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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 24248 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 14473 | . . . 4 ⊢ ℜ:ℂ⟶ℝ | |
2 | mbfres2cn.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | fco 6533 | . . . 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 6105 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐵) = (ℜ ∘ (𝐹 ↾ 𝐵)) | |
6 | mbfres2cn.b | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
7 | fresin 6549 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ) | |
8 | ismbfcn 24232 | . . . . . . 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 2919 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
13 | resco 6105 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐶) = (ℜ ∘ (𝐹 ↾ 𝐶)) | |
14 | mbfres2cn.c | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
15 | fresin 6549 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶ℂ) | |
16 | ismbfcn 24232 | . . . . . . 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 2919 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
21 | mbfres2cn.a | . . 3 ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) | |
22 | 4, 12, 20, 21 | mbfres2 24248 | . 2 ⊢ (𝜑 → (ℜ ∘ 𝐹) ∈ MblFn) |
23 | imf 14474 | . . . 4 ⊢ ℑ:ℂ⟶ℝ | |
24 | fco 6533 | . . . 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 6105 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐵) = (ℑ ∘ (𝐹 ↾ 𝐵)) | |
27 | 10 | simprd 498 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
28 | 26, 27 | eqeltrid 2919 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
29 | resco 6105 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐶) = (ℑ ∘ (𝐹 ↾ 𝐶)) | |
30 | 18 | simprd 498 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
31 | 29, 30 | eqeltrid 2919 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
32 | 25, 28, 31, 21 | mbfres2 24248 | . 2 ⊢ (𝜑 → (ℑ ∘ 𝐹) ∈ MblFn) |
33 | ismbfcn 24232 | . . 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 3936 ∩ cin 3937 ↾ cres 5559 ∘ ccom 5561 ⟶wf 6353 ℂcc 10537 ℝcr 10538 ℜcre 14458 ℑcim 14459 MblFncmbf 24217 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-xmet 20540 df-met 20541 df-ovol 24067 df-vol 24068 df-mbf 24222 |
This theorem is referenced by: iblsplit 42258 |
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