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Mirrors > Home > MPE Home > Th. List > mbfres2 | 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. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
mbfres2.1 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
mbfres2.2 | ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) |
mbfres2.3 | ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) |
mbfres2.4 | ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) |
Ref | Expression |
---|---|
mbfres2 | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfres2.4 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ∪ 𝐶) = 𝐴) | |
2 | 1 | reseq2d 5991 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∪ 𝐶)) = (𝐹 ↾ 𝐴)) |
3 | mbfres2.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
4 | ffn 6730 | . . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴) | |
5 | fnresdm 6682 | . . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
6 | 3, 4, 5 | 3syl 18 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹) |
7 | 2, 6 | eqtr2d 2767 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
8 | 7 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
9 | resundi 6005 | . . . . . . . . 9 ⊢ (𝐹 ↾ (𝐵 ∪ 𝐶)) = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) | |
10 | 8, 9 | eqtrdi 2782 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
11 | 10 | cnveqd 5884 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
12 | cnvun 6156 | . . . . . . 7 ⊢ ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) | |
13 | 11, 12 | eqtrdi 2782 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶))) |
14 | 13 | imaeq1d 6070 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥)) |
15 | imaundir 6164 | . . . . 5 ⊢ ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) | |
16 | 14, 15 | eqtrdi 2782 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥))) |
17 | mbfres2.2 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
18 | ssun1 4173 | . . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
19 | 18, 1 | sseqtrid 4032 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
20 | 3, 19 | fssresd 6771 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶ℝ) |
21 | ismbf 25651 | . . . . . . . 8 ⊢ ((𝐹 ↾ 𝐵):𝐵⟶ℝ → ((𝐹 ↾ 𝐵) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol)) | |
22 | 20, 21 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol)) |
23 | 17, 22 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol) |
24 | 23 | r19.21bi 3239 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol) |
25 | mbfres2.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
26 | ssun2 4174 | . . . . . . . . . 10 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶) | |
27 | 26, 1 | sseqtrid 4032 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
28 | 3, 27 | fssresd 6771 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶ℝ) |
29 | ismbf 25651 | . . . . . . . 8 ⊢ ((𝐹 ↾ 𝐶):𝐶⟶ℝ → ((𝐹 ↾ 𝐶) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol)) | |
30 | 28, 29 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ((𝐹 ↾ 𝐶) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol)) |
31 | 25, 30 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) |
32 | 31 | r19.21bi 3239 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) |
33 | unmbl 25560 | . . . . 5 ⊢ (((◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol ∧ (◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) → ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) ∈ dom vol) | |
34 | 24, 32, 33 | syl2anc 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) ∈ dom vol) |
35 | 16, 34 | eqeltrd 2826 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) ∈ dom vol) |
36 | 35 | ralrimiva 3136 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol) |
37 | ismbf 25651 | . . 3 ⊢ (𝐹:𝐴⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) | |
38 | 3, 37 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol)) |
39 | 36, 38 | mpbird 256 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∪ cun 3945 ◡ccnv 5683 dom cdm 5684 ran crn 5685 ↾ cres 5686 “ cima 5687 Fn wfn 6551 ⟶wf 6552 ℝcr 11159 (,)cioo 13380 volcvol 25486 MblFncmbf 25637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-er 8736 df-map 8859 df-pm 8860 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9487 df-inf 9488 df-oi 9555 df-dju 9946 df-card 9984 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12613 df-uz 12877 df-q 12987 df-rp 13031 df-xadd 13149 df-ioo 13384 df-ico 13386 df-icc 13387 df-fz 13541 df-fzo 13684 df-fl 13814 df-seq 14024 df-exp 14084 df-hash 14350 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-clim 15492 df-sum 15693 df-xmet 21338 df-met 21339 df-ovol 25487 df-vol 25488 df-mbf 25642 |
This theorem is referenced by: mbfss 25669 mbfresfi 37369 mbfposadd 37370 mbfres2cn 45597 |
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