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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 25594 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 15099 | . . . 4 ⊢ ℜ:ℂ⟶ℝ | |
2 | mbfres2cn.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | fco 6752 | . . . 4 ⊢ ((ℜ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℜ ∘ 𝐹):𝐴⟶ℝ) | |
4 | 1, 2, 3 | sylancr 585 | . . 3 ⊢ (𝜑 → (ℜ ∘ 𝐹):𝐴⟶ℝ) |
5 | resco 6259 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐵) = (ℜ ∘ (𝐹 ↾ 𝐵)) | |
6 | mbfres2cn.b | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
7 | fresin 6771 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐵):(𝐴 ∩ 𝐵)⟶ℂ) | |
8 | ismbfcn 25578 | . . . . . . 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 231 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn)) |
11 | 10 | simpld 493 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
12 | 5, 11 | eqeltrid 2833 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
13 | resco 6259 | . . . 4 ⊢ ((ℜ ∘ 𝐹) ↾ 𝐶) = (ℜ ∘ (𝐹 ↾ 𝐶)) | |
14 | mbfres2cn.c | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
15 | fresin 6771 | . . . . . . 7 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ 𝐶):(𝐴 ∩ 𝐶)⟶ℂ) | |
16 | ismbfcn 25578 | . . . . . . 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 231 | . . . . 5 ⊢ (𝜑 → ((ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn ∧ (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn)) |
19 | 18 | simpld 493 | . . . 4 ⊢ (𝜑 → (ℜ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
20 | 13, 19 | eqeltrid 2833 | . . 3 ⊢ (𝜑 → ((ℜ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
21 | mbfres2cn.a | . . 3 ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) | |
22 | 4, 12, 20, 21 | mbfres2 25594 | . 2 ⊢ (𝜑 → (ℜ ∘ 𝐹) ∈ MblFn) |
23 | imf 15100 | . . . 4 ⊢ ℑ:ℂ⟶ℝ | |
24 | fco 6752 | . . . 4 ⊢ ((ℑ:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (ℑ ∘ 𝐹):𝐴⟶ℝ) | |
25 | 23, 2, 24 | sylancr 585 | . . 3 ⊢ (𝜑 → (ℑ ∘ 𝐹):𝐴⟶ℝ) |
26 | resco 6259 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐵) = (ℑ ∘ (𝐹 ↾ 𝐵)) | |
27 | 10 | simprd 494 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐵)) ∈ MblFn) |
28 | 26, 27 | eqeltrid 2833 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐵) ∈ MblFn) |
29 | resco 6259 | . . . 4 ⊢ ((ℑ ∘ 𝐹) ↾ 𝐶) = (ℑ ∘ (𝐹 ↾ 𝐶)) | |
30 | 18 | simprd 494 | . . . 4 ⊢ (𝜑 → (ℑ ∘ (𝐹 ↾ 𝐶)) ∈ MblFn) |
31 | 29, 30 | eqeltrid 2833 | . . 3 ⊢ (𝜑 → ((ℑ ∘ 𝐹) ↾ 𝐶) ∈ MblFn) |
32 | 25, 28, 31, 21 | mbfres2 25594 | . 2 ⊢ (𝜑 → (ℑ ∘ 𝐹) ∈ MblFn) |
33 | ismbfcn 25578 | . . 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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cun 3947 ∩ cin 3948 ↾ cres 5684 ∘ ccom 5686 ⟶wf 6549 ℂcc 11144 ℝcr 11145 ℜcre 15084 ℑcim 15085 MblFncmbf 25563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 df-mbf 25568 |
This theorem is referenced by: iblsplit 45383 |
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