Mathbox for Glauco Siliprandi |
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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 38563 | . . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
3 | cncfshiftioo.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
4 | 3 | recnd 9947 | . . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
5 | eqeq1 2614 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇))) | |
6 | 5 | rexbidv 3034 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇))) |
7 | oveq1 6556 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇)) | |
8 | 7 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇))) |
9 | 8 | cbvrexv 3148 | . . . . 5 ⊢ (∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇)) |
10 | 6, 9 | syl6bb 275 | . . . 4 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇))) |
11 | 10 | cbvrabv 3172 | . . 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 6559 | . . . 4 ⊢ (𝐶–cn→ℂ) = ((𝐴(,)𝐵)–cn→ℂ) |
15 | 12, 14 | syl6eleq 2698 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
16 | eqid 2610 | . . 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 38759 | . 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 38595 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
23 | 19, 22 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
24 | 23 | mpteq1d 4666 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
25 | 18, 24 | syl5eq 2656 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
26 | 23 | oveq1d 6564 | . 2 ⊢ (𝜑 → (𝐷–cn→ℂ) = ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ)) |
27 | 17, 25, 26 | 3eltr4d 2703 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 + caddc 9818 − cmin 10145 (,)cioo 12046 –cn→ccncf 22487 |
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-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 |
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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 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-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-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 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-ioo 12050 df-cncf 22489 |
This theorem is referenced by: fourierdlem90 39089 |
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