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Mirrors > Home > ILE Home > Th. List > isumclim | GIF version |
Description: An infinite sum equals the value its series converges to. (Contributed by NM, 25-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumclim.1 | β’ π = (β€β₯βπ) |
isumclim.2 | β’ (π β π β β€) |
isumclim.3 | β’ ((π β§ π β π) β (πΉβπ) = π΄) |
isumclim.4 | β’ ((π β§ π β π) β π΄ β β) |
isumclim.6 | β’ (π β seqπ( + , πΉ) β π΅) |
Ref | Expression |
---|---|
isumclim | β’ (π β Ξ£π β π π΄ = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumclim.1 | . . 3 β’ π = (β€β₯βπ) | |
2 | isumclim.2 | . . 3 β’ (π β π β β€) | |
3 | isumclim.3 | . . 3 β’ ((π β§ π β π) β (πΉβπ) = π΄) | |
4 | isumclim.4 | . . 3 β’ ((π β§ π β π) β π΄ β β) | |
5 | 1, 2, 3, 4 | isum 11411 | . 2 β’ (π β Ξ£π β π π΄ = ( β βseqπ( + , πΉ))) |
6 | fclim 11320 | . . . 4 β’ β :dom β βΆβ | |
7 | ffun 5383 | . . . 4 β’ ( β :dom β βΆβ β Fun β ) | |
8 | 6, 7 | ax-mp 5 | . . 3 β’ Fun β |
9 | isumclim.6 | . . 3 β’ (π β seqπ( + , πΉ) β π΅) | |
10 | funbrfv 5570 | . . 3 β’ (Fun β β (seqπ( + , πΉ) β π΅ β ( β βseqπ( + , πΉ)) = π΅)) | |
11 | 8, 9, 10 | mpsyl 65 | . 2 β’ (π β ( β βseqπ( + , πΉ)) = π΅) |
12 | 5, 11 | eqtrd 2222 | 1 β’ (π β Ξ£π β π π΄ = π΅) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1364 β wcel 2160 class class class wbr 4018 dom cdm 4641 Fun wfun 5225 βΆwf 5227 βcfv 5231 βcc 7827 + caddc 7832 β€cz 9271 β€β₯cuz 9546 seqcseq 10463 β cli 11304 Ξ£csu 11379 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 ax-arch 7948 ax-caucvg 7949 |
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 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-en 6759 df-dom 6760 df-fin 6761 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-2 8996 df-3 8997 df-4 8998 df-n0 9195 df-z 9272 df-uz 9547 df-q 9638 df-rp 9672 df-fz 10027 df-fzo 10161 df-seqfrec 10464 df-exp 10538 df-ihash 10774 df-cj 10869 df-re 10870 df-im 10871 df-rsqrt 11025 df-abs 11026 df-clim 11305 df-sumdc 11380 |
This theorem is referenced by: isummulc2 11452 isumadd 11457 isumsplit 11517 trirecip 11527 geolim2 11538 geoisum 11543 geoisumr 11544 geoisum1 11545 eftlub 11716 eflegeo 11727 |
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