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Mirrors > Home > ILE Home > Th. List > geolim2 | Unicode version |
Description: The partial sums in the geometric series ... converge to . (Contributed by NM, 6-Jun-2006.) (Revised by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
geolim.1 | |
geolim.2 | |
geolim2.3 | |
geolim2.4 |
Ref | Expression |
---|---|
geolim2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2117 | . . 3 | |
2 | geolim2.3 | . . . 4 | |
3 | 2 | nn0zd 9129 | . . 3 |
4 | geolim2.4 | . . 3 | |
5 | geolim.1 | . . . . 5 | |
6 | 5 | adantr 274 | . . . 4 |
7 | eluznn0 9349 | . . . . 5 | |
8 | 2, 7 | sylan 281 | . . . 4 |
9 | 6, 8 | expcld 10379 | . . 3 |
10 | eluzelz 9291 | . . . . . . . . 9 | |
11 | 10 | adantl 275 | . . . . . . . 8 |
12 | 0red 7735 | . . . . . . . . 9 | |
13 | 3 | adantr 274 | . . . . . . . . . 10 |
14 | 13 | zred 9131 | . . . . . . . . 9 |
15 | 11 | zred 9131 | . . . . . . . . 9 |
16 | 2 | nn0ge0d 8991 | . . . . . . . . . 10 |
17 | 16 | adantr 274 | . . . . . . . . 9 |
18 | eluzle 9294 | . . . . . . . . . 10 | |
19 | 18 | adantl 275 | . . . . . . . . 9 |
20 | 12, 14, 15, 17, 19 | letrd 7854 | . . . . . . . 8 |
21 | elnn0z 9025 | . . . . . . . 8 | |
22 | 11, 20, 21 | sylanbrc 413 | . . . . . . 7 |
23 | 5 | adantr 274 | . . . . . . . 8 |
24 | 23, 22 | expcld 10379 | . . . . . . 7 |
25 | oveq2 5750 | . . . . . . . 8 | |
26 | eqid 2117 | . . . . . . . 8 | |
27 | 25, 26 | fvmptg 5465 | . . . . . . 7 |
28 | 22, 24, 27 | syl2anc 408 | . . . . . 6 |
29 | 28, 24 | eqeltrd 2194 | . . . . 5 |
30 | oveq2 5750 | . . . . . . . 8 | |
31 | 30, 26 | fvmptg 5465 | . . . . . . 7 |
32 | 8, 9, 31 | syl2anc 408 | . . . . . 6 |
33 | 32, 4 | eqtr4d 2153 | . . . . 5 |
34 | addcl 7713 | . . . . . 6 | |
35 | 34 | adantl 275 | . . . . 5 |
36 | 3, 29, 33, 35 | seq3feq 10200 | . . . 4 |
37 | seqex 10175 | . . . . . 6 | |
38 | ax-1cn 7681 | . . . . . . . 8 | |
39 | subcl 7929 | . . . . . . . 8 | |
40 | 38, 5, 39 | sylancr 410 | . . . . . . 7 |
41 | 1cnd 7750 | . . . . . . . 8 | |
42 | 1red 7749 | . . . . . . . . . 10 | |
43 | geolim.2 | . . . . . . . . . 10 | |
44 | 5, 42, 43 | absltap 11233 | . . . . . . . . 9 # |
45 | apsym 8335 | . . . . . . . . . 10 # # | |
46 | 5, 41, 45 | syl2anc 408 | . . . . . . . . 9 # # |
47 | 44, 46 | mpbid 146 | . . . . . . . 8 # |
48 | 41, 5, 47 | subap0d 8373 | . . . . . . 7 # |
49 | 40, 48 | recclapd 8508 | . . . . . 6 |
50 | simpr 109 | . . . . . . . 8 | |
51 | 5 | adantr 274 | . . . . . . . . 9 |
52 | 51, 50 | expcld 10379 | . . . . . . . 8 |
