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Mirrors > Home > ILE Home > Th. List > climshft2 | GIF 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 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climshft2.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climshft2.3 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
climshft2.5 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
climshft2.6 | ⊢ (𝜑 → 𝐺 ∈ 𝑋) |
climshft2.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 𝐾)) = (𝐹‘𝑘)) |
Ref | Expression |
---|---|
climshft2 | ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climshft2.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climshft2.6 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑋) | |
3 | climshft2.3 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
4 | 3 | zcnd 8361 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
5 | 4 | negcld 7309 | . . . 4 ⊢ (𝜑 → -𝐾 ∈ ℂ) |
6 | ovshftex 9420 | . . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ -𝐾 ∈ ℂ) → (𝐺 shift -𝐾) ∈ V) | |
7 | 2, 5, 6 | syl2anc 391 | . . 3 ⊢ (𝜑 → (𝐺 shift -𝐾) ∈ V) |
8 | climshft2.5 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
9 | climshft2.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | funi 4932 | . . . . . . . 8 ⊢ Fun I | |
11 | elex 2566 | . . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑋 → 𝐺 ∈ V) | |
12 | 2, 11 | syl 14 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ V) |
13 | dmi 4550 | . . . . . . . . 9 ⊢ dom I = V | |
14 | 12, 13 | syl6eleqr 2131 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ dom I ) |
15 | funfvex 5192 | . . . . . . . 8 ⊢ ((Fun I ∧ 𝐺 ∈ dom I ) → ( I ‘𝐺) ∈ V) | |
16 | 10, 14, 15 | sylancr 393 | . . . . . . 7 ⊢ (𝜑 → ( I ‘𝐺) ∈ V) |
17 | 16 | adantr 261 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) ∈ V) |
18 | 4 | adantr 261 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐾 ∈ ℂ) |
19 | eluzelz 8482 | . . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
20 | 19, 1 | eleq2s 2132 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
21 | 20 | zcnd 8361 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ) |
22 | 21 | adantl 262 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℂ) |
23 | shftval4g 9438 | . . . . . 6 ⊢ ((( I ‘𝐺) ∈ V ∧ 𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) | |
24 | 17, 18, 22, 23 | syl3anc 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) |
25 | fvi 5230 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝑋 → ( I ‘𝐺) = 𝐺) | |
26 | 2, 25 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → ( I ‘𝐺) = 𝐺) |
27 | 26 | adantr 261 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) = 𝐺) |
28 | 27 | oveq1d 5527 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺) shift -𝐾) = (𝐺 shift -𝐾)) |
29 | 28 | fveq1d 5180 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = ((𝐺 shift -𝐾)‘𝑘)) |
30 | addcom 7150 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) | |
31 | 4, 21, 30 | syl2an 273 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) |
32 | 27, 31 | fveq12d 5184 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺)‘(𝐾 + 𝑘)) = (𝐺‘(𝑘 + 𝐾))) |
33 | 24, 29, 32 | 3eqtr3d 2080 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺 shift -𝐾)‘𝑘) = (𝐺‘(𝑘 + 𝐾))) |
34 | climshft2.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 𝐾)) = (𝐹‘𝑘)) | |
35 | 33, 34 | eqtrd 2072 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺 shift -𝐾)‘𝑘) = (𝐹‘𝑘)) |
36 | 1, 7, 8, 9, 35 | climeq 9820 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
37 | 3 | znegcld 8362 | . . 3 ⊢ (𝜑 → -𝐾 ∈ ℤ) |
38 | climshft 9825 | . . 3 ⊢ ((-𝐾 ∈ ℤ ∧ 𝐺 ∈ 𝑋) → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | |
39 | 37, 2, 38 | syl2anc 391 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
40 | 36, 39 | bitr3d 179 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 class class class wbr 3764 I cid 4025 dom cdm 4345 Fun wfun 4896 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 + caddc 6892 -cneg 7183 ℤcz 8245 ℤ≥cuz 8473 shift cshi 9415 ⇝ cli 9799 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-if 3332 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-shft 9416 df-clim 9800 |
This theorem is referenced by: (None) |
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