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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfres2cn | Structured version Visualization version GIF version |
Description: Measurability of a piecewise function: if 𝐹 is measurable on subsets 𝐵 and 𝐶 of its domain, and these pieces make up all of 𝐴, then 𝐹 is measurable on the whole domain. Similar to mbfres2 23753 but here the theorem is extended to complex-valued functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
mbfres2cn.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
mbfres2cn.b | ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) |
mbfres2cn.c | ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) |
mbfres2cn.a | ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) |
Ref | Expression |
---|---|
mbfres2cn | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ref 14193 | . . . 4 ⊢ ℜ:ℂ⟶ℝ | |
2 | mbfres2cn.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | fco 6273 | . . . 4 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℜ ∘ 𝐹):𝐴⟶ℝ) | |
4 | 1, 2, 3 | sylancr 582 | . . 3 ⊢ (𝜑 → (ℜ ∘ 𝐹):𝐴⟶ℝ) |
5 | resco 5858 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐵) = (ℜ ∘ (𝐹 ↾ 𝐵)) | |
6 | mbfres2cn.b | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
7 | fresin 6288 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ) | |
8 | ismbfcn 23737 | . . . . . . 7 ⊢ ((𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ → ((𝐹 ↾ 𝐵) ∈ MblFn ↔ ((ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn))) | |
9 | 2, 7, 8 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) ∈ MblFn ↔ ((ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn))) |
10 | 6, 9 | mpbid 224 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn)) |
11 | 10 | simpld 489 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
12 | 5, 11 | syl5eqel 2882 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
13 | resco 5858 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐶) = (ℜ ∘ (𝐹 ↾ 𝐶)) | |
14 | mbfres2cn.c | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
15 | fresin 6288 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶ℂ) | |
16 | ismbfcn 23737 | . . . . . . 7 ⊢ ((𝐹 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶ℂ → ((𝐹 ↾ 𝐶) ∈ MblFn ↔ ((ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn))) | |
17 | 2, 15, 16 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∈ MblFn ↔ ((ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn))) |
18 | 14, 17 | mpbid 224 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn)) |
19 | 18 | simpld 489 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
20 | 13, 19 | syl5eqel 2882 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
21 | mbfres2cn.a | . . 3 ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) | |
22 | 4, 12, 20, 21 | mbfres2 23753 | . 2 ⊢ (𝜑 → (ℜ ∘ 𝐹) ∈ MblFn) |
23 | imf 14194 | . . . 4 ⊢ ℑ:ℂ⟶ℝ | |
24 | fco 6273 | . . . 4 ⊢ ((ℑ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℑ ∘ 𝐹):𝐴⟶ℝ) | |
25 | 23, 2, 24 | sylancr 582 | . . 3 ⊢ (𝜑 → (ℑ ∘ 𝐹):𝐴⟶ℝ) |
26 | resco 5858 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐵) = (ℑ ∘ (𝐹 ↾ 𝐵)) | |
27 | 10 | simprd 490 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
28 | 26, 27 | syl5eqel 2882 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
29 | resco 5858 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐶) = (ℑ ∘ (𝐹 ↾ 𝐶)) | |
30 | 18 | simprd 490 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
31 | 29, 30 | syl5eqel 2882 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
32 | 25, 28, 31, 21 | mbfres2 23753 | . 2 ⊢ (𝜑 → (ℑ ∘ 𝐹) ∈ MblFn) |
33 | ismbfcn 23737 | . . 3 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))) | |
34 | 2, 33 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 ∈ MblFn ↔ ((ℜ ∘ 𝐹) ∈ MblFn ∧ (ℑ ∘ 𝐹) ∈ MblFn))) |
35 | 22, 32, 34 | mpbir2and 705 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∪ cun 3767 ∩ cin 3768 ↾ cres 5314 ∘ ccom 5316 ⟶wf 6097 ℂcc 10222 ℝcr 10223 ℜcre 14178 ℑcim 14179 MblFncmbf 23722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-inf 8591 df-oi 8657 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-rp 12075 df-xadd 12194 df-ioo 12428 df-ico 12430 df-icc 12431 df-fz 12581 df-fzo 12721 df-fl 12848 df-seq 13056 df-exp 13115 df-hash 13371 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-clim 14560 df-sum 14758 df-xmet 20061 df-met 20062 df-ovol 23572 df-vol 23573 df-mbf 23727 |
This theorem is referenced by: iblsplit 40925 |
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