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Mirrors > Home > MPE Home > Th. List > geo2sum | Unicode version |
Description: The value of the finite
geometric series 2 -u 1 2 -u 2 ...
2 -u N , multiplied by a constant. (Contributed by Mario
Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro,
26-Apr-2014.) |
Ref | Expression |
---|---|
geo2sum |
N
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1zzd 10920 | . . 3 | |
2 | nnz 10911 | . . . 4 | |
3 | 2 | adantr 465 | . . 3 |
4 | simplr 755 | . . . 4 | |
5 | 2nn 10718 | . . . . . 6 | |
6 | elfznn 11743 | . . . . . . . 8 | |
7 | 6 | adantl 466 | . . . . . . 7 |
8 | 7 | nnnn0d 10877 | . . . . . 6 |
9 | nnexpcl 12179 | . . . . . 6 | |
10 | 5, 8, 9 | sylancr 663 | . . . . 5 |
11 | 10 | nncnd 10577 | . . . 4 |
12 | 10 | nnne0d 10605 | . . . 4 |
13 | 4, 11, 12 | divcld 10345 | . . 3 |
14 | oveq2 6304 | . . . 4 | |
15 | 14 | oveq2d 6312 | . . 3 |
16 | 1, 1, 3, 13, 15 | fsumshftm 13596 | . 2 |
17 | 1m1e0 10629 | . . . . 5 | |
18 | 17 | oveq1i 6306 | . . . 4 |
19 | 18 | sumeq1i 13520 | . . 3 |
20 | halfcn 10780 | . . . . . . . . . 10 | |
21 | elfznn0 11800 | . . . . . . . . . . 11 | |
22 | 21 | adantl 466 | . . . . . . . . . 10 |
23 | expcl 12184 | . . . . . . . . . 10 | |
24 | 20, 22, 23 | sylancr 663 | . . . . . . . . 9 |
25 | 2cnd 10633 | . . . . . . . . 9 | |
26 | 2ne0 10653 | . . . . . . . . . 10 | |
27 | 26 | a1i 11 | . . . . . . . . 9 |
28 | 24, 25, 27 | divrecd 10348 | . . . . . . . 8 |
29 | expp1 12173 | . . . . . . . . 9 | |
30 | 20, 22, 29 | sylancr 663 | . . . . . . . 8 |
31 | elfzelz 11717 | . . . . . . . . . . 11 | |
32 | 31 | peano2zd 10997 | . . . . . . . . . 10 |
33 | 32 | adantl 466 | . . . . . . . . 9 |
34 | 25, 27, 33 | exprecd 12318 | . . . . . . . 8 |
35 | 28, 30, 34 | 3eqtr2rd 2505 | . . . . . . 7 |
36 | 35 | oveq2d 6312 | . . . . . 6 |
37 | simplr 755 | . . . . . . 7 | |
38 | peano2nn0 10861 | . . . . . . . . . 10 | |
39 | 22, 38 | syl 16 | . . . . . . . . 9 |
40 | nnexpcl 12179 | . . . . . . . . 9 | |
41 | 5, 39, 40 | sylancr 663 | . . . . . . . 8 |
42 | 41 | nncnd 10577 | . . . . . . 7 |
43 | 41 | nnne0d 10605 | . . . . . . 7 |
44 | 37, 42, 43 | divrecd 10348 | . . . . . 6 |
45 | 24, 37, 25, 27 | div12d 10381 | . . . . . 6 |
46 | 36, 44, 45 | 3eqtr4d 2508 | . . . . 5 |
47 | 46 | sumeq2dv 13525 | . . . 4 |
48 | fzfid 12083 | . . . . 5 | |
49 | halfcl 10789 | . . . . . 6 | |
50 | 49 | adantl 466 | . . . . 5 |
51 | 48, 50, 24 | fsummulc1 13600 | . . . 4 |
52 | 47, 51 | eqtr4d 2501 | . . 3 |
53 | 19, 52 | syl5eq 2510 | . 2 |
54 | 2cnd 10633 | . . . . . . . 8 | |
55 | 26 | a1i 11 | . . . . . . . 8 |
56 | 54, 55, 3 | exprecd 12318 | . . . . . . 7 |
57 | 56 | oveq2d 6312 | . . . . . 6 |
58 | 1mhlfehlf 10783 | . . . . . . 7 | |
59 | 58 | a1i 11 | . . . . . 6 |
60 | 57, 59 | oveq12d 6314 | . . . . 5 |
61 | simpr 461 | . . . . . 6 | |
62 | 61, 54, 55 | divrec2d 10349 | . . . . 5 |
63 | 60, 62 | oveq12d 6314 | . . . 4 |
64 | ax-1cn 9571 | . . . . . . 7 | |
65 | nnnn0 10827 | . . . . . . . . . . 11 | |
