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Mirrors > Home > ILE Home > Th. List > ser3le | Unicode version |
Description: Comparison of partial sums of two infinite series of reals. (Contributed by NM, 27-Dec-2005.) (Revised by Jim Kingdon, 23-Apr-2023.) |
Ref | Expression |
---|---|
ser3ge0.1 | |
ser3ge0.2 | |
ser3le.3 | |
serle.4 |
Ref | Expression |
---|---|
ser3le |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ser3ge0.1 | . . . 4 | |
2 | vex 2715 | . . . . . 6 | |
3 | ser3le.3 | . . . . . . 7 | |
4 | ser3ge0.2 | . . . . . . 7 | |
5 | 3, 4 | resubcld 8256 | . . . . . 6 |
6 | fveq2 5468 | . . . . . . . 8 | |
7 | fveq2 5468 | . . . . . . . 8 | |
8 | 6, 7 | oveq12d 5842 | . . . . . . 7 |
9 | eqid 2157 | . . . . . . 7 | |
10 | 8, 9 | fvmptg 5544 | . . . . . 6 |
11 | 2, 5, 10 | sylancr 411 | . . . . 5 |
12 | 11, 5 | eqeltrd 2234 | . . . 4 |
13 | elfzuz 9924 | . . . . . . 7 | |
14 | serle.4 | . . . . . . 7 | |
15 | 13, 14 | sylan2 284 | . . . . . 6 |
16 | 3, 4 | subge0d 8410 | . . . . . . 7 |
17 | 13, 16 | sylan2 284 | . . . . . 6 |
18 | 15, 17 | mpbird 166 | . . . . 5 |
19 | 13, 11 | sylan2 284 | . . . . 5 |
20 | 18, 19 | breqtrrd 3992 | . . . 4 |
21 | 1, 12, 20 | ser3ge0 10416 | . . 3 |
22 | 3 | recnd 7906 | . . . 4 |
23 | 4 | recnd 7906 | . . . 4 |
24 | 1, 22, 23, 11 | ser3sub 10405 | . . 3 |
25 | 21, 24 | breqtrd 3990 | . 2 |
26 | eqid 2157 | . . . . 5 | |
27 | eluzel2 9444 | . . . . . 6 | |
28 | 1, 27 | syl 14 | . . . . 5 |
29 | 26, 28, 3 | serfre 10374 | . . . 4 |
30 | 29, 1 | ffvelrnd 5603 | . . 3 |
31 | 26, 28, 4 | serfre 10374 | . . . 4 |
32 | 31, 1 | ffvelrnd 5603 | . . 3 |
33 | 30, 32 | subge0d 8410 | . 2 |
34 | 25, 33 | mpbid 146 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cvv 2712 class class class wbr 3965 cmpt 4025 cfv 5170 (class class class)co 5824 cr 7731 cc0 7732 caddc 7735 cle 7913 cmin 8046 cz 9167 cuz 9439 cfz 9912 cseq 10344 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-n0 9091 df-z 9168 df-uz 9440 df-fz 9913 df-fzo 10042 df-seqfrec 10345 |
This theorem is referenced by: iserle 11239 cvgcmpub 11373 |
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