Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvresntr | Structured version Visualization version GIF version |
Description: Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dvresntr.s | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
dvresntr.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
dvresntr.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvresntr.j | ⊢ 𝐽 = (𝐾 ↾t 𝑆) |
dvresntr.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
dvresntr.i | ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) |
Ref | Expression |
---|---|
dvresntr | ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹 ↾ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvresntr.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
2 | dvresntr.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
3 | dvresntr.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
4 | dvresntr.k | . . . 4 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
5 | dvresntr.j | . . . 4 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
6 | 4, 5 | dvres 24438 | . . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝑋⟶ℂ) ∧ (𝑋 ⊆ 𝑆 ∧ 𝑋 ⊆ 𝑆)) → (𝑆 D (𝐹 ↾ 𝑋)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋))) |
7 | 1, 2, 3, 3, 6 | syl22anc 834 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋))) |
8 | ffn 6508 | . . . 4 ⊢ (𝐹:𝑋⟶ℂ → 𝐹 Fn 𝑋) | |
9 | fnresdm 6460 | . . . 4 ⊢ (𝐹 Fn 𝑋 → (𝐹 ↾ 𝑋) = 𝐹) | |
10 | 2, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹) |
11 | 10 | oveq2d 7161 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = (𝑆 D 𝐹)) |
12 | 4 | cnfldtopon 23320 | . . . . . . . . 9 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
13 | resttopon 21699 | . . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
14 | 12, 1, 13 | sylancr 587 | . . . . . . . 8 ⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
15 | 5, 14 | eqeltrid 2917 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) |
16 | topontop 21451 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) |
18 | toponuni 21452 | . . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽) | |
19 | 15, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 = ∪ 𝐽) |
20 | 3, 19 | sseqtrd 4006 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽) |
21 | eqid 2821 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
22 | 21 | ntridm 21606 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
23 | 17, 20, 22 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
24 | dvresntr.i | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) | |
25 | 24 | fveq2d 6668 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌)) |
26 | 23, 25, 24 | 3eqtr3d 2864 | . . . 4 ⊢ (𝜑 → ((int‘𝐽)‘𝑌) = 𝑌) |
27 | 26 | reseq2d 5847 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌)) = ((𝑆 D 𝐹) ↾ 𝑌)) |
28 | 21 | ntrss2 21595 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
29 | 17, 20, 28 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
30 | 24, 29 | eqsstrrd 4005 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
31 | 30, 3 | sstrd 3976 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
32 | 4, 5 | dvres 24438 | . . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝑋⟶ℂ) ∧ (𝑋 ⊆ 𝑆 ∧ 𝑌 ⊆ 𝑆)) → (𝑆 D (𝐹 ↾ 𝑌)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌))) |
33 | 1, 2, 3, 31, 32 | syl22anc 834 | . . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑌)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌))) |
34 | 24 | reseq2d 5847 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = ((𝑆 D 𝐹) ↾ 𝑌)) |
35 | 27, 33, 34 | 3eqtr4rd 2867 | . 2 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = (𝑆 D (𝐹 ↾ 𝑌))) |
36 | 7, 11, 35 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ∪ cuni 4832 ↾ cres 5551 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ↾t crest 16684 TopOpenctopn 16685 ℂfldccnfld 20475 Topctop 21431 TopOnctopon 21448 intcnt 21555 D cdv 24390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fi 8864 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-fz 12883 df-seq 13360 df-exp 13420 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-plusg 16568 df-mulr 16569 df-starv 16570 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-rest 16686 df-topn 16687 df-topgen 16707 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-cnfld 20476 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-cnp 21766 df-xms 22859 df-ms 22860 df-limc 24393 df-dv 24394 |
This theorem is referenced by: fourierdlem73 42345 |
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