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Mirrors > Home > MPE Home > Th. List > climshft2 | Structured version Visualization version 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 | ovex 6577 | . . . 4 ⊢ (𝐺 shift -𝐾) ∈ V | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐺 shift -𝐾) ∈ V) |
4 | climshft2.5 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
5 | climshft2.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | climshft2.3 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
7 | 6 | zcnd 11359 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℂ) |
8 | eluzelz 11573 | . . . . . . . 8 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | |
9 | 8, 1 | eleq2s 2706 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) |
10 | 9 | zcnd 11359 | . . . . . 6 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ) |
11 | fvex 6113 | . . . . . . 7 ⊢ ( I ‘𝐺) ∈ V | |
12 | 11 | shftval4 13665 | . . . . . 6 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) |
13 | 7, 10, 12 | syl2an 493 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = (( I ‘𝐺)‘(𝐾 + 𝑘))) |
14 | climshft2.6 | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝑋) | |
15 | fvi 6165 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝑋 → ( I ‘𝐺) = 𝐺) | |
16 | 14, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ( I ‘𝐺) = 𝐺) |
17 | 16 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ( I ‘𝐺) = 𝐺) |
18 | 17 | oveq1d 6564 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺) shift -𝐾) = (𝐺 shift -𝐾)) |
19 | 18 | fveq1d 6105 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((( I ‘𝐺) shift -𝐾)‘𝑘) = ((𝐺 shift -𝐾)‘𝑘)) |
20 | addcom 10101 | . . . . . . 7 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) | |
21 | 7, 10, 20 | syl2an 493 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐾 + 𝑘) = (𝑘 + 𝐾)) |
22 | 17, 21 | fveq12d 6109 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (( I ‘𝐺)‘(𝐾 + 𝑘)) = (𝐺‘(𝑘 + 𝐾))) |
23 | 13, 19, 22 | 3eqtr3d 2652 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺 shift -𝐾)‘𝑘) = (𝐺‘(𝑘 + 𝐾))) |
24 | climshft2.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘(𝑘 + 𝐾)) = (𝐹‘𝑘)) | |
25 | 23, 24 | eqtrd 2644 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐺 shift -𝐾)‘𝑘) = (𝐹‘𝑘)) |
26 | 1, 3, 4, 5, 25 | climeq 14146 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
27 | 6 | znegcld 11360 | . . 3 ⊢ (𝜑 → -𝐾 ∈ ℤ) |
28 | climshft 14155 | . . 3 ⊢ ((-𝐾 ∈ ℤ ∧ 𝐺 ∈ 𝑋) → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | |
29 | 27, 14, 28 | syl2anc 691 | . 2 ⊢ (𝜑 → ((𝐺 shift -𝐾) ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
30 | 26, 29 | bitr3d 269 | 1 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 I cid 4948 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 + caddc 9818 -cneg 10146 ℤcz 11254 ℤ≥cuz 11563 shift cshi 13654 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-shft 13655 df-clim 14067 |
This theorem is referenced by: isercoll2 14247 divcnvshft 14426 trireciplem 14433 divcnvlin 30871 |
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