|   | Mathbox for Glauco Siliprandi | < Previous  
      Next > Nearby theorems | |
| 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 25947 | . . 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 838 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋))) | 
| 8 | ffn 6735 | . . . 4 ⊢ (𝐹:𝑋⟶ℂ → 𝐹 Fn 𝑋) | |
| 9 | fnresdm 6686 | . . . 4 ⊢ (𝐹 Fn 𝑋 → (𝐹 ↾ 𝑋) = 𝐹) | |
| 10 | 2, 8, 9 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑋) = 𝐹) | 
| 11 | 10 | oveq2d 7448 | . 2 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑋)) = (𝑆 D 𝐹)) | 
| 12 | 4 | cnfldtopon 24804 | . . . . . . . . 9 ⊢ 𝐾 ∈ (TopOn‘ℂ) | 
| 13 | resttopon 23170 | . . . . . . . . 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 2844 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) | 
| 16 | topontop 22920 | . . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top) | |
| 17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top) | 
| 18 | toponuni 22921 | . . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽) | |
| 19 | 15, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 = ∪ 𝐽) | 
| 20 | 3, 19 | sseqtrd 4019 | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽) | 
| 21 | eqid 2736 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 22 | 21 | ntridm 23077 | . . . . . 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 6909 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌)) | 
| 26 | 23, 25, 24 | 3eqtr3d 2784 | . . . 4 ⊢ (𝜑 → ((int‘𝐽)‘𝑌) = 𝑌) | 
| 27 | 26 | reseq2d 5996 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌)) = ((𝑆 D 𝐹) ↾ 𝑌)) | 
| 28 | 21 | ntrss2 23066 | . . . . . . 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 4018 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | 
| 31 | 30, 3 | sstrd 3993 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) | 
| 32 | 4, 5 | dvres 25947 | . . . 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 838 | . . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ↾ 𝑌)) = ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑌))) | 
| 34 | 24 | reseq2d 5996 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = ((𝑆 D 𝐹) ↾ 𝑌)) | 
| 35 | 27, 33, 34 | 3eqtr4rd 2787 | . 2 ⊢ (𝜑 → ((𝑆 D 𝐹) ↾ ((int‘𝐽)‘𝑋)) = (𝑆 D (𝐹 ↾ 𝑌))) | 
| 36 | 7, 11, 35 | 3eqtr3d 2784 | 1 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑆 D (𝐹 ↾ 𝑌))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ∪ cuni 4906 ↾ cres 5686 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ↾t crest 17466 TopOpenctopn 17467 ℂfldccnfld 21365 Topctop 22900 TopOnctopon 22917 intcnt 23026 D cdv 25899 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fi 9452 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-fz 13549 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-cnp 23237 df-xms 24331 df-ms 24332 df-limc 25902 df-dv 25903 | 
| This theorem is referenced by: fourierdlem73 46199 | 
| Copyright terms: Public domain | W3C validator |