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Mirrors > Home > MPE Home > Th. List > divcnv | Structured version Visualization version GIF version |
Description: The sequence of reciprocals of positive integers, multiplied by the factor π΄, converges to zero. (Contributed by NM, 6-Feb-2008.) (Revised by Mario Carneiro, 18-Sep-2014.) |
Ref | Expression |
---|---|
divcnv | β’ (π΄ β β β (π β β β¦ (π΄ / π)) β 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnrp 12988 | . . . . 5 β’ (π β β β π β β+) | |
2 | 1 | ssriv 3981 | . . . 4 β’ β β β+ |
3 | 2 | a1i 11 | . . 3 β’ (π΄ β β β β β β+) |
4 | divrcnv 15802 | . . 3 β’ (π΄ β β β (π β β+ β¦ (π΄ / π)) βπ 0) | |
5 | 3, 4 | rlimres2 15509 | . 2 β’ (π΄ β β β (π β β β¦ (π΄ / π)) βπ 0) |
6 | nnuz 12866 | . . 3 β’ β = (β€β₯β1) | |
7 | 1zzd 12594 | . . 3 β’ (π΄ β β β 1 β β€) | |
8 | simpl 482 | . . . . 5 β’ ((π΄ β β β§ π β β) β π΄ β β) | |
9 | nncn 12221 | . . . . . 6 β’ (π β β β π β β) | |
10 | 9 | adantl 481 | . . . . 5 β’ ((π΄ β β β§ π β β) β π β β) |
11 | nnne0 12247 | . . . . . 6 β’ (π β β β π β 0) | |
12 | 11 | adantl 481 | . . . . 5 β’ ((π΄ β β β§ π β β) β π β 0) |
13 | 8, 10, 12 | divcld 11991 | . . . 4 β’ ((π΄ β β β§ π β β) β (π΄ / π) β β) |
14 | 13 | fmpttd 7109 | . . 3 β’ (π΄ β β β (π β β β¦ (π΄ / π)):ββΆβ) |
15 | 6, 7, 14 | rlimclim 15494 | . 2 β’ (π΄ β β β ((π β β β¦ (π΄ / π)) βπ 0 β (π β β β¦ (π΄ / π)) β 0)) |
16 | 5, 15 | mpbid 231 | 1 β’ (π΄ β β β (π β β β¦ (π΄ / π)) β 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β wcel 2098 β wne 2934 β wss 3943 class class class wbr 5141 β¦ cmpt 5224 (class class class)co 7404 βcc 11107 0cc0 11109 1c1 11110 / cdiv 11872 βcn 12213 β+crp 12977 β cli 15432 βπ crli 15433 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 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 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fl 13760 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 |
This theorem is referenced by: divcnvshft 15805 supcvg 15806 expcnv 15814 plyeq0lem 26095 leibpi 26825 emcllem4 26882 basellem6 26969 circum 35187 divcnvlin 35236 hashnzfzclim 43638 clim1fr1 44870 divcnvg 44896 fprodsubrecnncnvlem 45176 fprodaddrecnncnvlem 45178 stirlinglem1 45343 |
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