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 25075 | . . 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 836 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋))) |
8 | ffn 6600 | . . . 4 ⊢ (𝐹:𝑋⟶ℂ → 𝐹 Fn 𝑋) | |
9 | fnresdm 6551 | . . . 4 ⊢ (𝐹 Fn 𝑋 → (𝐹 ↾ 𝑋) = 𝐹) | |
10 | 2, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹) |
11 | 10 | oveq2d 7291 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = (𝑆 D 𝐹)) |
12 | 4 | cnfldtopon 23946 | . . . . . . . . 9 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
13 | resttopon 22312 | . . . . . . . . 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 2843 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) |
16 | topontop 22062 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) |
18 | toponuni 22063 | . . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽) | |
19 | 15, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 = ∪ 𝐽) |
20 | 3, 19 | sseqtrd 3961 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽) |
21 | eqid 2738 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
22 | 21 | ntridm 22219 | . . . . . 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 6778 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌)) |
26 | 23, 25, 24 | 3eqtr3d 2786 | . . . 4 ⊢ (𝜑 → ((int‘𝐽)‘𝑌) = 𝑌) |
27 | 26 | reseq2d 5891 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌)) = ((𝑆 D 𝐹) ↾ 𝑌)) |
28 | 21 | ntrss2 22208 | . . . . . . 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 3960 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
31 | 30, 3 | sstrd 3931 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
32 | 4, 5 | dvres 25075 | . . . 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 836 | . . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑌)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌))) |
34 | 24 | reseq2d 5891 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = ((𝑆 D 𝐹) ↾ 𝑌)) |
35 | 27, 33, 34 | 3eqtr4rd 2789 | . 2 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = (𝑆 D (𝐹 ↾ 𝑌))) |
36 | 7, 11, 35 | 3eqtr3d 2786 | 1 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹 ↾ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ∪ cuni 4839 ↾ cres 5591 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ↾t crest 17131 TopOpenctopn 17132 ℂfldccnfld 20597 Topctop 22042 TopOnctopon 22059 intcnt 22168 D cdv 25027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-starv 16977 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-rest 17133 df-topn 17134 df-topgen 17154 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-cnp 22379 df-xms 23473 df-ms 23474 df-limc 25030 df-dv 25031 |
This theorem is referenced by: fourierdlem73 43720 |
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