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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 5936 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∪ 𝐶)) = (𝐹 ↾ 𝐴)) |
3 | mbfres2.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
4 | ffn 6666 | . . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴) | |
5 | fnresdm 6618 | . . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
6 | 3, 4, 5 | 3syl 18 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹) |
7 | 2, 6 | eqtr2d 2779 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
8 | 7 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
9 | resundi 5950 | . . . . . . . . 9 ⊢ (𝐹 ↾ (𝐵 ∪ 𝐶)) = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) | |
10 | 8, 9 | eqtrdi 2794 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
11 | 10 | cnveqd 5830 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
12 | cnvun 6094 | . . . . . . 7 ⊢ ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) | |
13 | 11, 12 | eqtrdi 2794 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶))) |
14 | 13 | imaeq1d 6011 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥)) |
15 | imaundir 6102 | . . . . 5 ⊢ ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) | |
16 | 14, 15 | eqtrdi 2794 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥))) |
17 | mbfres2.2 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
18 | ssun1 4131 | . . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
19 | 18, 1 | sseqtrid 3995 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
20 | 3, 19 | fssresd 6707 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶ℝ) |
21 | ismbf 24944 | . . . . . . . 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 3233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol) |
25 | mbfres2.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
26 | ssun2 4132 | . . . . . . . . . 10 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶) | |
27 | 26, 1 | sseqtrid 3995 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
28 | 3, 27 | fssresd 6707 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶ℝ) |
29 | ismbf 24944 | . . . . . . . 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 3233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) |
33 | unmbl 24853 | . . . . 5 ⊢ (((◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol ∧ (◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) → ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) ∈ dom vol) | |
34 | 24, 32, 33 | syl2anc 585 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) ∈ dom vol) |
35 | 16, 34 | eqeltrd 2839 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) ∈ dom vol) |
36 | 35 | ralrimiva 3142 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol) |
37 | ismbf 24944 | . . 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 257 | 1 ⊢ (𝜑 → 𝐹 ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ∪ cun 3907 ◡ccnv 5631 dom cdm 5632 ran crn 5633 ↾ cres 5634 “ cima 5635 Fn wfn 6489 ⟶wf 6490 ℝcr 11009 (,)cioo 13219 volcvol 24779 MblFncmbf 24930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-inf2 9536 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-se 5588 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7610 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8607 df-map 8726 df-pm 8727 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-sup 9337 df-inf 9338 df-oi 9405 df-dju 9796 df-card 9834 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-n0 12373 df-z 12459 df-uz 12723 df-q 12829 df-rp 12871 df-xadd 12989 df-ioo 13223 df-ico 13225 df-icc 13226 df-fz 13380 df-fzo 13523 df-fl 13652 df-seq 13862 df-exp 13923 df-hash 14185 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-clim 15330 df-sum 15531 df-xmet 20742 df-met 20743 df-ovol 24780 df-vol 24781 df-mbf 24935 |
This theorem is referenced by: mbfss 24962 mbfresfi 36056 mbfposadd 36057 mbfres2cn 44094 |
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