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 24762 | . . 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 839 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋))) |
8 | ffn 6523 | . . . 4 ⊢ (𝐹:𝑋⟶ℂ → 𝐹 Fn 𝑋) | |
9 | fnresdm 6474 | . . . 4 ⊢ (𝐹 Fn 𝑋 → (𝐹 ↾ 𝑋) = 𝐹) | |
10 | 2, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹) |
11 | 10 | oveq2d 7207 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = (𝑆 D 𝐹)) |
12 | 4 | cnfldtopon 23634 | . . . . . . . . 9 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
13 | resttopon 22012 | . . . . . . . . 9 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
14 | 12, 1, 13 | sylancr 590 | . . . . . . . 8 ⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
15 | 5, 14 | eqeltrid 2835 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) |
16 | topontop 21764 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) |
18 | toponuni 21765 | . . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽) | |
19 | 15, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 = ∪ 𝐽) |
20 | 3, 19 | sseqtrd 3927 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽) |
21 | eqid 2736 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
22 | 21 | ntridm 21919 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
23 | 17, 20, 22 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
24 | dvresntr.i | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) | |
25 | 24 | fveq2d 6699 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌)) |
26 | 23, 25, 24 | 3eqtr3d 2779 | . . . 4 ⊢ (𝜑 → ((int‘𝐽)‘𝑌) = 𝑌) |
27 | 26 | reseq2d 5836 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌)) = ((𝑆 D 𝐹) ↾ 𝑌)) |
28 | 21 | ntrss2 21908 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
29 | 17, 20, 28 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
30 | 24, 29 | eqsstrrd 3926 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
31 | 30, 3 | sstrd 3897 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
32 | 4, 5 | dvres 24762 | . . . 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 839 | . . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑌)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌))) |
34 | 24 | reseq2d 5836 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = ((𝑆 D 𝐹) ↾ 𝑌)) |
35 | 27, 33, 34 | 3eqtr4rd 2782 | . 2 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = (𝑆 D (𝐹 ↾ 𝑌))) |
36 | 7, 11, 35 | 3eqtr3d 2779 | 1 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ∪ cuni 4805 ↾ cres 5538 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ↾t crest 16879 TopOpenctopn 16880 ℂfldccnfld 20317 Topctop 21744 TopOnctopon 21761 intcnt 21868 D cdv 24714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fi 9005 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-fz 13061 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-starv 16764 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-rest 16881 df-topn 16882 df-topgen 16902 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cld 21870 df-ntr 21871 df-cls 21872 df-cnp 22079 df-xms 23172 df-ms 23173 df-limc 24717 df-dv 24718 |
This theorem is referenced by: fourierdlem73 43338 |
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