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Mirrors > Home > ILE Home > Th. List > dvbss | GIF version |
Description: The set of differentiable points is a subset of the domain of the function. (Contributed by Mario Carneiro, 6-Aug-2014.) (Revised by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
dvcl.s | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
dvcl.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
dvcl.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
Ref | Expression |
---|---|
dvbss | ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcl.s | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
2 | dvcl.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | dvcl.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | eqid 2177 | . . 3 ⊢ ((MetOpen‘(abs ∘ − )) ↾t 𝑆) = ((MetOpen‘(abs ∘ − )) ↾t 𝑆) | |
5 | eqid 2177 | . . 3 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
6 | 1, 2, 3, 4, 5 | dvbssntrcntop 14084 | . 2 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘((MetOpen‘(abs ∘ − )) ↾t 𝑆))‘𝐴)) |
7 | 5 | cntoptop 13964 | . . . 4 ⊢ (MetOpen‘(abs ∘ − )) ∈ Top |
8 | cnex 7934 | . . . . 5 ⊢ ℂ ∈ V | |
9 | ssexg 4142 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ ℂ ∈ V) → 𝑆 ∈ V) | |
10 | 1, 8, 9 | sylancl 413 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
11 | resttop 13601 | . . . 4 ⊢ (((MetOpen‘(abs ∘ − )) ∈ Top ∧ 𝑆 ∈ V) → ((MetOpen‘(abs ∘ − )) ↾t 𝑆) ∈ Top) | |
12 | 7, 10, 11 | sylancr 414 | . . 3 ⊢ (𝜑 → ((MetOpen‘(abs ∘ − )) ↾t 𝑆) ∈ Top) |
13 | 5 | cntoptopon 13963 | . . . . . 6 ⊢ (MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) |
14 | resttopon 13602 | . . . . . 6 ⊢ (((MetOpen‘(abs ∘ − )) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((MetOpen‘(abs ∘ − )) ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
15 | 13, 1, 14 | sylancr 414 | . . . . 5 ⊢ (𝜑 → ((MetOpen‘(abs ∘ − )) ↾t 𝑆) ∈ (TopOn‘𝑆)) |
16 | toponuni 13446 | . . . . 5 ⊢ (((MetOpen‘(abs ∘ − )) ↾t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ ((MetOpen‘(abs ∘ − )) ↾t 𝑆)) | |
17 | 15, 16 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑆 = ∪ ((MetOpen‘(abs ∘ − )) ↾t 𝑆)) |
18 | 3, 17 | sseqtrd 3193 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ ((MetOpen‘(abs ∘ − )) ↾t 𝑆)) |
19 | eqid 2177 | . . . 4 ⊢ ∪ ((MetOpen‘(abs ∘ − )) ↾t 𝑆) = ∪ ((MetOpen‘(abs ∘ − )) ↾t 𝑆) | |
20 | 19 | ntrss2 13552 | . . 3 ⊢ ((((MetOpen‘(abs ∘ − )) ↾t 𝑆) ∈ Top ∧ 𝐴 ⊆ ∪ ((MetOpen‘(abs ∘ − )) ↾t 𝑆)) → ((int‘((MetOpen‘(abs ∘ − )) ↾t 𝑆))‘𝐴) ⊆ 𝐴) |
21 | 12, 18, 20 | syl2anc 411 | . 2 ⊢ (𝜑 → ((int‘((MetOpen‘(abs ∘ − )) ↾t 𝑆))‘𝐴) ⊆ 𝐴) |
22 | 6, 21 | sstrd 3165 | 1 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 ∪ cuni 3809 dom cdm 4626 ∘ ccom 4630 ⟶wf 5212 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 − cmin 8126 abscabs 11001 ↾t crest 12682 MetOpencmopn 13376 Topctop 13428 TopOnctopon 13441 intcnt 13524 D cdv 14055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-map 6649 df-pm 6650 df-sup 6982 df-inf 6983 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-xneg 9770 df-xadd 9771 df-seqfrec 10443 df-exp 10517 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-rest 12684 df-topgen 12703 df-psmet 13378 df-xmet 13379 df-met 13380 df-bl 13381 df-mopn 13382 df-top 13429 df-topon 13442 df-bases 13474 df-ntr 13527 df-limced 14056 df-dvap 14057 |
This theorem is referenced by: dvbsssg 14086 dvidlemap 14091 dviaddf 14100 dvimulf 14101 dvcoapbr 14102 dvcjbr 14103 dvrecap 14108 |
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