Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmym1 | Structured version Visualization version GIF version |
Description: Subtraction of 1 formula for Y sequence. Part 2 of equation 2.10 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 19-Oct-2014.) |
Ref | Expression |
---|---|
rmym1 | β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π β 1)) = (((π΄ Yrm π) Β· π΄) β (π΄ Xrm π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12366 | . . . . . 6 β’ (π β β€ β π β β) | |
2 | 1 | adantl 483 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β π β β) |
3 | ax-1cn 10971 | . . . . 5 β’ 1 β β | |
4 | negsub 11311 | . . . . 5 β’ ((π β β β§ 1 β β) β (π + -1) = (π β 1)) | |
5 | 2, 3, 4 | sylancl 587 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π + -1) = (π β 1)) |
6 | 5 | eqcomd 2742 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π β 1) = (π + -1)) |
7 | 6 | oveq2d 7319 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π β 1)) = (π΄ Yrm (π + -1))) |
8 | neg1z 12398 | . . 3 β’ -1 β β€ | |
9 | rmyadd 40790 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ -1 β β€) β (π΄ Yrm (π + -1)) = (((π΄ Yrm π) Β· (π΄ Xrm -1)) + ((π΄ Xrm π) Β· (π΄ Yrm -1)))) | |
10 | 8, 9 | mp3an3 1450 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π + -1)) = (((π΄ Yrm π) Β· (π΄ Xrm -1)) + ((π΄ Xrm π) Β· (π΄ Yrm -1)))) |
11 | 1z 12392 | . . . . . . . 8 β’ 1 β β€ | |
12 | rmxneg 40783 | . . . . . . . 8 β’ ((π΄ β (β€β₯β2) β§ 1 β β€) β (π΄ Xrm -1) = (π΄ Xrm 1)) | |
13 | 11, 12 | mpan2 689 | . . . . . . 7 β’ (π΄ β (β€β₯β2) β (π΄ Xrm -1) = (π΄ Xrm 1)) |
14 | rmx1 40785 | . . . . . . 7 β’ (π΄ β (β€β₯β2) β (π΄ Xrm 1) = π΄) | |
15 | 13, 14 | eqtrd 2776 | . . . . . 6 β’ (π΄ β (β€β₯β2) β (π΄ Xrm -1) = π΄) |
16 | 15 | adantr 482 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm -1) = π΄) |
17 | 16 | oveq2d 7319 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Yrm π) Β· (π΄ Xrm -1)) = ((π΄ Yrm π) Β· π΄)) |
18 | rmyneg 40787 | . . . . . . . . 9 β’ ((π΄ β (β€β₯β2) β§ 1 β β€) β (π΄ Yrm -1) = -(π΄ Yrm 1)) | |
19 | 11, 18 | mpan2 689 | . . . . . . . 8 β’ (π΄ β (β€β₯β2) β (π΄ Yrm -1) = -(π΄ Yrm 1)) |
20 | rmy1 40789 | . . . . . . . . 9 β’ (π΄ β (β€β₯β2) β (π΄ Yrm 1) = 1) | |
21 | 20 | negeqd 11257 | . . . . . . . 8 β’ (π΄ β (β€β₯β2) β -(π΄ Yrm 1) = -1) |
22 | 19, 21 | eqtrd 2776 | . . . . . . 7 β’ (π΄ β (β€β₯β2) β (π΄ Yrm -1) = -1) |
23 | 22 | adantr 482 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm -1) = -1) |
24 | 23 | oveq2d 7319 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) Β· (π΄ Yrm -1)) = ((π΄ Xrm π) Β· -1)) |
25 | frmx 40772 | . . . . . . . 8 β’ Xrm :((β€β₯β2) Γ β€)βΆβ0 | |
26 | 25 | fovcl 7430 | . . . . . . 7 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) β β0) |
27 | 26 | nn0cnd 12337 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Xrm π) β β) |
28 | neg1cn 12129 | . . . . . 6 β’ -1 β β | |
29 | mulcom 10999 | . . . . . 6 β’ (((π΄ Xrm π) β β β§ -1 β β) β ((π΄ Xrm π) Β· -1) = (-1 Β· (π΄ Xrm π))) | |
30 | 27, 28, 29 | sylancl 587 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) Β· -1) = (-1 Β· (π΄ Xrm π))) |
31 | 27 | mulm1d 11469 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (-1 Β· (π΄ Xrm π)) = -(π΄ Xrm π)) |
32 | 24, 30, 31 | 3eqtrd 2780 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Xrm π) Β· (π΄ Yrm -1)) = -(π΄ Xrm π)) |
33 | 17, 32 | oveq12d 7321 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Yrm π) Β· (π΄ Xrm -1)) + ((π΄ Xrm π) Β· (π΄ Yrm -1))) = (((π΄ Yrm π) Β· π΄) + -(π΄ Xrm π))) |
34 | frmy 40773 | . . . . . . 7 β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | |
