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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 9378 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
5 | 4 | negcld 8257 | . . . 4 ⊢ (𝜑 → -𝐾 ∈ ℂ) |
6 | ovshftex 10830 | . . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ -𝐾 ∈ ℂ) → (𝐺 shift -𝐾) ∈ V) | |
7 | 2, 5, 6 | syl2anc 411 | . . 3 ⊢ (𝜑 → (𝐺 shift -𝐾) ∈ V) |
8 | climshft2.5 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
9 | climshft2.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | funi 5250 | . . . . . . . 8 ⊢ Fun I | |
11 | elex 2750 | . . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑋 → 𝐺 ∈ V) | |
12 | 2, 11 | syl 14 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ V) |
13 | dmi 4844 | . . . . . . . . 9 ⊢ dom I = V | |
14 | 12, 13 | eleqtrrdi 2271 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ dom I ) |
15 | funfvex 5534 | . . . . . . . 8 ⊢ ((Fun I ∧ 𝐺 ∈ dom I ) → ( I ‘𝐺) ∈ V) | |
16 | 10, 14, 15 | sylancr 414 | . . . . . . 7 ⊢ (𝜑 → ( I ‘𝐺) ∈ V) |
17 | 16 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) ∈ V) |
18 | 4 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐾 ∈ ℂ) |
19 | eluzelz 9539 | . . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
20 | 19, 1 | eleq2s 2272 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
21 | 20 | zcnd 9378 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ) |
22 | 21 | adantl 277 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℂ) |
23 | shftval4g 10848 | . . . . . 6 ⊢ ((( I ‘𝐺) ∈ V ∧ 𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) | |
24 | 17, 18, 22, 23 | syl3anc 1238 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) |
25 | fvi 5575 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝑋 → ( I ‘𝐺) = 𝐺) | |
26 | 2, 25 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → ( I ‘𝐺) = 𝐺) |
27 | 26 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) = 𝐺) |
28 | 27 | oveq1d 5892 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺) shift -𝐾) = (𝐺 shift -𝐾)) |
29 | 28 | fveq1d 5519 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = ((𝐺 shift -𝐾)‘𝑘)) |
30 | addcom 8096 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) | |
31 | 4, 21, 30 | syl2an 289 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) |
32 | 27, 31 | fveq12d 5524 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺)‘(𝐾 + 𝑘)) = (𝐺‘(𝑘 + 𝐾))) |
33 | 24, 29, 32 | 3eqtr3d 2218 | . . . 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 2210 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺 shift -𝐾)‘𝑘) = (𝐹‘𝑘)) |
36 | 1, 7, 8, 9, 35 | climeq 11309 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
37 | 3 | znegcld 9379 | . . 3 ⊢ (𝜑 → -𝐾 ∈ ℤ) |
38 | climshft 11314 | . . 3 ⊢ ((-𝐾 ∈ ℤ ∧ 𝐺 ∈ 𝑋) → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | |
39 | 37, 2, 38 | syl2anc 411 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
40 | 36, 39 | bitr3d 190 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2739 class class class wbr 4005 I cid 4290 dom cdm 4628 Fun wfun 5212 ‘cfv 5218 (class class class)co 5877 ℂcc 7811 + caddc 7816 -cneg 8131 ℤcz 9255 ℤ≥cuz 9530 shift cshi 10825 ⇝ cli 11288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 df-uz 9531 df-shft 10826 df-clim 11289 |
This theorem is referenced by: trireciplem 11510 |
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