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Mirrors > Home > ILE Home > Th. List > cvgcmpub | Unicode version |
Description: An upper bound for the limit of a real infinite series. This theorem can also be used to compare two infinite series. (Contributed by Mario Carneiro, 24-Mar-2014.) |
Ref | Expression |
---|---|
cvgcmp.1 | |
cvgcmp.2 | |
cvgcmp.3 | |
cvgcmp.4 | |
cvgcmpub.5 | |
cvgcmpub.6 | |
cvgcmpub.7 |
Ref | Expression |
---|---|
cvgcmpub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvgcmp.1 | . 2 | |
2 | cvgcmp.2 | . . . 4 | |
3 | 2, 1 | eleqtrdi 2233 | . . 3 |
4 | eluzel2 9355 | . . 3 | |
5 | 3, 4 | syl 14 | . 2 |
6 | cvgcmpub.6 | . 2 | |
7 | cvgcmpub.5 | . 2 | |
8 | cvgcmp.4 | . . . 4 | |
9 | 1, 5, 8 | serfre 10279 | . . 3 |
10 | 9 | ffvelrnda 5563 | . 2 |
11 | cvgcmp.3 | . . . 4 | |
12 | 1, 5, 11 | serfre 10279 | . . 3 |
13 | 12 | ffvelrnda 5563 | . 2 |
14 | simpr 109 | . . . 4 | |
15 | 14, 1 | eleqtrdi 2233 | . . 3 |
16 | simpl 108 | . . . 4 | |
17 | 1 | eleq2i 2207 | . . . . 5 |
18 | 17 | biimpri 132 | . . . 4 |
19 | 16, 18, 8 | syl2an 287 | . . 3 |
20 | 16, 18, 11 | syl2an 287 | . . 3 |
21 | cvgcmpub.7 | . . . 4 | |
22 | 16, 18, 21 | syl2an 287 | . . 3 |
23 | 15, 19, 20, 22 | ser3le 10322 | . 2 |
24 | 1, 5, 6, 7, 10, 13, 23 | climle 11135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1332 wcel 1481 class class class wbr 3937 cfv 5131 cr 7643 caddc 7647 cle 7825 cz 9078 cuz 9350 cseq 10249 cli 11079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 |
This theorem is referenced by: (None) |
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