53 | oveq2 5750 | . . . . . . . . 9 | |
54 | 53, 26 | fvmptg 5465 | . . . . . . . 8 |
55 | 50, 52, 54 | syl2anc 408 | . . . . . . 7 |
56 | 5, 43, 55 | geolim 11235 | . . . . . 6 |
57 | breldmg 4715 | . . . . . 6 | |
58 | 37, 49, 56, 57 | mp3an2i 1305 | . . . . 5 |
59 | nn0uz 9316 | . . . . . 6 | |
60 | expcl 10266 | . . . . . . . 8 | |
61 | 5, 60 | sylan 281 | . . . . . . 7 |
62 | 55, 61 | eqeltrd 2194 | . . . . . 6 |
63 | 59, 2, 62 | iserex 11063 | . . . . 5 |
64 | 58, 63 | mpbid 146 | . . . 4 |
65 | 36, 64 | eqeltrrd 2195 | . . 3 |
66 | 1, 3, 4, 9, 65 | isumclim2 11146 | . 2 |
67 | simpr 109 | . . . . . . . 8 | |
68 | 5 | adantr 274 | . . . . . . . . 9 |
69 | 68, 67 | expcld 10379 | . . . . . . . 8 |
70 | 67, 69, 31 | syl2anc 408 | . . . . . . 7 |
71 | expcl 10266 | . . . . . . . 8 | |
72 | 5, 71 | sylan 281 | . . . . . . 7 |
73 | 59, 1, 2, 70, 72, 58 | isumsplit 11215 | . . . . . 6 |
74 | 0zd 9024 | . . . . . . 7 | |
75 | 59, 74, 70, 72, 56 | isumclim 11145 | . . . . . 6 |
76 | 73, 75 | eqtr3d 2152 | . . . . 5 |
77 | 5, 44, 2 | geoserap 11231 | . . . . . 6 |
78 | 77 | oveq1d 5757 | . . . . 5 |
79 | 76, 78 | eqtr3d 2152 | . . . 4 |
80 | 79 | oveq1d 5757 | . . 3 |
81 | 5, 2 | expcld 10379 | . . . . . 6 |
82 | subcl 7929 | . . . . . 6 | |
83 | 38, 81, 82 | sylancr 410 | . . . . 5 |
84 | 41, 83, 40, 48 | divsubdirapd 8557 | . . . 4 |
85 | nncan 7959 | . . . . . 6 | |
86 | 38, 81, 85 | sylancr 410 | . . . . 5 |
87 | 86 | oveq1d 5757 | . . . 4 |
88 | 84, 87 | eqtr3d 2152 | . . 3 |
89 | 83, 40, 48 | divclapd 8517 | . . . 4 |
90 | 1, 3, 32, 9, 64 | isumcl 11149 | . . . 4 |
91 | 89, 90 | pncan2d 8043 | . . 3 |
92 | 80, 88, 91 | 3eqtr3rd 2159 | . 2 |
93 | 66, 92 | breqtrd 3924 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cvv 2660 class class class wbr 3899 cmpt 3959 cdm 4509 cfv 5093 (class class class)co 5742 cc 7586 cc0 7588 c1 7589 caddc 7591 clt 7768 cle 7769 cmin 7901 # cap 8310 cdiv 8399 cn0 8935 cz 9012 cuz 9282 cfz 9745 cseq 10173 cexp 10247 cabs 10724 cli 11002 csu 11077 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-frec 6256 df-1o 6281 df-oadd 6285 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-3 8744 df-4 8745 df-n0 8936 df-z 9013 df-uz 9283 df-q 9368 df-rp 9398 df-fz 9746 df-fzo 9875 df-seqfrec 10174 df-exp 10248 df-ihash 10477 df-cj 10569 df-re 10570 df-im 10571 df-rsqrt 10725 df-abs 10726 df-clim 11003 df-sumdc 11078 |
This theorem is referenced by: geoisum1 11243 geoisum1c 11244 trilpolemisumle 13127 |
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