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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 24246 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 14471 | . . . 4 ⊢ ℜ:ℂ⟶ℝ | |
2 | mbfres2cn.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | fco 6531 | . . . 4 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℜ ∘ 𝐹):𝐴⟶ℝ) | |
4 | 1, 2, 3 | sylancr 589 | . . 3 ⊢ (𝜑 → (ℜ ∘ 𝐹):𝐴⟶ℝ) |
5 | resco 6103 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐵) = (ℜ ∘ (𝐹 ↾ 𝐵)) | |
6 | mbfres2cn.b | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
7 | fresin 6547 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ) | |
8 | ismbfcn 24230 | . . . . . . 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 234 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn)) |
11 | 10 | simpld 497 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
12 | 5, 11 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
13 | resco 6103 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐶) = (ℜ ∘ (𝐹 ↾ 𝐶)) | |
14 | mbfres2cn.c | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
15 | fresin 6547 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶ℂ) | |
16 | ismbfcn 24230 | . . . . . . 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 234 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn)) |
19 | 18 | simpld 497 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
20 | 13, 19 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
21 | mbfres2cn.a | . . 3 ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) | |
22 | 4, 12, 20, 21 | mbfres2 24246 | . 2 ⊢ (𝜑 → (ℜ ∘ 𝐹) ∈ MblFn) |
23 | imf 14472 | . . . 4 ⊢ ℑ:ℂ⟶ℝ | |
24 | fco 6531 | . . . 4 ⊢ ((ℑ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℑ ∘ 𝐹):𝐴⟶ℝ) | |
25 | 23, 2, 24 | sylancr 589 | . . 3 ⊢ (𝜑 → (ℑ ∘ 𝐹):𝐴⟶ℝ) |
26 | resco 6103 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐵) = (ℑ ∘ (𝐹 ↾ 𝐵)) | |
27 | 10 | simprd 498 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
28 | 26, 27 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
29 | resco 6103 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐶) = (ℑ ∘ (𝐹 ↾ 𝐶)) | |
30 | 18 | simprd 498 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
31 | 29, 30 | eqeltrid 2917 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
32 | 25, 28, 31, 21 | mbfres2 24246 | . 2 ⊢ (𝜑 → (ℑ ∘ 𝐹) ∈ MblFn) |
33 | ismbfcn 24230 | . . 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 711 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ∩ cin 3935 ↾ cres 5557 ∘ ccom 5559 ⟶wf 6351 ℂcc 10535 ℝcr 10536 ℜcre 14456 ℑcim 14457 MblFncmbf 24215 |
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-inf2 9104 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-fal 1550 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-se 5515 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-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 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-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 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-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xadd 12509 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-xmet 20538 df-met 20539 df-ovol 24065 df-vol 24066 df-mbf 24220 |
This theorem is referenced by: iblsplit 42271 |
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