Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > telfsumo | Unicode version |
Description: Sum of a telescoping series, using half-open intervals. (Contributed by Mario Carneiro, 2-May-2016.) |
Ref | Expression |
---|---|
telfsumo.1 | |
telfsumo.2 | |
telfsumo.3 | |
telfsumo.4 | |
telfsumo.5 | |
telfsumo.6 |
Ref | Expression |
---|---|
telfsumo | ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | telfsumo.3 | . . . . . . . 8 | |
2 | 1 | eleq1d 2206 | . . . . . . 7 |
3 | telfsumo.6 | . . . . . . . 8 | |
4 | 3 | ralrimiva 2503 | . . . . . . 7 |
5 | telfsumo.5 | . . . . . . . 8 | |
6 | eluzfz1 9804 | . . . . . . . 8 | |
7 | 5, 6 | syl 14 | . . . . . . 7 |
8 | 2, 4, 7 | rspcdva 2789 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | 9 | subidd 8054 | . . . 4 |
11 | sum0 11150 | . . . 4 | |
12 | 10, 11 | syl6reqr 2189 | . . 3 |
13 | oveq2 5775 | . . . . . 6 ..^ ..^ | |
14 | 13 | adantl 275 | . . . . 5 ..^ ..^ |
15 | fzo0 9938 | . . . . 5 ..^ | |
16 | 14, 15 | syl6eq 2186 | . . . 4 ..^ |
17 | 16 | sumeq1d 11128 | . . 3 ..^ |
18 | eqeq1 2144 | . . . . . . . 8 | |
19 | telfsumo.4 | . . . . . . . . 9 | |
20 | 19 | eqeq1d 2146 | . . . . . . . 8 |
21 | 18, 20 | imbi12d 233 | . . . . . . 7 |
22 | 21, 1 | vtoclg 2741 | . . . . . 6 |
23 | 22 | imp 123 | . . . . 5 |
24 | 5, 23 | sylan 281 | . . . 4 |
25 | 24 | oveq2d 5783 | . . 3 |
26 | 12, 17, 25 | 3eqtr4d 2180 | . 2 ..^ |
27 | eluzel2 9324 | . . . . . . 7 | |
28 | 5, 27 | syl 14 | . . . . . 6 |
29 | eluzelz 9328 | . . . . . . 7 | |
30 | 5, 29 | syl 14 | . . . . . 6 |
31 | fzofig 10198 | . . . . . 6 ..^ | |
32 | 28, 30, 31 | syl2anc 408 | . . . . 5 ..^ |
33 | telfsumo.1 | . . . . . . 7 | |
34 | 33 | eleq1d 2206 | . . . . . 6 |
35 | 4 | adantr 274 | . . . . . 6 ..^ |
36 | elfzofz 9932 | . . . . . . 7 ..^ | |
37 | 36 | adantl 275 | . . . . . 6 ..^ |
38 | 34, 35, 37 | rspcdva 2789 | . . . . 5 ..^ |
39 | telfsumo.2 | . . . . . . 7 | |
40 | 39 | eleq1d 2206 | . . . . . 6 |
41 | fzofzp1 9997 | . . . . . . 7 ..^ | |
42 | 41 | adantl 275 | . . . . . 6 ..^ |
43 | 40, 35, 42 | rspcdva 2789 | . . . . 5 ..^ |
44 | 32, 38, 43 | fsumsub 11214 | . . . 4 ..^ ..^ ..^ |
45 | 44 | adantr 274 | . . 3 ..^ ..^ ..^ |
46 | 33 | cbvsumv 11123 | . . . . 5 ..^ ..^ |
47 | eluzp1m1 9342 | . . . . . . . 8 | |
48 | 28, 47 | sylan 281 | . . . . . . 7 |
49 | 30 | adantr 274 | . . . . . . . . . . 11 |
50 | fzoval 9918 | . . . . . . . . . . 11 ..^ | |
51 | 49, 50 | syl 14 | . . . . . . . . . 10 ..^ |
52 | fzossfz 9935 | . . . . . . . . . 10 ..^ | |
53 | 51, 52 | eqsstrrdi 3145 | . . . . . . . . 9 |
54 | 53 | sselda 3092 | . . . . . . . 8 |
55 | 3 | adantlr 468 | . . . . . . . 8 |
56 | 54, 55 | syldan 280 | . . . . . . 7 |
57 | 48, 56, 1 | fsum1p 11180 | . . . . . 6 |
