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Mirrors > Home > MPE Home > Th. List > dvbsss | Structured version Visualization version GIF version |
Description: The set of differentiable points is a subset of the ambient topology. (Contributed by Mario Carneiro, 18-Mar-2015.) |
Ref | Expression |
---|---|
dvbsss | ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dv 24392 | . . . . . . . . . . 11 ⊢ D = (𝑠 ∈ 𝒫 ℂ, 𝑓 ∈ (ℂ ↑pm 𝑠) ↦ ∪ 𝑥 ∈ ((int‘((TopOpen‘ℂfld) ↾t 𝑠))‘dom 𝑓)({𝑥} × ((𝑧 ∈ (dom 𝑓 ∖ {𝑥}) ↦ (((𝑓‘𝑧) − (𝑓‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥))) | |
2 | 1 | reldmmpo 7274 | . . . . . . . . . 10 ⊢ Rel dom D |
3 | df-rel 5555 | . . . . . . . . . 10 ⊢ (Rel dom D ↔ dom D ⊆ (V × V)) | |
4 | 2, 3 | mpbi 231 | . . . . . . . . 9 ⊢ dom D ⊆ (V × V) |
5 | 4 | sseli 3960 | . . . . . . . 8 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → 〈𝑆, 𝐹〉 ∈ (V × V)) |
6 | opelxp1 5589 | . . . . . . . 8 ⊢ (〈𝑆, 𝐹〉 ∈ (V × V) → 𝑆 ∈ V) | |
7 | 5, 6 | syl 17 | . . . . . . 7 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → 𝑆 ∈ V) |
8 | opeq1 4795 | . . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → 〈𝑠, 𝐹〉 = 〈𝑆, 𝐹〉) | |
9 | 8 | eleq1d 2894 | . . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (〈𝑠, 𝐹〉 ∈ dom D ↔ 〈𝑆, 𝐹〉 ∈ dom D )) |
10 | eleq1 2897 | . . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝒫 ℂ ↔ 𝑆 ∈ 𝒫 ℂ)) | |
11 | oveq2 7153 | . . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (ℂ ↑pm 𝑠) = (ℂ ↑pm 𝑆)) | |
12 | 11 | eleq2d 2895 | . . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝐹 ∈ (ℂ ↑pm 𝑠) ↔ 𝐹 ∈ (ℂ ↑pm 𝑆))) |
13 | 10, 12 | anbi12d 630 | . . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑠)) ↔ (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))) |
14 | 9, 13 | imbi12d 346 | . . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((〈𝑠, 𝐹〉 ∈ dom D → (𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑠))) ↔ (〈𝑆, 𝐹〉 ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆))))) |
15 | 1 | dmmpossx 7753 | . . . . . . . . . 10 ⊢ dom D ⊆ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)) |
16 | 15 | sseli 3960 | . . . . . . . . 9 ⊢ (〈𝑠, 𝐹〉 ∈ dom D → 〈𝑠, 𝐹〉 ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠))) |
17 | opeliunxp 5612 | . . . . . . . . 9 ⊢ (〈𝑠, 𝐹〉 ∈ ∪ 𝑠 ∈ 𝒫 ℂ({𝑠} × (ℂ ↑pm 𝑠)) ↔ (𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑠))) | |
18 | 16, 17 | sylib 219 | . . . . . . . 8 ⊢ (〈𝑠, 𝐹〉 ∈ dom D → (𝑠 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑠))) |
19 | 14, 18 | vtoclg 3565 | . . . . . . 7 ⊢ (𝑆 ∈ V → (〈𝑆, 𝐹〉 ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)))) |
