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| 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 13449 | . . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℂ | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
| 3 | cncfshiftioo.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
| 4 | 3 | recnd 11289 | . . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
| 5 | eqeq1 2741 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇))) | |
| 6 | 5 | rexbidv 3179 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇))) |
| 7 | oveq1 7438 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇)) | |
| 8 | 7 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇))) |
| 9 | 8 | cbvrexvw 3238 | . . . . 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 3447 | . . 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 7441 | . . . 4 ⊢ (𝐶–cn→ℂ) = ((𝐴(,)𝐵)–cn→ℂ) |
| 15 | 12, 14 | eleqtrdi 2851 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
| 16 | eqid 2737 | . . 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 45889 | . 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 45535 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
| 23 | 19, 22 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
| 24 | 23 | mpteq1d 5237 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
| 25 | 18, 24 | eqtrid 2789 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
| 26 | 23 | oveq1d 7446 | . 2 ⊢ (𝜑 → (𝐷–cn→ℂ) = ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ)) |
| 27 | 17, 25, 26 | 3eltr4d 2856 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 + caddc 11158 − cmin 11492 (,)cioo 13387 –cn→ccncf 24902 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-ioo 13391 df-cncf 24904 |
| This theorem is referenced by: fourierdlem90 46211 |
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