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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 23481, 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 5346 | . . 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 2610 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
4 | eqid 2610 | . . . . 5 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆) | |
5 | eqid 2610 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ (𝐴 ∖ {𝑥}) ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) | |
6 | simp1l 1078 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝑆 ⊆ ℂ) | |
7 | simp1r 1079 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝐹:𝐴⟶ℂ) | |
8 | simp2l 1080 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝐴 ⊆ 𝑆) | |
9 | simp2r 1081 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝐵 ⊆ 𝑆) | |
10 | simp3l 1082 | . . . . . 6 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝑥(𝑆 D 𝐹)𝑦) | |
11 | 6, 7, 8 | dvcl 23469 | . . . . . 6 ⊢ ((((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) ∧ 𝑥(𝑆 D 𝐹)𝑦) → 𝑦 ∈ ℂ) |
12 | 10, 11 | mpdan 699 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝑦 ∈ ℂ) |
13 | simp3r 1083 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵) | |
14 | 3, 4, 5, 6, 7, 8, 9, 12, 10, 13 | dvres2lem 23480 | . . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆) ∧ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) → 𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦) |
15 | 14 | 3expia 1259 | . . 3 ⊢ (((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → ((𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵) → 𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦)) |
16 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
17 | 16 | brres 5323 | . . . 4 ⊢ (𝑥((𝑆 D 𝐹) ↾ 𝐵)𝑦 ↔ (𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵)) |
18 | df-br 4584 | . . . 4 ⊢ (𝑥((𝑆 D 𝐹) ↾ 𝐵)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ((𝑆 D 𝐹) ↾ 𝐵)) | |
19 | 17, 18 | bitr3i 265 | . . 3 ⊢ ((𝑥(𝑆 D 𝐹)𝑦 ∧ 𝑥 ∈ 𝐵) ↔ 〈𝑥, 𝑦〉 ∈ ((𝑆 D 𝐹) ↾ 𝐵)) |
20 | df-br 4584 | . . 3 ⊢ (𝑥(𝐵 D (𝐹 ↾ 𝐵))𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (𝐵 D (𝐹 ↾ 𝐵))) | |
21 | 15, 19, 20 | 3imtr3g 283 | . 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 5135 | 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 383 ∧ w3a 1031 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 ↾ cres 5040 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 − cmin 10145 / cdiv 10563 ↾t crest 15904 TopOpenctopn 15905 ℂfldccnfld 19567 D cdv 23433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-cnp 20842 df-xms 21935 df-ms 21936 df-limc 23436 df-dv 23437 |
This theorem is referenced by: dvres3 23483 dvres3a 23484 |
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