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Mirrors > Home > ILE Home > Th. List > isummulc2 | Unicode version |
Description: An infinite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumcl.1 |
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isumcl.2 |
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isumcl.3 |
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isumcl.4 |
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isumcl.5 |
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summulc.6 |
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Ref | Expression |
---|---|
isummulc2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumcl.1 |
. . 3
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2 | isumcl.2 |
. . 3
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3 | eqidd 2190 |
. . 3
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4 | summulc.6 |
. . . . . . 7
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5 | 4 | adantr 276 |
. . . . . 6
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6 | isumcl.4 |
. . . . . 6
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7 | 5, 6 | mulcld 8008 |
. . . . 5
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8 | 7 | fmpttd 5692 |
. . . 4
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9 | 8 | ffvelcdmda 5672 |
. . 3
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10 | isumcl.3 |
. . . . 5
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11 | isumcl.5 |
. . . . 5
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12 | 1, 2, 10, 6, 11 | isumclim2 11462 |
. . . 4
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13 | 10, 6 | eqeltrd 2266 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13 | ralrimiva 2563 |
. . . . 5
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15 | fveq2 5534 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | eleq1d 2258 |
. . . . . 6
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17 | 16 | rspccva 2855 |
. . . . 5
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18 | 14, 17 | sylan 283 |
. . . 4
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19 | simpr 110 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | eqid 2189 |
. . . . . . . . 9
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21 | 20 | fvmpt2 5620 |
. . . . . . . 8
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22 | 19, 7, 21 | syl2anc 411 |
. . . . . . 7
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23 | 10 | oveq2d 5912 |
. . . . . . 7
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24 | 22, 23 | eqtr4d 2225 |
. . . . . 6
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25 | 24 | ralrimiva 2563 |
. . . . 5
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26 | nffvmpt1 5545 |
. . . . . . 7
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27 | 26 | nfeq1 2342 |
. . . . . 6
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28 | fveq2 5534 |
. . . . . . 7
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29 | 15 | oveq2d 5912 |
. . . . . . 7
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30 | 28, 29 | eqeq12d 2204 |
. . . . . 6
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31 | 27, 30 | rspc 2850 |
. . . . 5
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32 | 25, 31 | mpan9 281 |
. . . 4
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33 | 1, 2, 4, 12, 18, 32 | isermulc2 11380 |
. . 3
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34 | 1, 2, 3, 9, 33 | isumclim 11461 |
. 2
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35 | 7 | ralrimiva 2563 |
. . 3
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36 | sumfct 11414 |
. . 3
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37 | 35, 36 | syl 14 |
. 2
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38 | 34, 37 | eqtr3d 2224 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-frec 6416 df-1o 6441 df-oadd 6445 df-er 6559 df-en 6767 df-dom 6768 df-fin 6769 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-fz 10039 df-fzo 10173 df-seqfrec 10477 df-exp 10551 df-ihash 10788 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 df-clim 11319 df-sumdc 11394 |
This theorem is referenced by: isummulc1 11467 trirecip 11541 geoisum1c 11560 |
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