20 | 7, 19 | mpcom 38 | . . . . . 6 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → (𝑆 ∈ 𝒫 ℂ ∧ 𝐹 ∈ (ℂ ↑pm 𝑆))) |
21 | 20 | simpld 495 | . . . . 5 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → 𝑆 ∈ 𝒫 ℂ) |
22 | 21 | elpwid 4549 | . . . 4 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → 𝑆 ⊆ ℂ) |
23 | 20 | simprd 496 | . . . . . 6 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
24 | cnex 10606 | . . . . . . 7 ⊢ ℂ ∈ V | |
25 | elpm2g 8412 | . . . . . . 7 ⊢ ((ℂ ∈ V ∧ 𝑆 ∈ 𝒫 ℂ) → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))) | |
26 | 24, 21, 25 | sylancr 587 | . . . . . 6 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → (𝐹 ∈ (ℂ ↑pm 𝑆) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))) |
27 | 23, 26 | mpbid 233 | . . . . 5 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆)) |
28 | 27 | simpld 495 | . . . 4 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → 𝐹:dom 𝐹⟶ℂ) |
29 | 27 | simprd 496 | . . . 4 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → dom 𝐹 ⊆ 𝑆) |
30 | 22, 28, 29 | dvbss 24426 | . . 3 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → dom (𝑆 D 𝐹) ⊆ dom 𝐹) |
31 | 30, 29 | sstrd 3974 | . 2 ⊢ (〈𝑆, 𝐹〉 ∈ dom D → dom (𝑆 D 𝐹) ⊆ 𝑆) |
32 | df-ov 7148 | . . . . . 6 ⊢ (𝑆 D 𝐹) = ( D ‘〈𝑆, 𝐹〉) | |
33 | ndmfv 6693 | . . . . . 6 ⊢ (¬ 〈𝑆, 𝐹〉 ∈ dom D → ( D ‘〈𝑆, 𝐹〉) = ∅) | |
34 | 32, 33 | syl5eq 2865 | . . . . 5 ⊢ (¬ 〈𝑆, 𝐹〉 ∈ dom D → (𝑆 D 𝐹) = ∅) |
35 | 34 | dmeqd 5767 | . . . 4 ⊢ (¬ 〈𝑆, 𝐹〉 ∈ dom D → dom (𝑆 D 𝐹) = dom ∅) |
36 | dm0 5783 | . . . 4 ⊢ dom ∅ = ∅ | |
37 | 35, 36 | syl6eq 2869 | . . 3 ⊢ (¬ 〈𝑆, 𝐹〉 ∈ dom D → dom (𝑆 D 𝐹) = ∅) |
38 | 0ss 4347 | . . 3 ⊢ ∅ ⊆ 𝑆 | |
39 | 37, 38 | eqsstrdi 4018 | . 2 ⊢ (¬ 〈𝑆, 𝐹〉 ∈ dom D → dom (𝑆 D 𝐹) ⊆ 𝑆) |
40 | 31, 39 | pm2.61i 183 | 1 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 ∅c0 4288 𝒫 cpw 4535 {csn 4557 〈cop 4563 ∪ ciun 4910 ↦ cmpt 5137 × cxp 5546 dom cdm 5548 Rel wrel 5553 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑pm cpm 8396 ℂcc 10523 − cmin 10858 / cdiv 11285 ↾t crest 16682 TopOpenctopn 16683 ℂfldccnfld 20473 intcnt 21553 limℂ climc 24387 D cdv 24388 |
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 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-topn 16685 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-ntr 21556 df-cnp 21764 df-xms 22857 df-ms 22858 df-limc 24391 df-dv 24392 |
This theorem is referenced by: dvaddf 24466 dvmulf 24467 dvcmulf 24469 dvcof 24472 dvmptres2 24486 dvmptcmul 24488 dvmptcj 24492 dvcnvlem 24500 dvcnv 24501 dvef 24504 dvcnvrelem1 24541 dvcnvrelem2 24542 dvcnvre 24543 ulmdvlem1 24915 ulmdvlem3 24917 ulmdv 24918 fperdvper 42079 dvmulcncf 42086 dvdivcncf 42088 |
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