Β· (-((π΄β2) β 1) Β· (π΄ Yrm π)))) |
24 | 15, 23 | oveq12d 7433 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Xrm π)β2) + -(((π΄β2) β 1) Β· ((π΄ Yrm π)β2))) = (((π΄ Xrm π) Β· (π΄ Xrm π)) + ((π΄ Yrm π) Β· (-((π΄β2) β 1) Β· (π΄ Yrm π))))) |
25 | rmxynorm 42403 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Xrm π)β2) β (((π΄β2) β 1) Β· ((π΄ Yrm π)β2))) = 1) | |
26 | 14, 24, 25 | 3eqtr3d 2773 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Xrm π) Β· (π΄ Xrm π)) + ((π΄ Yrm π) Β· (-((π΄β2) β 1) Β· (π΄ Yrm π)))) = 1) |
27 | 2 | nn0zd 12612 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) β β€) |
28 | 20, 10 | zmulcld 12700 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (-((π΄β2) β 1) Β· (π΄ Yrm π)) β β€) |
29 | bezoutr1 16537 | . . 3 β’ ((((π΄ Xrm π) β β€ β§ (π΄ Yrm π) β β€) β§ ((π΄ Xrm π) β β€ β§ (-((π΄β2) β 1) Β· (π΄ Yrm π)) β β€)) β ((((π΄ Xrm π) Β· (π΄ Xrm π)) + ((π΄ Yrm π) Β· (-((π΄β2) β 1) Β· (π΄ Yrm π)))) = 1 β ((π΄ Xrm π) gcd (π΄ Yrm π)) = 1)) | |
30 | 27, 10, 27, 28, 29 | syl22anc 837 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((((π΄ Xrm π) Β· (π΄ Xrm π)) + ((π΄ Yrm π) Β· (-((π΄β2) β 1) Β· (π΄ Yrm π)))) = 1 β ((π΄ Xrm π) gcd (π΄ Yrm π)) = 1)) |
31 | 26, 30 | mpd 15 | 1 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) gcd (π΄ Yrm π)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6542 (class class class)co 7415 1c1 11137 + caddc 11139 Β· cmul 11141 β cmin 11472 -cneg 11473 βcn 12240 2c2 12295 β0cn0 12500 β€cz 12586 β€β₯cuz 12850 βcexp 14056 gcd cgcd 16466 β»NNcsquarenn 42320 Xrm crmx 42384 Yrm crmy 42385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-omul 8488 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-acn 9963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-xnn0 12573 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-sum 15663 df-ef 16041 df-sin 16043 df-cos 16044 df-pi 16046 df-dvds 16229 df-gcd 16467 df-numer 16704 df-denom 16705 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-mulg 19026 df-cntz 19270 df-cmn 19739 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-limc 25811 df-dv 25812 df-log 26506 df-squarenn 42325 df-pell1qr 42326 df-pell14qr 42327 df-pell1234qr 42328 df-pellfund 42329 df-rmx 42386 df-rmy 42387 |
This theorem is referenced by: jm2.19lem2 42475 jm2.20nn 42482 |
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