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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 5997 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∪ 𝐶)) = (𝐹 ↾ 𝐴)) |
| 3 | mbfres2.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
| 4 | ffn 6736 | . . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴) | |
| 5 | fnresdm 6687 | . . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
| 6 | 3, 4, 5 | 3syl 18 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹) |
| 7 | 2, 6 | eqtr2d 2778 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
| 8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
| 9 | resundi 6011 | . . . . . . . . 9 ⊢ (𝐹 ↾ (𝐵 ∪ 𝐶)) = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) | |
| 10 | 8, 9 | eqtrdi 2793 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
| 11 | 10 | cnveqd 5886 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
| 12 | cnvun 6162 | . . . . . . 7 ⊢ ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) | |
| 13 | 11, 12 | eqtrdi 2793 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶))) |
| 14 | 13 | imaeq1d 6077 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥)) |
| 15 | imaundir 6170 | . . . . 5 ⊢ ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) | |
| 16 | 14, 15 | eqtrdi 2793 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥))) |
| 17 | mbfres2.2 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
| 18 | ssun1 4178 | . . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
| 19 | 18, 1 | sseqtrid 4026 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| 20 | 3, 19 | fssresd 6775 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶ℝ) |
| 21 | ismbf 25663 | . . . . . . . 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 232 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol) |
| 24 | 23 | r19.21bi 3251 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol) |
| 25 | mbfres2.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
| 26 | ssun2 4179 | . . . . . . . . . 10 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶) | |
| 27 | 26, 1 | sseqtrid 4026 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
| 28 | 3, 27 | fssresd 6775 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶ℝ) |
| 29 | ismbf 25663 | . . . . . . . 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 232 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) |
| 32 | 31 | r19.21bi 3251 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) |
| 33 | unmbl 25572 | . . . . 5 ⊢ (((◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol ∧ (◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) → ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) ∈ dom vol) | |
| 34 | 24, 32, 33 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) ∈ dom vol) |
| 35 | 16, 34 | eqeltrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) ∈ dom vol) |
| 36 | 35 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol) |
| 37 | ismbf 25663 | . . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∪ cun 3949 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ℝcr 11154 (,)cioo 13387 volcvol 25498 MblFncmbf 25649 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500 df-mbf 25654 |
| This theorem is referenced by: mbfss 25681 mbfresfi 37673 mbfposadd 37674 mbfres2cn 45973 |
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