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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 24249 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 14463 | . . . 4 ⊢ ℜ:ℂ⟶ℝ | |
2 | mbfres2cn.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | fco 6505 | . . . 4 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℜ ∘ 𝐹):𝐴⟶ℝ) | |
4 | 1, 2, 3 | sylancr 590 | . . 3 ⊢ (𝜑 → (ℜ ∘ 𝐹):𝐴⟶ℝ) |
5 | resco 6070 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐵) = (ℜ ∘ (𝐹 ↾ 𝐵)) | |
6 | mbfres2cn.b | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
7 | fresin 6521 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ) | |
8 | ismbfcn 24233 | . . . . . . 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 235 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn)) |
11 | 10 | simpld 498 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
12 | 5, 11 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
13 | resco 6070 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐶) = (ℜ ∘ (𝐹 ↾ 𝐶)) | |
14 | mbfres2cn.c | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
15 | fresin 6521 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶ℂ) | |
16 | ismbfcn 24233 | . . . . . . 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 235 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn)) |
19 | 18 | simpld 498 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
20 | 13, 19 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
21 | mbfres2cn.a | . . 3 ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) | |
22 | 4, 12, 20, 21 | mbfres2 24249 | . 2 ⊢ (𝜑 → (ℜ ∘ 𝐹) ∈ MblFn) |
23 | imf 14464 | . . . 4 ⊢ ℑ:ℂ⟶ℝ | |
24 | fco 6505 | . . . 4 ⊢ ((ℑ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℑ ∘ 𝐹):𝐴⟶ℝ) | |
25 | 23, 2, 24 | sylancr 590 | . . 3 ⊢ (𝜑 → (ℑ ∘ 𝐹):𝐴⟶ℝ) |
26 | resco 6070 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐵) = (ℑ ∘ (𝐹 ↾ 𝐵)) | |
27 | 10 | simprd 499 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
28 | 26, 27 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
29 | resco 6070 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐶) = (ℑ ∘ (𝐹 ↾ 𝐶)) | |
30 | 18 | simprd 499 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
31 | 29, 30 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
32 | 25, 28, 31, 21 | mbfres2 24249 | . 2 ⊢ (𝜑 → (ℑ ∘ 𝐹) ∈ MblFn) |
33 | ismbfcn 24233 | . . 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 712 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ↾ cres 5521 ∘ ccom 5523 ⟶wf 6320 ℂcc 10524 ℝcr 10525 ℜcre 14448 ℑcim 14449 MblFncmbf 24218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 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 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 |
This theorem is referenced by: iblsplit 42608 |
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