![]() |
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 13392 | . . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
3 | cncfshiftioo.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
4 | 3 | recnd 11246 | . . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
5 | eqeq1 2730 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇))) | |
6 | 5 | rexbidv 3172 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇))) |
7 | oveq1 7412 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇)) | |
8 | 7 | eqeq2d 2737 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇))) |
9 | 8 | cbvrexvw 3229 | . . . . 5 ⊢ (∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇)) |
10 | 6, 9 | bitrdi 287 | . . . 4 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇))) |
11 | 10 | cbvrabv 3436 | . . 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 7415 | . . . 4 ⊢ (𝐶–cn→ℂ) = ((𝐴(,)𝐵)–cn→ℂ) |
15 | 12, 14 | eleqtrdi 2837 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
16 | eqid 2726 | . . 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 45162 | . 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 44807 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
23 | 19, 22 | eqtrid 2778 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
24 | 23 | mpteq1d 5236 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
25 | 18, 24 | eqtrid 2778 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
26 | 23 | oveq1d 7420 | . 2 ⊢ (𝜑 → (𝐷–cn→ℂ) = ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ)) |
27 | 17, 25, 26 | 3eltr4d 2842 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 {crab 3426 ⊆ wss 3943 ↦ cmpt 5224 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℝcr 11111 + caddc 11115 − cmin 11448 (,)cioo 13330 –cn→ccncf 24751 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-ioo 13334 df-cncf 24753 |
This theorem is referenced by: fourierdlem90 45484 |
Copyright terms: Public domain | W3C validator |