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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 8932 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
5 | 4 | negcld 7843 | . . . 4 ⊢ (𝜑 → -𝐾 ∈ ℂ) |
6 | ovshftex 10316 | . . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ -𝐾 ∈ ℂ) → (𝐺 shift -𝐾) ∈ V) | |
7 | 2, 5, 6 | syl2anc 404 | . . 3 ⊢ (𝜑 → (𝐺 shift -𝐾) ∈ V) |
8 | climshft2.5 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
9 | climshft2.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | funi 5061 | . . . . . . . 8 ⊢ Fun I | |
11 | elex 2633 | . . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑋 → 𝐺 ∈ V) | |
12 | 2, 11 | syl 14 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ V) |
13 | dmi 4666 | . . . . . . . . 9 ⊢ dom I = V | |
14 | 12, 13 | syl6eleqr 2182 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ dom I ) |
15 | funfvex 5337 | . . . . . . . 8 ⊢ ((Fun I ∧ 𝐺 ∈ dom I ) → ( I ‘𝐺) ∈ V) | |
16 | 10, 14, 15 | sylancr 406 | . . . . . . 7 ⊢ (𝜑 → ( I ‘𝐺) ∈ V) |
17 | 16 | adantr 271 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) ∈ V) |
18 | 4 | adantr 271 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐾 ∈ ℂ) |
19 | eluzelz 9091 | . . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
20 | 19, 1 | eleq2s 2183 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
21 | 20 | zcnd 8932 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ) |
22 | 21 | adantl 272 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℂ) |
23 | shftval4g 10334 | . . . . . 6 ⊢ ((( I ‘𝐺) ∈ V ∧ 𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) | |
24 | 17, 18, 22, 23 | syl3anc 1175 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) |
25 | fvi 5376 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝑋 → ( I ‘𝐺) = 𝐺) | |
26 | 2, 25 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → ( I ‘𝐺) = 𝐺) |
27 | 26 | adantr 271 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) = 𝐺) |
28 | 27 | oveq1d 5683 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺) shift -𝐾) = (𝐺 shift -𝐾)) |
29 | 28 | fveq1d 5322 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = ((𝐺 shift -𝐾)‘𝑘)) |
30 | addcom 7682 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) | |
31 | 4, 21, 30 | syl2an 284 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) |
32 | 27, 31 | fveq12d 5327 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺)‘(𝐾 + 𝑘)) = (𝐺‘(𝑘 + 𝐾))) |
33 | 24, 29, 32 | 3eqtr3d 2129 | . . . 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 2121 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺 shift -𝐾)‘𝑘) = (𝐹‘𝑘)) |
36 | 1, 7, 8, 9, 35 | climeq 10750 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
37 | 3 | znegcld 8933 | . . 3 ⊢ (𝜑 → -𝐾 ∈ ℤ) |
38 | climshft 10755 | . . 3 ⊢ ((-𝐾 ∈ ℤ ∧ 𝐺 ∈ 𝑋) → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | |
39 | 37, 2, 38 | syl2anc 404 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
40 | 36, 39 | bitr3d 189 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 Vcvv 2622 class class class wbr 3853 I cid 4126 dom cdm 4454 Fun wfun 5024 ‘cfv 5030 (class class class)co 5668 ℂcc 7411 + caddc 7416 -cneg 7717 ℤcz 8813 ℤ≥cuz 9082 shift cshi 10311 ⇝ cli 10729 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-inn 8486 df-n0 8737 df-z 8814 df-uz 9083 df-shft 10312 df-clim 10730 |
This theorem is referenced by: trireciplem 10957 |
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