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Mathbox for Stefan O'Rear |
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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 12568 | . . . 4 β’ 0 β β€ | |
2 | rmxyval 41644 | . . . 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 689 | . . 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 41648 | . . . . 5 β’ (π΄ β (β€β₯β2) β (π΄ + (ββ((π΄β2) β 1))) β β+) | |
5 | 4 | rpcnd 13017 | . . . 4 β’ (π΄ β (β€β₯β2) β (π΄ + (ββ((π΄β2) β 1))) β β) |
6 | 5 | exp0d 14104 | . . 3 β’ (π΄ β (β€β₯β2) β ((π΄ + (ββ((π΄β2) β 1)))β0) = 1) |
7 | rmspecpos 41645 | . . . . . . . 8 β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β β+) | |
8 | 7 | rpcnd 13017 | . . . . . . 7 β’ (π΄ β (β€β₯β2) β ((π΄β2) β 1) β β) |
9 | 8 | sqrtcld 15383 | . . . . . 6 β’ (π΄ β (β€β₯β2) β (ββ((π΄β2) β 1)) β β) |
10 | 9 | mul01d 11412 | . . . . 5 β’ (π΄ β (β€β₯β2) β ((ββ((π΄β2) β 1)) Β· 0) = 0) |
11 | 10 | oveq2d 7424 | . . . 4 β’ (π΄ β (β€β₯β2) β (1 + ((ββ((π΄β2) β 1)) Β· 0)) = (1 + 0)) |
12 | 1p0e1 12335 | . . . 4 β’ (1 + 0) = 1 | |
13 | 11, 12 | eqtr2di 2789 | . . 3 β’ (π΄ β (β€β₯β2) β 1 = (1 + ((ββ((π΄β2) β 1)) Β· 0))) |
14 | 3, 6, 13 | 3eqtrd 2776 | . 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 41634 | . . 3 β’ (π΄ β (β€β₯β2) β (ββ((π΄β2) β 1)) β (β β β)) | |
16 | nn0ssq 12940 | . . . 4 β’ β0 β β | |
17 | frmx 41642 | . . . . . 6 β’ Xrm :((β€β₯β2) Γ β€)βΆβ0 | |
18 | 17 | fovcl 7536 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ 0 β β€) β (π΄ Xrm 0) β β0) |
19 | 1, 18 | mpan2 689 | . . . 4 β’ (π΄ β (β€β₯β2) β (π΄ Xrm 0) β β0) |
20 | 16, 19 | sselid 3980 | . . 3 β’ (π΄ β (β€β₯β2) β (π΄ Xrm 0) β β) |
21 | zssq 12939 | . . . 4 β’ β€ β β | |
22 | frmy 41643 | . . . . . 6 β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | |
23 | 22 | fovcl 7536 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ 0 β β€) β (π΄ Yrm 0) β β€) |
24 | 1, 23 | mpan2 689 | . . . 4 β’ (π΄ β (β€β₯β2) β (π΄ Yrm 0) β β€) |
25 | 21, 24 | sselid 3980 | . . 3 β’ (π΄ β (β€β₯β2) β (π΄ Yrm 0) β β) |
26 | 1z 12591 | . . . . 5 β’ 1 β β€ | |
27 | 21, 26 | sselii 3979 | . . . 4 β’ 1 β β |
28 | 27 | a1i 11 | . . 3 β’ (π΄ β (β€β₯β2) β 1 β β) |
29 | 21, 1 | sselii 3979 | . . . 4 β’ 0 β β |
30 | 29 | a1i 11 | . . 3 β’ (π΄ β (β€β₯β2) β 0 β β) |
31 | qirropth 41636 | . . 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 1379 | . 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 396 = wceq 1541 β wcel 2106 β cdif 3945 βcfv 6543 (class class class)co 7408 βcc 11107 0cc0 11109 1c1 11110 + caddc 11112 Β· cmul 11114 β cmin 11443 2c2 12266 β0cn0 12471 β€cz 12557 β€β₯cuz 12821 βcq 12931 βcexp 14026 βcsqrt 15179 Xrm crmx 41628 Yrm crmy 41629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-oadd 8469 df-omul 8470 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-xnn0 12544 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 df-dvds 16197 df-gcd 16435 df-numer 16670 df-denom 16671 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-submnd 18671 df-mulg 18950 df-cntz 19180 df-cmn 19649 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-limc 25382 df-dv 25383 df-log 26064 df-squarenn 41569 df-pell1qr 41570 df-pell14qr 41571 df-pell1234qr 41572 df-pellfund 41573 df-rmx 41630 df-rmy 41631 |
This theorem is referenced by: rmx0 41654 rmy0 41658 |
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