![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfshiftioo | Structured version Visualization version GIF version |
Description: A periodic continuous function stays continuous if the domain is an open interval that is shifted a period. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
cncfshiftioo.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
cncfshiftioo.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
cncfshiftioo.c | ⊢ 𝐶 = (𝐴(,)𝐵) |
cncfshiftioo.t | ⊢ (𝜑 → 𝑇 ∈ ℝ) |
cncfshiftioo.d | ⊢ 𝐷 = ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) |
cncfshiftioo.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ)) |
cncfshiftioo.g | ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇))) |
Ref | Expression |
---|---|
cncfshiftioo | ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioosscn 13418 | . . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
3 | cncfshiftioo.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
4 | 3 | recnd 11272 | . . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
5 | eqeq1 2729 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇))) | |
6 | 5 | rexbidv 3169 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇))) |
7 | oveq1 7423 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇)) | |
8 | 7 | eqeq2d 2736 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇))) |
9 | 8 | cbvrexvw 3226 | . . . . 5 ⊢ (∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇)) |
10 | 6, 9 | bitrdi 286 | . . . 4 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇))) |
11 | 10 | cbvrabv 3430 | . . 3 ⊢ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} = {𝑥 ∈ ℂ ∣ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇)} |
12 | cncfshiftioo.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐶–cn→ℂ)) | |
13 | cncfshiftioo.c | . . . . 5 ⊢ 𝐶 = (𝐴(,)𝐵) | |
14 | 13 | oveq1i 7426 | . . . 4 ⊢ (𝐶–cn→ℂ) = ((𝐴(,)𝐵)–cn→ℂ) |
15 | 12, 14 | eleqtrdi 2835 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
16 | eqid 2725 | . . 3 ⊢ (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇))) | |
17 | 2, 4, 11, 15, 16 | cncfshift 45325 | . 2 ⊢ (𝜑 → (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇))) ∈ ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ)) |
18 | cncfshiftioo.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇))) | |
19 | cncfshiftioo.d | . . . . 5 ⊢ 𝐷 = ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) | |
20 | cncfshiftioo.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
21 | cncfshiftioo.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
22 | 20, 21, 3 | iooshift 44970 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
23 | 19, 22 | eqtrid 2777 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
24 | 23 | mpteq1d 5238 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
25 | 18, 24 | eqtrid 2777 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
26 | 23 | oveq1d 7431 | . 2 ⊢ (𝜑 → (𝐷–cn→ℂ) = ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ)) |
27 | 17, 25, 26 | 3eltr4d 2840 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 {crab 3419 ⊆ wss 3939 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7416 ℂcc 11136 ℝcr 11137 + caddc 11141 − cmin 11474 (,)cioo 13356 –cn→ccncf 24814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-ioo 13360 df-cncf 24816 |
This theorem is referenced by: fourierdlem90 45647 |
Copyright terms: Public domain | W3C validator |