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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 11406 | . 2 β’ (π β Ξ£π β π π΄ = ( β βseqπ( + , πΉ))) |
6 | fclim 11315 | . . . 4 β’ β :dom β βΆβ | |
7 | ffun 5380 | . . . 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 5567 | . . 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 2220 | 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 1363 β wcel 2158 class class class wbr 4015 dom cdm 4638 Fun wfun 5222 βΆwf 5224 βcfv 5228 βcc 7822 + caddc 7827 β€cz 9266 β€β₯cuz 9541 seqcseq 10458 β cli 11299 Ξ£csu 11374 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-frec 6405 df-1o 6430 df-oadd 6434 df-er 6548 df-en 6754 df-dom 6755 df-fin 6756 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-fz 10022 df-fzo 10156 df-seqfrec 10459 df-exp 10533 df-ihash 10769 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-clim 11300 df-sumdc 11375 |
This theorem is referenced by: isummulc2 11447 isumadd 11452 isumsplit 11512 trirecip 11522 geolim2 11533 geoisum 11538 geoisumr 11539 geoisum1 11540 eftlub 11711 eflegeo 11722 |
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