58 | 51 | sumeq1d 11128 | . . . . . 6 ..^ |
59 | fzoval 9918 | . . . . . . . . 9 ..^ | |
60 | 49, 59 | syl 14 | . . . . . . . 8 ..^ |
61 | 60 | sumeq1d 11128 | . . . . . . 7 ..^ |
62 | 61 | oveq2d 5783 | . . . . . 6 ..^ |
63 | 57, 58, 62 | 3eqtr4d 2180 | . . . . 5 ..^ ..^ |
64 | 46, 63 | syl5eqr 2184 | . . . 4 ..^ ..^ |
65 | simpr 109 | . . . . . 6 | |
66 | fzp1ss 9846 | . . . . . . . . . 10 | |
67 | 28, 66 | syl 14 | . . . . . . . . 9 |
68 | 67 | sselda 3092 | . . . . . . . 8 |
69 | 68, 3 | syldan 280 | . . . . . . 7 |
70 | 69 | adantlr 468 | . . . . . 6 |
71 | 65, 70, 19 | fsumm1 11178 | . . . . 5 |
72 | 1zzd 9074 | . . . . . . . 8 | |
73 | 28 | peano2zd 9169 | . . . . . . . 8 |
74 | 72, 73, 30, 69, 39 | fsumshftm 11207 | . . . . . . 7 |
75 | 28 | zcnd 9167 | . . . . . . . . . . 11 |
76 | ax-1cn 7706 | . . . . . . . . . . 11 | |
77 | pncan 7961 | . . . . . . . . . . 11 | |
78 | 75, 76, 77 | sylancl 409 | . . . . . . . . . 10 |
79 | 78 | oveq1d 5782 | . . . . . . . . 9 |
80 | 30, 50 | syl 14 | . . . . . . . . 9 ..^ |
81 | 79, 80 | eqtr4d 2173 | . . . . . . . 8 ..^ |
82 | 81 | sumeq1d 11128 | . . . . . . 7 ..^ |
83 | 74, 82 | eqtrd 2170 | . . . . . 6 ..^ |
84 | 83 | adantr 274 | . . . . 5 ..^ |
85 | 30, 59 | syl 14 | . . . . . . . . 9 ..^ |
86 | 85 | sumeq1d 11128 | . . . . . . . 8 ..^ |
87 | 86 | oveq1d 5782 | . . . . . . 7 ..^ |
88 | fzofig 10198 | . . . . . . . . . 10 ..^ | |
89 | 73, 30, 88 | syl2anc 408 | . . . . . . . . 9 ..^ |
90 | elfzofz 9932 | . . . . . . . . . 10 ..^ | |
91 | 90, 69 | sylan2 284 | . . . . . . . . 9 ..^ |
92 | 89, 91 | fsumcl 11162 | . . . . . . . 8 ..^ |
93 | 19 | eleq1d 2206 | . . . . . . . . 9 |
94 | eluzfz2 9805 | . . . . . . . . . 10 | |
95 | 5, 94 | syl 14 | . . . . . . . . 9 |
96 | 93, 4, 95 | rspcdva 2789 | . . . . . . . 8 |
97 | 92, 96 | addcomd 7906 | . . . . . . 7 ..^ ..^ |
98 | 87, 97 | eqtr3d 2172 | . . . . . 6 ..^ |
99 | 98 | adantr 274 | . . . . 5 ..^ |
100 | 71, 84, 99 | 3eqtr3d 2178 | . . . 4 ..^ ..^ |
101 | 64, 100 | oveq12d 5785 | . . 3 ..^ ..^ ..^ ..^ |
102 | 8, 96, 92 | pnpcan2d 8104 | . . . 4 ..^ ..^ |
103 | 102 | adantr 274 | . . 3 ..^ ..^ |
104 | 45, 101, 103 | 3eqtrd 2174 | . 2 ..^ |
105 | uzp1 9352 | . . 3 | |
106 | 5, 105 | syl 14 | . 2 |
107 | 26, 104, 106 | mpjaodan 787 | 1 ..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 wral 2414 wss 3066 c0 3358 cfv 5118 (class class class)co 5767 cfn 6627 cc 7611 cc0 7613 c1 7614 caddc 7616 cmin 7926 cz 9047 cuz 9319 cfz 9783 ..^cfzo 9912 csu 11115 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-ihash 10515 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 |
This theorem is referenced by: telfsumo2 11229 telfsum 11230 geosergap 11268 |
Copyright terms: Public domain | W3C validator |