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Mirrors > Home > MPE Home > Th. List > dvres2 | Structured version Visualization version GIF version |
Description: Restriction of the base set of a derivative. The primary application of this theorem says that if a function is complex-differentiable then it is also real-differentiable. Unlike dvres 24512, there is no simple reverse relation relating real-differentiable functions to complex differentiability, and indeed there are functions like ℜ(𝑥) which are everywhere real-differentiable but nowhere complex-differentiable.) (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
dvres2 | ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝑆 D 𝐹) ↾ 𝐵) ⊆ (𝐵 D (𝐹 ↾ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5885 | . . 3 ⊢ Rel ((𝑆 D 𝐹) ↾ 𝐵) | |
2 | 1 | a1i 11 | . 2 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → Rel ((𝑆 D 𝐹) ↾ 𝐵)) |
3 | eqid 2824 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
4 | eqid 2824 | . . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆) | |
5 | eqid 2824 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) | |
6 | simp1l 1193 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝑆 ⊆ ℂ) | |
7 | simp1r 1194 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝐹:𝐴⟶ℂ) | |
8 | simp2l 1195 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝐴 ⊆ 𝑆) | |
9 | simp2r 1196 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝐵 ⊆ 𝑆) | |
10 | simp3r 1198 | . . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝑥(𝑆 D 𝐹)𝑦) | |
11 | 6, 7, 8 | dvcl 24500 | . . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ) |
12 | 10, 11 | mpdan 685 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝑦 ∈ ℂ) |
13 | simp3l 1197 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝑥 ∈ 𝐵) | |
14 | 3, 4, 5, 6, 7, 8, 9, 12, 10, 13 | dvres2lem 24511 | . . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) → 𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦) |
15 | 14 | 3expia 1117 | . . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦)) |
16 | vex 3500 | . . . . 5 ⊢ 𝑦 ∈ V | |
17 | 16 | brresi 5865 | . . . 4 ⊢ (𝑥((𝑆 D 𝐹) ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦)) |
18 | df-br 5070 | . . . 4 ⊢ (𝑥((𝑆 D 𝐹) ↾ 𝐵)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ((𝑆 D 𝐹) ↾ 𝐵)) | |
19 | 17, 18 | bitr3i 279 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥(𝑆 D 𝐹)𝑦) ↔ 〈𝑥, 𝑦〉 ∈ ((𝑆 D 𝐹) ↾ 𝐵)) |
20 | df-br 5070 | . . 3 ⊢ (𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (𝐵 D (𝐹 ↾ 𝐵))) | |
21 | 15, 19, 20 | 3imtr3g 297 | . 2 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (〈𝑥, 𝑦〉 ∈ ((𝑆 D 𝐹) ↾ 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝐵 D (𝐹 ↾ 𝐵)))) |
22 | 2, 21 | relssdv 5664 | 1 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝑆 D 𝐹) ↾ 𝐵) ⊆ (𝐵 D (𝐹 ↾ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2113 ∖ cdif 3936 ⊆ wss 3939 {csn 4570 〈cop 4576 class class class wbr 5069 ↦ cmpt 5149 ↾ cres 5560 Rel wrel 5563 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 − cmin 10873 / cdiv 11300 ↾t crest 16697 TopOpenctopn 16698 ℂfldccnfld 20548 D cdv 24464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-rest 16699 df-topn 16700 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-cnp 21839 df-xms 22933 df-ms 22934 df-limc 24467 df-dv 24468 |
This theorem is referenced by: dvres3 24514 dvres3a 24515 |
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