Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > geo2sum | Unicode version |
Description: The value of the finite geometric series ... , multiplied by a constant. (Contributed by Mario Carneiro, 17-Mar-2014.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
geo2sum |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1zzd 9081 | . . 3 | |
2 | nnz 9073 | . . . 4 | |
3 | 2 | adantr 274 | . . 3 |
4 | simplr 519 | . . . 4 | |
5 | 2nn 8881 | . . . . . 6 | |
6 | elfznn 9834 | . . . . . . . 8 | |
7 | 6 | adantl 275 | . . . . . . 7 |
8 | 7 | nnnn0d 9030 | . . . . . 6 |
9 | nnexpcl 10306 | . . . . . 6 | |
10 | 5, 8, 9 | sylancr 410 | . . . . 5 |
11 | 10 | nncnd 8734 | . . . 4 |
12 | 10 | nnap0d 8766 | . . . 4 # |
13 | 4, 11, 12 | divclapd 8550 | . . 3 |
14 | oveq2 5782 | . . . 4 | |
15 | 14 | oveq2d 5790 | . . 3 |
16 | 1, 1, 3, 13, 15 | fsumshftm 11214 | . 2 |
17 | 1m1e0 8789 | . . . . 5 | |
18 | 17 | oveq1i 5784 | . . . 4 |
19 | 18 | sumeq1i 11132 | . . 3 |
20 | halfcn 8934 | . . . . . . . . . 10 | |
21 | elfznn0 9894 | . . . . . . . . . . 11 | |
22 | 21 | adantl 275 | . . . . . . . . . 10 |
23 | expcl 10311 | . . . . . . . . . 10 | |
24 | 20, 22, 23 | sylancr 410 | . . . . . . . . 9 |
25 | 2cnd 8793 | . . . . . . . . 9 | |
26 | 2ap0 8813 | . . . . . . . . . 10 # | |
27 | 26 | a1i 9 | . . . . . . . . 9 # |
28 | 24, 25, 27 | divrecapd 8553 | . . . . . . . 8 |
29 | expp1 10300 | . . . . . . . . 9 | |
30 | 20, 22, 29 | sylancr 410 | . . . . . . . 8 |
31 | elfzelz 9806 | . . . . . . . . . . 11 | |
32 | 31 | peano2zd 9176 | . . . . . . . . . 10 |
33 | 32 | adantl 275 | . . . . . . . . 9 |
34 | 25, 27, 33 | exprecapd 10432 | . . . . . . . 8 |
35 | 28, 30, 34 | 3eqtr2rd 2179 | . . . . . . 7 |
36 | 35 | oveq2d 5790 | . . . . . 6 |
37 | simplr 519 | . . . . . . 7 | |
38 | peano2nn0 9017 | . . . . . . . . . 10 | |
39 | 22, 38 | syl 14 | . . . . . . . . 9 |
40 | nnexpcl 10306 | . . . . . . . . 9 | |
41 | 5, 39, 40 | sylancr 410 | . . . . . . . 8 |
42 | 41 | nncnd 8734 | . . . . . . 7 |
43 | 41 | nnap0d 8766 | . . . . . . 7 # |
44 | 37, 42, 43 | divrecapd 8553 | . . . . . 6 |
45 | 24, 37, 25, 27 | div12apd 8587 | . . . . . 6 |
46 | 36, 44, 45 | 3eqtr4d 2182 | . . . . 5 |
47 | 46 | sumeq2dv 11137 | . . . 4 |
48 | 0zd 9066 | . . . . . 6 | |
49 | 3, 1 | zsubcld 9178 | . . . . . 6 |
50 | 48, 49 | fzfigd 10204 | . . . . 5 |
51 | halfcl 8946 | . . . . . 6 | |
52 | 51 | adantl 275 | . . . . 5 |
53 | 50, 52, 24 | fsummulc1 11218 | . . . 4 |
54 | 47, 53 | eqtr4d 2175 | . . 3 |
55 | 19, 54 | syl5eq 2184 | . 2 |
56 | 2cnd 8793 | . . . . . . . 8 | |
57 | 26 | a1i 9 | . . . . . . . 8 # |
58 | 56, 57, 3 | exprecapd 10432 | . . . . . . 7 |
59 | 58 | oveq2d 5790 | . . . . . 6 |
60 | 1mhlfehlf 8938 | . . . . . . 7 | |
61 | 60 | a1i 9 | . . . . . 6 |
62 | 59, 61 | oveq12d 5792 | . . . . 5 |
63 | simpr 109 | . . . . . 6 | |
64 | 63, 56, 57 | divrecap2d 8554 | . . . . 5 |
65 | 62, 64 | oveq12d 5792 | . . . 4 |
66 | ax-1cn 7713 | . . . . . . 7 | |
67 | nnnn0 8984 | . . . . . . . . . . 11 | |
68 | 67 | adantr 274 | . . . . . . . . . 10 |
69 | nnexpcl 10306 | . . . . . . . . . 10 | |
70 | 5, 68, 69 | sylancr 410 | . . . . . . . . 9 |
71 | 70 | nnrecred 8767 | . . . . . . . 8 |
72 | 71 | recnd 7794 | . . . . . . 7 |
73 | subcl 7961 | . . . . . . 7 | |
74 | 66, 72, 73 | sylancr 410 | . . . . . 6 |
75 | 20 | a1i 9 | . . . . . 6 |
76 | 56, 57 | recap0d 8542 | . . . . . 6 # |
77 | 74, 75, 76 | divclapd 8550 | . . . . 5 |
78 | 77, 75, 63 | mulassd 7789 | . . . 4 |
79 | 74, 75, 76 | divcanap1d 8551 | . . . . 5 |
80 | 79 | oveq1d 5789 | . . . 4 |
81 | 65, 78, 80 | 3eqtr2d 2178 | . . 3 |
82 | halfre 8933 | . . . . . . 7 | |
83 | 1re 7765 | . . . . . . 7 | |
84 | halflt1 8937 | . . . . . . 7 | |
85 | 82, 83, 84 | ltapii 8397 | . . . . . 6 # |
86 | 85 | a1i 9 | . . . . 5 # |
87 | 75, 86, 68 | geoserap 11276 | . . . 4 |
88 | 87 | oveq1d 5789 | . . 3 |
89 | mulid2 7764 | . . . . . . 7 | |
90 | 89 | adantl 275 | . . . . . 6 |
91 | 90 | eqcomd 2145 | . . . . 5 |
92 | 70 | nncnd 8734 | . . . . . 6 |
93 | 70 | nnap0d 8766 | . . . . . 6 # |
94 | 63, 92, 93 | divrecap2d 8554 | . . . . 5 |
95 | 91, 94 | oveq12d 5792 | . . . 4 |
96 | 66 | a1i 9 | . . . . 5 |
97 | 96, 72, 63 | subdird 8177 | . . . 4 |
98 | 95, 97 | eqtr4d 2175 | . . 3 |
99 | 81, 88, 98 | 3eqtr4d 2182 | . 2 |
100 | 16, 55, 99 | 3eqtrd 2176 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 (class class class)co 5774 cc 7618 cc0 7620 c1 7621 caddc 7623 cmul 7625 cmin 7933 # cap 8343 cdiv 8432 cn 8720 c2 8771 cn0 8977 cz 9054 cfz 9790 cexp 10292 csu 11122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 |
This theorem is referenced by: geo2lim 11285 trilpolemlt1 13234 |
Copyright terms: Public domain | W3C validator |