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Mirrors > Home > ILE Home > Th. List > climshft2 | Unicode version |
Description: A shifted function converges iff the original function converges. (Contributed by Paul Chapman, 21-Nov-2007.) (Revised by Mario Carneiro, 6-Feb-2014.) |
Ref | Expression |
---|---|
climshft2.1 | |
climshft2.2 | |
climshft2.3 | |
climshft2.5 | |
climshft2.6 | |
climshft2.7 |
Ref | Expression |
---|---|
climshft2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climshft2.1 | . . 3 | |
2 | climshft2.6 | . . . 4 | |
3 | climshft2.3 | . . . . . 6 | |
4 | 3 | zcnd 9142 | . . . . 5 |
5 | 4 | negcld 8028 | . . . 4 |
6 | ovshftex 10559 | . . . 4 | |
7 | 2, 5, 6 | syl2anc 408 | . . 3 |
8 | climshft2.5 | . . 3 | |
9 | climshft2.2 | . . 3 | |
10 | funi 5125 | . . . . . . . 8 | |
11 | elex 2671 | . . . . . . . . . 10 | |
12 | 2, 11 | syl 14 | . . . . . . . . 9 |
13 | dmi 4724 | . . . . . . . . 9 | |
14 | 12, 13 | eleqtrrdi 2211 | . . . . . . . 8 |
15 | funfvex 5406 | . . . . . . . 8 | |
16 | 10, 14, 15 | sylancr 410 | . . . . . . 7 |
17 | 16 | adantr 274 | . . . . . 6 |
18 | 4 | adantr 274 | . . . . . 6 |
19 | eluzelz 9303 | . . . . . . . . 9 | |
20 | 19, 1 | eleq2s 2212 | . . . . . . . 8 |
21 | 20 | zcnd 9142 | . . . . . . 7 |
22 | 21 | adantl 275 | . . . . . 6 |
23 | shftval4g 10577 | . . . . . 6 | |
24 | 17, 18, 22, 23 | syl3anc 1201 | . . . . 5 |
25 | fvi 5446 | . . . . . . . . 9 | |
26 | 2, 25 | syl 14 | . . . . . . . 8 |
27 | 26 | adantr 274 | . . . . . . 7 |
28 | 27 | oveq1d 5757 | . . . . . 6 |
29 | 28 | fveq1d 5391 | . . . . 5 |
30 | addcom 7867 | . . . . . . 7 | |
31 | 4, 21, 30 | syl2an 287 | . . . . . 6 |
32 | 27, 31 | fveq12d 5396 | . . . . 5 |
33 | 24, 29, 32 | 3eqtr3d 2158 | . . . 4 |
34 | climshft2.7 | . . . 4 | |
35 | 33, 34 | eqtrd 2150 | . . 3 |
36 | 1, 7, 8, 9, 35 | climeq 11036 | . 2 |
37 | 3 | znegcld 9143 | . . 3 |
38 | climshft 11041 | . . 3 | |
39 | 37, 2, 38 | syl2anc 408 | . 2 |
40 | 36, 39 | bitr3d 189 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cvv 2660 class class class wbr 3899 cid 4180 cdm 4509 wfun 5087 cfv 5093 (class class class)co 5742 cc 7586 caddc 7591 cneg 7902 cz 9022 cuz 9294 cshi 10554 cli 11015 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-shft 10555 df-clim 11016 |
This theorem is referenced by: trireciplem 11237 |
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