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Mirrors > Home > MPE Home > Th. List > iserle | Structured version Visualization version GIF version |
Description: Comparison of the limits of two infinite series. (Contributed by Paul Chapman, 12-Nov-2007.) (Revised by Mario Carneiro, 3-Feb-2014.) |
Ref | Expression |
---|---|
clim2ser.1 | β’ π = (β€β₯βπ) |
iserle.2 | β’ (π β π β β€) |
iserle.4 | β’ (π β seqπ( + , πΉ) β π΄) |
iserle.5 | β’ (π β seqπ( + , πΊ) β π΅) |
iserle.6 | β’ ((π β§ π β π) β (πΉβπ) β β) |
iserle.7 | β’ ((π β§ π β π) β (πΊβπ) β β) |
iserle.8 | β’ ((π β§ π β π) β (πΉβπ) β€ (πΊβπ)) |
Ref | Expression |
---|---|
iserle | β’ (π β π΄ β€ π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2ser.1 | . 2 β’ π = (β€β₯βπ) | |
2 | iserle.2 | . 2 β’ (π β π β β€) | |
3 | iserle.4 | . 2 β’ (π β seqπ( + , πΉ) β π΄) | |
4 | iserle.5 | . 2 β’ (π β seqπ( + , πΊ) β π΅) | |
5 | iserle.6 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) β β) | |
6 | 1, 2, 5 | serfre 13943 | . . 3 β’ (π β seqπ( + , πΉ):πβΆβ) |
7 | 6 | ffvelcdmda 7036 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β β) |
8 | iserle.7 | . . . 4 β’ ((π β§ π β π) β (πΊβπ) β β) | |
9 | 1, 2, 8 | serfre 13943 | . . 3 β’ (π β seqπ( + , πΊ):πβΆβ) |
10 | 9 | ffvelcdmda 7036 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΊ)βπ) β β) |
11 | simpr 486 | . . . 4 β’ ((π β§ π β π) β π β π) | |
12 | 11, 1 | eleqtrdi 2844 | . . 3 β’ ((π β§ π β π) β π β (β€β₯βπ)) |
13 | simpll 766 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β π) | |
14 | elfzuz 13443 | . . . . . 6 β’ (π β (π...π) β π β (β€β₯βπ)) | |
15 | 14, 1 | eleqtrrdi 2845 | . . . . 5 β’ (π β (π...π) β π β π) |
16 | 15 | adantl 483 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β π β π) |
17 | 13, 16, 5 | syl2anc 585 | . . 3 β’ (((π β§ π β π) β§ π β (π...π)) β (πΉβπ) β β) |
18 | 13, 16, 8 | syl2anc 585 | . . 3 β’ (((π β§ π β π) β§ π β (π...π)) β (πΊβπ) β β) |
19 | iserle.8 | . . . 4 β’ ((π β§ π β π) β (πΉβπ) β€ (πΊβπ)) | |
20 | 13, 16, 19 | syl2anc 585 | . . 3 β’ (((π β§ π β π) β§ π β (π...π)) β (πΉβπ) β€ (πΊβπ)) |
21 | 12, 17, 18, 20 | serle 13969 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β€ (seqπ( + , πΊ)βπ)) |
22 | 1, 2, 3, 4, 7, 10, 21 | climle 15528 | 1 β’ (π β π΄ β€ π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 class class class wbr 5106 βcfv 6497 (class class class)co 7358 βcr 11055 + caddc 11059 β€ cle 11195 β€cz 12504 β€β₯cuz 12768 ...cfz 13430 seqcseq 13912 β cli 15372 |
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-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-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-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-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-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 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-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-clim 15376 df-rlim 15377 |
This theorem is referenced by: iserge0 15551 isumle 15734 ege2le3 15977 |
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