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Mirrors > Home > MPE Home > Th. List > isumrecl | Structured version Visualization version GIF version |
Description: The sum of a converging infinite real series is a real number. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumrecl.1 | β’ π = (β€β₯βπ) |
isumrecl.2 | β’ (π β π β β€) |
isumrecl.3 | β’ ((π β§ π β π) β (πΉβπ) = π΄) |
isumrecl.4 | β’ ((π β§ π β π) β π΄ β β) |
isumrecl.5 | β’ (π β seqπ( + , πΉ) β dom β ) |
Ref | Expression |
---|---|
isumrecl | β’ (π β Ξ£π β π π΄ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumrecl.1 | . 2 β’ π = (β€β₯βπ) | |
2 | isumrecl.2 | . 2 β’ (π β π β β€) | |
3 | isumrecl.3 | . . 3 β’ ((π β§ π β π) β (πΉβπ) = π΄) | |
4 | isumrecl.4 | . . . 4 β’ ((π β§ π β π) β π΄ β β) | |
5 | 4 | recnd 11188 | . . 3 β’ ((π β§ π β π) β π΄ β β) |
6 | isumrecl.5 | . . 3 β’ (π β seqπ( + , πΉ) β dom β ) | |
7 | 1, 2, 3, 5, 6 | isumclim2 15648 | . 2 β’ (π β seqπ( + , πΉ) β Ξ£π β π π΄) |
8 | 3, 4 | eqeltrd 2834 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) β β) |
9 | 1, 2, 8 | serfre 13943 | . . 3 β’ (π β seqπ( + , πΉ):πβΆβ) |
10 | 9 | ffvelcdmda 7036 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β β) |
11 | 1, 2, 7, 10 | climrecl 15471 | 1 β’ (π β Ξ£π β π π΄ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 dom cdm 5634 βcfv 6497 βcr 11055 + caddc 11059 β€cz 12504 β€β₯cuz 12768 seqcseq 13912 β cli 15372 Ξ£csu 15576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-fz 13431 df-fzo 13574 df-fl 13703 df-seq 13913 df-exp 13974 df-hash 14237 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 df-sum 15577 |
This theorem is referenced by: isumrpcl 15733 isumltss 15738 climcnds 15741 harmonic 15749 mertenslem1 15774 mertenslem2 15775 reefcl 15974 reeftlcl 15995 rpnnen2lem6 16106 prmreclem5 16797 prmreclem6 16798 ovoliun2 24886 abelthlem7 25813 log2tlbnd 26311 esumpcvgval 32734 esumcvg 32742 eulerpartlems 33017 knoppf 35044 geomcau 36264 stirlinglem12 44412 |
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