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 39716 | . . . 4 ⊢ (𝐴(,)𝐵) ⊆ ℂ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
3 | cncfshiftioo.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ℝ) | |
4 | 3 | recnd 10068 | . . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
5 | eqeq1 2626 | . . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑧 + 𝑇))) | |
6 | 5 | rexbidv 3052 | . . . . 5 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇))) |
7 | oveq1 6657 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 + 𝑇) = (𝑦 + 𝑇)) | |
8 | 7 | eqeq2d 2632 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑥 = (𝑧 + 𝑇) ↔ 𝑥 = (𝑦 + 𝑇))) |
9 | 8 | cbvrexv 3172 | . . . . 5 ⊢ (∃𝑧 ∈ (𝐴(,)𝐵)𝑥 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇)) |
10 | 6, 9 | syl6bb 276 | . . . 4 ⊢ (𝑤 = 𝑥 → (∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇) ↔ ∃𝑦 ∈ (𝐴(,)𝐵)𝑥 = (𝑦 + 𝑇))) |
11 | 10 | cbvrabv 3199 | . . 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 6660 | . . . 4 ⊢ (𝐶–cn→ℂ) = ((𝐴(,)𝐵)–cn→ℂ) |
15 | 12, 14 | syl6eleq 2711 | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
16 | eqid 2622 | . . 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 40087 | . 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 39748 | . . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑇)(,)(𝐵 + 𝑇)) = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
23 | 19, 22 | syl5eq 2668 | . . . 4 ⊢ (𝜑 → 𝐷 = {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}) |
24 | 23 | mpteq1d 4738 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐹‘(𝑥 − 𝑇))) = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
25 | 18, 24 | syl5eq 2668 | . 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ {𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)} ↦ (𝐹‘(𝑥 − 𝑇)))) |
26 | 23 | oveq1d 6665 | . 2 ⊢ (𝜑 → (𝐷–cn→ℂ) = ({𝑤 ∈ ℂ ∣ ∃𝑧 ∈ (𝐴(,)𝐵)𝑤 = (𝑧 + 𝑇)}–cn→ℂ)) |
27 | 17, 25, 26 | 3eltr4d 2716 | 1 ⊢ (𝜑 → 𝐺 ∈ (𝐷–cn→ℂ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 + caddc 9939 − cmin 10266 (,)cioo 12175 –cn→ccncf 22679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-ioo 12179 df-cncf 22681 |
This theorem is referenced by: fourierdlem90 40413 |
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