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Mirrors > Home > ILE Home > Th. List > isumdivapc | Unicode version |
Description: An infinite sum divided by a constant. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumcl.1 | |
isumcl.2 | |
isumcl.3 | |
isumcl.4 | |
isumcl.5 | |
summulc.6 | |
isumdivapc.7 | # |
Ref | Expression |
---|---|
isumdivapc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumcl.1 | . . 3 | |
2 | isumcl.2 | . . 3 | |
3 | isumcl.3 | . . 3 | |
4 | isumcl.4 | . . 3 | |
5 | isumcl.5 | . . 3 | |
6 | summulc.6 | . . . 4 | |
7 | isumdivapc.7 | . . . 4 # | |
8 | 6, 7 | recclapd 8546 | . . 3 |
9 | 1, 2, 3, 4, 5, 8 | isummulc1 11201 | . 2 |
10 | 1, 2, 3, 4, 5 | isumcl 11199 | . . 3 |
11 | 10, 6, 7 | divrecapd 8558 | . 2 |
12 | 6 | adantr 274 | . . . 4 |
13 | 7 | adantr 274 | . . . 4 # |
14 | 4, 12, 13 | divrecapd 8558 | . . 3 |
15 | 14 | sumeq2dv 11142 | . 2 |
16 | 9, 11, 15 | 3eqtr4d 2182 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 class class class wbr 3929 cdm 4539 cfv 5123 (class class class)co 5774 cc 7623 cc0 7625 c1 7626 caddc 7628 cmul 7630 # cap 8348 cdiv 8437 cz 9059 cuz 9331 cseq 10223 cli 11052 csu 11127 |
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 7716 ax-resscn 7717 ax-1cn 7718 ax-1re 7719 ax-icn 7720 ax-addcl 7721 ax-addrcl 7722 ax-mulcl 7723 ax-mulrcl 7724 ax-addcom 7725 ax-mulcom 7726 ax-addass 7727 ax-mulass 7728 ax-distr 7729 ax-i2m1 7730 ax-0lt1 7731 ax-1rid 7732 ax-0id 7733 ax-rnegex 7734 ax-precex 7735 ax-cnre 7736 ax-pre-ltirr 7737 ax-pre-ltwlin 7738 ax-pre-lttrn 7739 ax-pre-apti 7740 ax-pre-ltadd 7741 ax-pre-mulgt0 7742 ax-pre-mulext 7743 ax-arch 7744 ax-caucvg 7745 |
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 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 df-sub 7940 df-neg 7941 df-reap 8342 df-ap 8349 df-div 8438 df-inn 8726 df-2 8784 df-3 8785 df-4 8786 df-n0 8983 df-z 9060 df-uz 9332 df-q 9417 df-rp 9447 df-fz 9796 df-fzo 9925 df-seqfrec 10224 df-exp 10298 df-ihash 10527 df-cj 10619 df-re 10620 df-im 10621 df-rsqrt 10775 df-abs 10776 df-clim 11053 df-sumdc 11128 |
This theorem is referenced by: (None) |
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