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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 13998 | . . 3 β’ (π β seqπ( + , πΉ):πβΆβ) |
7 | 6 | ffvelcdmda 7077 | . 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 13998 | . . 3 β’ (π β seqπ( + , πΊ):πβΆβ) |
10 | 9 | ffvelcdmda 7077 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΊ)βπ) β β) |
11 | simpr 484 | . . . 4 β’ ((π β§ π β π) β π β π) | |
12 | 11, 1 | eleqtrdi 2835 | . . 3 β’ ((π β§ π β π) β π β (β€β₯βπ)) |
13 | simpll 764 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β π) | |
14 | elfzuz 13498 | . . . . . 6 β’ (π β (π...π) β π β (β€β₯βπ)) | |
15 | 14, 1 | eleqtrrdi 2836 | . . . . 5 β’ (π β (π...π) β π β π) |
16 | 15 | adantl 481 | . . . 4 β’ (((π β§ π β π) β§ π β (π...π)) β π β π) |
17 | 13, 16, 5 | syl2anc 583 | . . 3 β’ (((π β§ π β π) β§ π β (π...π)) β (πΉβπ) β β) |
18 | 13, 16, 8 | syl2anc 583 | . . 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 583 | . . 3 β’ (((π β§ π β π) β§ π β (π...π)) β (πΉβπ) β€ (πΊβπ)) |
21 | 12, 17, 18, 20 | serle 14024 | . 2 β’ ((π β§ π β π) β (seqπ( + , πΉ)βπ) β€ (seqπ( + , πΊ)βπ)) |
22 | 1, 2, 3, 4, 7, 10, 21 | climle 15586 | 1 β’ (π β π΄ β€ π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 class class class wbr 5139 βcfv 6534 (class class class)co 7402 βcr 11106 + caddc 11110 β€ cle 11248 β€cz 12557 β€β₯cuz 12821 ...cfz 13485 seqcseq 13967 β cli 15430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 |
This theorem is referenced by: iserge0 15609 isumle 15792 ege2le3 16036 |
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