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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 9335 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
5 | 4 | negcld 8217 | . . . 4 ⊢ (𝜑 → -𝐾 ∈ ℂ) |
6 | ovshftex 10783 | . . . 4 ⊢ ((𝐺 ∈ 𝑋 ∧ -𝐾 ∈ ℂ) → (𝐺 shift -𝐾) ∈ V) | |
7 | 2, 5, 6 | syl2anc 409 | . . 3 ⊢ (𝜑 → (𝐺 shift -𝐾) ∈ V) |
8 | climshft2.5 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
9 | climshft2.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
10 | funi 5230 | . . . . . . . 8 ⊢ Fun I | |
11 | elex 2741 | . . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑋 → 𝐺 ∈ V) | |
12 | 2, 11 | syl 14 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ V) |
13 | dmi 4826 | . . . . . . . . 9 ⊢ dom I = V | |
14 | 12, 13 | eleqtrrdi 2264 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ dom I ) |
15 | funfvex 5513 | . . . . . . . 8 ⊢ ((Fun I ∧ 𝐺 ∈ dom I ) → ( I ‘𝐺) ∈ V) | |
16 | 10, 14, 15 | sylancr 412 | . . . . . . 7 ⊢ (𝜑 → ( I ‘𝐺) ∈ V) |
17 | 16 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) ∈ V) |
18 | 4 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐾 ∈ ℂ) |
19 | eluzelz 9496 | . . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
20 | 19, 1 | eleq2s 2265 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
21 | 20 | zcnd 9335 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ) |
22 | 21 | adantl 275 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℂ) |
23 | shftval4g 10801 | . . . . . 6 ⊢ ((( I ‘𝐺) ∈ V ∧ 𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) | |
24 | 17, 18, 22, 23 | syl3anc 1233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) |
25 | fvi 5553 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝑋 → ( I ‘𝐺) = 𝐺) | |
26 | 2, 25 | syl 14 | . . . . . . . 8 ⊢ (𝜑 → ( I ‘𝐺) = 𝐺) |
27 | 26 | adantr 274 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) = 𝐺) |
28 | 27 | oveq1d 5868 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺) shift -𝐾) = (𝐺 shift -𝐾)) |
29 | 28 | fveq1d 5498 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = ((𝐺 shift -𝐾)‘𝑘)) |
30 | addcom 8056 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) | |
31 | 4, 21, 30 | syl2an 287 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) |
32 | 27, 31 | fveq12d 5503 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺)‘(𝐾 + 𝑘)) = (𝐺‘(𝑘 + 𝐾))) |
33 | 24, 29, 32 | 3eqtr3d 2211 | . . . 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 2203 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺 shift -𝐾)‘𝑘) = (𝐹‘𝑘)) |
36 | 1, 7, 8, 9, 35 | climeq 11262 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
37 | 3 | znegcld 9336 | . . 3 ⊢ (𝜑 → -𝐾 ∈ ℤ) |
38 | climshft 11267 | . . 3 ⊢ ((-𝐾 ∈ ℤ ∧ 𝐺 ∈ 𝑋) → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | |
39 | 37, 2, 38 | syl2anc 409 | . 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 1348 ∈ wcel 2141 Vcvv 2730 class class class wbr 3989 I cid 4273 dom cdm 4611 Fun wfun 5192 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 + caddc 7777 -cneg 8091 ℤcz 9212 ℤ≥cuz 9487 shift cshi 10778 ⇝ cli 11241 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-shft 10779 df-clim 11242 |
This theorem is referenced by: trireciplem 11463 |
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