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Mirrors > Home > MPE Home > Th. List > ismbfcn | Structured version Visualization version GIF version |
Description: A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
ismbfcn | ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfdm 24156 | . . 3 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | |
2 | fdm 6516 | . . . 4 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴) | |
3 | 2 | eleq1d 2897 | . . 3 ⊢ (𝐹:𝐴⟶ℂ → (dom 𝐹 ∈ dom vol ↔ 𝐴 ∈ dom vol)) |
4 | 1, 3 | syl5ib 245 | . 2 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ MblFn → 𝐴 ∈ dom vol)) |
5 | mbfdm 24156 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ∈ MblFn → dom (ℜ ∘ 𝐹) ∈ dom vol) | |
6 | 5 | adantr 481 | . . 3 ⊢ (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn) → dom (ℜ ∘ 𝐹) ∈ dom vol) |
7 | ref 14461 | . . . . . 6 ⊢ ℜ:ℂ⟶ℝ | |
8 | fco 6525 | . . . . . 6 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℜ ∘ 𝐹):𝐴⟶ℝ) | |
9 | 7, 8 | mpan 686 | . . . . 5 ⊢ (𝐹:𝐴⟶ℂ → (ℜ ∘ 𝐹):𝐴⟶ℝ) |
10 | 9 | fdmd 6517 | . . . 4 ⊢ (𝐹:𝐴⟶ℂ → dom (ℜ ∘ 𝐹) = 𝐴) |
11 | 10 | eleq1d 2897 | . . 3 ⊢ (𝐹:𝐴⟶ℂ → (dom (ℜ ∘ 𝐹) ∈ dom vol ↔ 𝐴 ∈ dom vol)) |
12 | 6, 11 | syl5ib 245 | . 2 ⊢ (𝐹:𝐴⟶ℂ → (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn) → 𝐴 ∈ dom vol)) |
13 | 9 | adantr 481 | . . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → (ℜ ∘ 𝐹):𝐴⟶ℝ) |
14 | ismbf 24158 | . . . . . . . 8 ⊢ ((ℜ ∘ 𝐹):𝐴⟶ℝ → ((ℜ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → ((ℜ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol)) |
16 | imf 14462 | . . . . . . . . . 10 ⊢ ℑ:ℂ⟶ℝ | |
17 | fco 6525 | . . . . . . . . . 10 ⊢ ((ℑ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℑ ∘ 𝐹):𝐴⟶ℝ) | |
18 | 16, 17 | mpan 686 | . . . . . . . . 9 ⊢ (𝐹:𝐴⟶ℂ → (ℑ ∘ 𝐹):𝐴⟶ℝ) |
19 | 18 | adantr 481 | . . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → (ℑ ∘ 𝐹):𝐴⟶ℝ) |
20 | ismbf 24158 | . . . . . . . 8 ⊢ ((ℑ ∘ 𝐹):𝐴⟶ℝ → ((ℑ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → ((ℑ ∘ 𝐹) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)) |
22 | 15, 21 | anbi12d 630 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn) ↔ (∀𝑥 ∈ ran (,)(◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ∀𝑥 ∈ ran (,)(◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) |
23 | r19.26 3170 | . . . . . 6 ⊢ (∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) ↔ (∀𝑥 ∈ ran (,)(◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ ∀𝑥 ∈ ran (,)(◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)) | |
24 | 22, 23 | syl6bbr 290 | . . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn) ↔ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) |
25 | mblss 24061 | . . . . . . 7 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
26 | cnex 10607 | . . . . . . . 8 ⊢ ℂ ∈ V | |
27 | reex 10617 | . . . . . . . 8 ⊢ ℝ ∈ V | |
28 | elpm2r 8414 | . . . . . . . 8 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑pm ℝ)) | |
29 | 26, 27, 28 | mpanl12 698 | . . . . . . 7 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
30 | 25, 29 | sylan2 592 | . . . . . 6 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
31 | 30 | biantrurd 533 | . . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → (∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))) |
32 | 24, 31 | bitrd 280 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → (((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))) |
33 | ismbf1 24154 | . . . 4 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) | |
34 | 32, 33 | syl6rbbr 291 | . . 3 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ∈ dom vol) → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))) |
35 | 34 | ex 413 | . 2 ⊢ (𝐹:𝐴⟶ℂ → (𝐴 ∈ dom vol → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn)))) |
36 | 4, 12, 35 | pm5.21ndd 381 | 1 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ∀wral 3138 Vcvv 3495 ⊆ wss 3935 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 ∘ ccom 5553 ⟶wf 6345 (class class class)co 7145 ↑pm cpm 8397 ℂcc 10524 ℝcr 10525 (,)cioo 12728 ℜcre 14446 ℑcim 14447 volcvol 23993 MblFncmbf 24144 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-rp 12380 df-xadd 12498 df-ioo 12732 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-sum 15033 df-xmet 20468 df-met 20469 df-ovol 23994 df-vol 23995 df-mbf 24149 |
This theorem is referenced by: ismbfcn2 24168 mbfres 24174 mbfimaopnlem 24185 mbfresfi 34820 itgaddnc 34834 itgmulc2nc 34842 ftc1anclem5 34853 mbfres2cn 42123 |
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