66 | 65 | adantr 465 | . . . . . . . . . 10 |
67 | nnexpcl 12179 | . . . . . . . . . 10 | |
68 | 5, 66, 67 | sylancr 663 | . . . . . . . . 9 |
69 | 68 | nnrecred 10606 | . . . . . . . 8 |
70 | 69 | recnd 9643 | . . . . . . 7 |
71 | subcl 9842 | . . . . . . 7 | |
72 | 64, 70, 71 | sylancr 663 | . . . . . 6 |
73 | 20 | a1i 11 | . . . . . 6 |
74 | 0re 9617 | . . . . . . . 8 | |
75 | halfgt0 10781 | . . . . . . . 8 | |
76 | 74, 75 | gtneii 9717 | . . . . . . 7 |
77 | 76 | a1i 11 | . . . . . 6 |
78 | 72, 73, 77 | divcld 10345 | . . . . 5 |
79 | 78, 73, 61 | mulassd 9640 | . . . 4 |
80 | 72, 73, 77 | divcan1d 10346 | . . . . 5 |
81 | 80 | oveq1d 6311 | . . . 4 |
82 | 63, 79, 81 | 3eqtr2d 2504 | . . 3 |
83 | halfre 10779 | . . . . . . 7 | |
84 | halflt1 10782 | . . . . . . 7 | |
85 | 83, 84 | ltneii 9718 | . . . . . 6 |
86 | 85 | a1i 11 | . . . . 5 |
87 | 73, 86, 66 | geoser 13678 | . . . 4 |
88 | 87 | oveq1d 6311 | . . 3 |
89 | mulid2 9615 | . . . . . . 7 | |
90 | 89 | adantl 466 | . . . . . 6 |
91 | 90 | eqcomd 2465 | . . . . 5 |
92 | 68 | nncnd 10577 | . . . . . 6 |
93 | 68 | nnne0d 10605 | . . . . . 6 |
94 | 61, 92, 93 | divrec2d 10349 | . . . . 5 |
95 | 91, 94 | oveq12d 6314 | . . . 4 |
96 | 64 | a1i 11 | . . . . 5 |
97 | 96, 70, 61 | subdird 10038 | . . . 4 |
98 | 95, 97 | eqtr4d 2501 | . . 3 |
99 | 82, 88, 98 | 3eqtr4d 2508 | . 2 |
100 | 16, 53, 99 | 3eqtrd 2502 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 /\ wa 369
= wceq 1395 e. wcel 1818 =/= wne 2652
(class class class)co 6296 cc 9511 0 cc0 9513 1 c1 9514
caddc 9516 cmul 9518 cmin 9828 cdiv 10231 cn 10561 2 c2 10610 cn0 10820
cz 10889 cfz 11701 cexp 12166 sum_ csu 13508 |
This theorem is referenced by: geo2lim 13684 ovollb2lem 21899 ovoliunlem1 21913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 ax-inf2 8079 ax-cnex 9569 ax-resscn 9570 ax-1cn 9571 ax-icn 9572 ax-addcl 9573 ax-addrcl 9574 ax-mulcl 9575 ax-mulrcl 9576 ax-mulcom 9577 ax-addass 9578 ax-mulass 9579 ax-distr 9580 ax-i2m1 9581 ax-1ne0 9582 ax-1rid 9583 ax-rnegex 9584 ax-rrecex 9585 ax-cnre 9586 ax-pre-lttri 9587 ax-pre-lttrn 9588 ax-pre-ltadd 9589 ax-pre-mulgt0 9590 ax-pre-sup 9591 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-fal 1401 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-nel 2655 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-int 4287 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-se 4844 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-ov 6299 df-oprab 6300 df-mpt2 6301 df-om 6701 df-1st 6800 df-2nd 6801 df-recs 7061 df-rdg 7095 df-1o 7149 df-oadd 7153 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-fin 7540 df-sup 7921 df-oi 7956 df-card 8341 df-pnf 9651 df-mnf 9652 df-xr 9653 df-ltxr 9654 df-le 9655 df-sub 9830 df-neg 9831 df-div 10232 df-nn 10562 df-2 10619 df-3 10620 df-n0 10821 df-z 10890 df-uz 11111 df-rp 11250 df-fz 11702 df-fzo 11825 df-seq 12108 df-exp 12167 df-hash 12406 df-cj 12932 df-re 12933 df-im 12934 df-sqrt 13068 df-abs 13069 df-clim 13311 df-sum 13509 |
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