35 | 34 | fovcl 7430 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm π) β β€) |
36 | 35 | zcnd 12469 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm π) β β) |
37 | eluzelcn 12636 | . . . . . 6 β’ (π΄ β (β€β₯β2) β π΄ β β) | |
38 | 37 | adantr 482 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β π΄ β β) |
39 | 36, 38 | mulcld 11037 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Yrm π) Β· π΄) β β) |
40 | 39, 27 | negsubd 11380 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Yrm π) Β· π΄) + -(π΄ Xrm π)) = (((π΄ Yrm π) Β· π΄) β (π΄ Xrm π))) |
41 | 33, 40 | eqtrd 2776 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (((π΄ Yrm π) Β· (π΄ Xrm -1)) + ((π΄ Xrm π) Β· (π΄ Yrm -1))) = (((π΄ Yrm π) Β· π΄) β (π΄ Xrm π))) |
42 | 7, 10, 41 | 3eqtrd 2780 | 1 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm (π β 1)) = (((π΄ Yrm π) Β· π΄) β (π΄ Xrm π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1539 β wcel 2104 βcfv 6454 (class class class)co 7303 βcc 10911 1c1 10914 + caddc 10916 Β· cmul 10918 β cmin 11247 -cneg 11248 2c2 12070 β0cn0 12275 β€cz 12361 β€β₯cuz 12624 Xrm crmx 40758 Yrm crmy 40759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-inf2 9439 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 ax-addf 10992 ax-mulf 10993 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-of 7561 df-om 7741 df-1st 7859 df-2nd 7860 df-supp 8005 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-2o 8325 df-oadd 8328 df-omul 8329 df-er 8525 df-map 8644 df-pm 8645 df-ixp 8713 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-fsupp 9169 df-fi 9210 df-sup 9241 df-inf 9242 df-oi 9309 df-card 9737 df-acn 9740 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-4 12080 df-5 12081 df-6 12082 df-7 12083 df-8 12084 df-9 12085 df-n0 12276 df-xnn0 12348 df-z 12362 df-dec 12480 df-uz 12625 df-q 12731 df-rp 12773 df-xneg 12890 df-xadd 12891 df-xmul 12892 df-ioo 13125 df-ioc 13126 df-ico 13127 df-icc 13128 df-fz 13282 df-fzo 13425 df-fl 13554 df-mod 13632 df-seq 13764 df-exp 13825 df-fac 14030 df-bc 14059 df-hash 14087 df-shft 14819 df-cj 14851 df-re 14852 df-im 14853 df-sqrt 14987 df-abs 14988 df-limsup 15221 df-clim 15238 df-rlim 15239 df-sum 15439 df-ef 15818 df-sin 15820 df-cos 15821 df-pi 15823 df-dvds 16005 df-gcd 16243 df-numer 16480 df-denom 16481 df-struct 16889 df-sets 16906 df-slot 16924 df-ndx 16936 df-base 16954 df-ress 16983 df-plusg 17016 df-mulr 17017 df-starv 17018 df-sca 17019 df-vsca 17020 df-ip 17021 df-tset 17022 df-ple 17023 df-ds 17025 df-unif 17026 df-hom 17027 df-cco 17028 df-rest 17174 df-topn 17175 df-0g 17193 df-gsum 17194 df-topgen 17195 df-pt 17196 df-prds 17199 df-xrs 17254 df-qtop 17259 df-imas 17260 df-xps 17262 df-mre 17336 df-mrc 17337 df-acs 17339 df-mgm 18367 df-sgrp 18416 df-mnd 18427 df-submnd 18472 df-mulg 18742 df-cntz 18964 df-cmn 19429 df-psmet 20630 df-xmet 20631 df-met 20632 df-bl 20633 df-mopn 20634 df-fbas 20635 df-fg 20636 df-cnfld 20639 df-top 22084 df-topon 22101 df-topsp 22123 df-bases 22137 df-cld 22211 df-ntr 22212 df-cls 22213 df-nei 22290 df-lp 22328 df-perf 22329 df-cn 22419 df-cnp 22420 df-haus 22507 df-tx 22754 df-hmeo 22947 df-fil 23038 df-fm 23130 df-flim 23131 df-flf 23132 df-xms 23514 df-ms 23515 df-tms 23516 df-cncf 24082 df-limc 25071 df-dv 25072 df-log 25753 df-squarenn 40699 df-pell1qr 40700 df-pell14qr 40701 df-pell1234qr 40702 df-pellfund 40703 df-rmx 40760 df-rmy 40761 |
This theorem is referenced by: rmyluc 40796 jm2.24nn 40818 |
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