Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dvbssntr | Structured version Visualization version GIF version |
Description: The set of differentiable points is a subset of the interior of the domain of the function. (Contributed by Mario Carneiro, 7-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) |
dvbssntr.j | ⊢ 𝐽 = (𝐾 ↾t 𝑆) |
dvbssntr.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
dvbssntr | ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcl.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
2 | dvcl.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | dvcl.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | dvbssntr.j | . . . . 5 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
5 | dvbssntr.k | . . . . 5 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
6 | 4, 5 | dvfval 24495 | . . . 4 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝐽)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ))) |
7 | 1, 2, 3, 6 | syl3anc 1367 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝐽)‘𝐴)({𝑥} × ((𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) limℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ))) |
8 | dmss 5771 | . . 3 ⊢ ((𝑆 D 𝐹) ⊆ (((int‘𝐽)‘𝐴) × ℂ) → dom (𝑆 D 𝐹) ⊆ dom (((int‘𝐽)‘𝐴) × ℂ)) | |
9 | 7, 8 | simpl2im 506 | . 2 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ dom (((int‘𝐽)‘𝐴) × ℂ)) |
10 | dmxpss 6028 | . 2 ⊢ dom (((int‘𝐽)‘𝐴) × ℂ) ⊆ ((int‘𝐽)‘𝐴) | |
11 | 9, 10 | sstrdi 3979 | 1 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ ((int‘𝐽)‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 ∪ ciun 4919 ↦ cmpt 5146 × cxp 5553 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 − cmin 10870 / cdiv 11297 ↾t crest 16694 TopOpenctopn 16695 ℂfldccnfld 20545 intcnt 21625 limℂ climc 24460 D cdv 24461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cnp 21836 df-xms 22930 df-ms 22931 df-limc 24464 df-dv 24465 |
This theorem is referenced by: dvbss 24499 dvnres 24528 dvcmulf 24542 dvcjbr 24546 dvmptcmul 24561 dvcnvre 24616 ftc1cn 24640 taylthlem1 24961 taylthlem2 24962 ulmdvlem3 24990 ftc1cnnc 34981 |
Copyright terms: Public domain | W3C validator |