![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > rmxy0 | Structured version Visualization version GIF version |
Description: Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.) |
Ref | Expression |
---|---|
rmxy0 | β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 0) = 1 β§ (π΄ Yrm 0) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12593 | . . . 4 β’ 0 β β€ | |
2 | rmxyval 42330 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ 0 β β€) β ((π΄ Xrm 0) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm 0))) = ((π΄ + (ββ((π΄β2) β 1)))β0)) | |
3 | 1, 2 | mpan2 690 | . . 3 β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 0) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm 0))) = ((π΄ + (ββ((π΄β2) β 1)))β0)) |
4 | rmbaserp 42334 | . . . . 5 β’ (π΄ β (β€β₯β2) β (π΄ + (ββ((π΄β2) β 1))) β β+) | |
5 | 4 | rpcnd 13044 | . . . 4 β’ (π΄ β (β€β₯β2) β (π΄ + (ββ((π΄β2) β 1))) β β) |
6 | 5 | exp0d 14130 | . . 3 β’ (π΄ β (β€β₯β2) β ((π΄ + (ββ((π΄β2) β 1)))β0) = 1) |
7 | rmspecpos 42331 | . . . . . . . 8 β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β β+) | |
8 | 7 | rpcnd 13044 | . . . . . . 7 β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β β) |
9 | 8 | sqrtcld 15410 | . . . . . 6 β’ (π΄ β (β€β₯β2) β (ββ((π΄β2) β 1)) β β) |
10 | 9 | mul01d 11437 | . . . . 5 β’ (π΄ β (β€β₯β2) β ((ββ((π΄β2) β 1)) Β· 0) = 0) |
11 | 10 | oveq2d 7430 | . . . 4 β’ (π΄ β (β€β₯β2) β (1 + ((ββ((π΄β2) β 1)) Β· 0)) = (1 + 0)) |
12 | 1p0e1 12360 | . . . 4 β’ (1 + 0) = 1 | |
13 | 11, 12 | eqtr2di 2785 | . . 3 β’ (π΄ β (β€β₯β2) β 1 = (1 + ((ββ((π΄β2) β 1)) Β· 0))) |
14 | 3, 6, 13 | 3eqtrd 2772 | . 2 β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 0) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm 0))) = (1 + ((ββ((π΄β2) β 1)) Β· 0))) |
15 | rmspecsqrtnq 42320 | . . 3 β’ (π΄ β (β€β₯β2) β (ββ((π΄β2) β 1)) β (β β β)) | |
16 | nn0ssq 12965 | . . . 4 β’ β0 β β | |
17 | frmx 42328 | . . . . . 6 β’ Xrm :((β€β₯β2) Γ β€)βΆβ0 | |
18 | 17 | fovcl 7543 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ 0 β β€) β (π΄ Xrm 0) β β0) |
19 | 1, 18 | mpan2 690 | . . . 4 β’ (π΄ β (β€β₯β2) β (π΄ Xrm 0) β β0) |
20 | 16, 19 | sselid 3976 | . . 3 β’ (π΄ β (β€β₯β2) β (π΄ Xrm 0) β β) |
21 | zssq 12964 | . . . 4 β’ β€ β β | |
22 | frmy 42329 | . . . . . 6 β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | |
23 | 22 | fovcl 7543 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ 0 β β€) β (π΄ Yrm 0) β β€) |
24 | 1, 23 | mpan2 690 | . . . 4 β’ (π΄ β (β€β₯β2) β (π΄ Yrm 0) β β€) |
25 | 21, 24 | sselid 3976 | . . 3 β’ (π΄ β (β€β₯β2) β (π΄ Yrm 0) β β) |
26 | 1z 12616 | . . . . 5 β’ 1 β β€ | |
27 | 21, 26 | sselii 3975 | . . . 4 β’ 1 β β |
28 | 27 | a1i 11 | . . 3 β’ (π΄ β (β€β₯β2) β 1 β β) |
29 | 21, 1 | sselii 3975 | . . . 4 β’ 0 β β |
30 | 29 | a1i 11 | . . 3 β’ (π΄ β (β€β₯β2) β 0 β β) |
31 | qirropth 42322 | . . 3 β’ (((ββ((π΄β2) β 1)) β (β β β) β§ ((π΄ Xrm 0) β β β§ (π΄ Yrm 0) β β) β§ (1 β β β§ 0 β β)) β (((π΄ Xrm 0) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm 0))) = (1 + ((ββ((π΄β2) β 1)) Β· 0)) β ((π΄ Xrm 0) = 1 β§ (π΄ Yrm 0) = 0))) | |
32 | 15, 20, 25, 28, 30, 31 | syl122anc 1377 | . 2 β’ (π΄ β (β€β₯β2) β (((π΄ Xrm 0) + ((ββ((π΄β2) β 1)) Β· (π΄ Yrm 0))) = (1 + ((ββ((π΄β2) β 1)) Β· 0)) β ((π΄ Xrm 0) = 1 β§ (π΄ Yrm 0) = 0))) |
33 | 14, 32 | mpbid 231 | 1 β’ (π΄ β (β€β₯β2) β ((π΄ Xrm 0) = 1 β§ (π΄ Yrm 0) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 β cdif 3942 βcfv 6542 (class class class)co 7414 βcc 11130 0cc0 11132 1c1 11133 + caddc 11135 Β· cmul 11137 β cmin 11468 2c2 12291 β0cn0 12496 β€cz 12582 β€β₯cuz 12846 βcq 12956 βcexp 14052 βcsqrt 15206 Xrm crmx 42314 Yrm crmy 42315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-acn 9959 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-xnn0 12569 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ioc 13355 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-fl 13783 df-mod 13861 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-ef 16037 df-sin 16039 df-cos 16040 df-pi 16042 df-dvds 16225 df-gcd 16463 df-numer 16700 df-denom 16701 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-lp 23033 df-perf 23034 df-cn 23124 df-cnp 23125 df-haus 23212 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-xms 24219 df-ms 24220 df-tms 24221 df-cncf 24791 df-limc 25788 df-dv 25789 df-log 26483 df-squarenn 42255 df-pell1qr 42256 df-pell14qr 42257 df-pell1234qr 42258 df-pellfund 42259 df-rmx 42316 df-rmy 42317 |
This theorem is referenced by: rmx0 42340 rmy0 42344 |
Copyright terms: Public domain | W3C validator |