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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 5938 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ (𝐵 ∪ 𝐶)) = (𝐹 ↾ 𝐴)) |
| 3 | mbfres2.1 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
| 4 | ffn 6662 | . . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶ℝ → 𝐹 Fn 𝐴) | |
| 5 | fnresdm 6611 | . . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
| 6 | 3, 4, 5 | 3syl 18 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = 𝐹) |
| 7 | 2, 6 | eqtr2d 2772 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
| 8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = (𝐹 ↾ (𝐵 ∪ 𝐶))) |
| 9 | resundi 5952 | . . . . . . . . 9 ⊢ (𝐹 ↾ (𝐵 ∪ 𝐶)) = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) | |
| 10 | 8, 9 | eqtrdi 2787 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → 𝐹 = ((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
| 11 | 10 | cnveqd 5824 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶))) |
| 12 | cnvun 6100 | . . . . . . 7 ⊢ ◡((𝐹 ↾ 𝐵) ∪ (𝐹 ↾ 𝐶)) = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) | |
| 13 | 11, 12 | eqtrdi 2787 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → ◡𝐹 = (◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶))) |
| 14 | 13 | imaeq1d 6018 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥)) |
| 15 | imaundir 6108 | . . . . 5 ⊢ ((◡(𝐹 ↾ 𝐵) ∪ ◡(𝐹 ↾ 𝐶)) “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥)) | |
| 16 | 14, 15 | eqtrdi 2787 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) = ((◡(𝐹 ↾ 𝐵) “ 𝑥) ∪ (◡(𝐹 ↾ 𝐶) “ 𝑥))) |
| 17 | mbfres2.2 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐵) ∈ MblFn) | |
| 18 | ssun1 4130 | . . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) | |
| 19 | 18, 1 | sseqtrid 3976 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| 20 | 3, 19 | fssresd 6701 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶ℝ) |
| 21 | ismbf 25585 | . . . . . . . 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 3228 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐵) “ 𝑥) ∈ dom vol) |
| 25 | mbfres2.3 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ↾ 𝐶) ∈ MblFn) | |
| 26 | ssun2 4131 | . . . . . . . . . 10 ⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶) | |
| 27 | 26, 1 | sseqtrid 3976 | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
| 28 | 3, 27 | fssresd 6701 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶ℝ) |
| 29 | ismbf 25585 | . . . . . . . 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 3228 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡(𝐹 ↾ 𝐶) “ 𝑥) ∈ dom vol) |
| 33 | unmbl 25494 | . . . . 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 2836 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ran (,)) → (◡𝐹 “ 𝑥) ∈ dom vol) |
| 36 | 35 | ralrimiva 3128 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ran (,)(◡𝐹 “ 𝑥) ∈ dom vol) |
| 37 | ismbf 25585 | . . 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 1541 ∈ wcel 2113 ∀wral 3051 ∪ cun 3899 ◡ccnv 5623 dom cdm 5624 ran crn 5625 ↾ cres 5626 “ cima 5627 Fn wfn 6487 ⟶wf 6488 ℝcr 11025 (,)cioo 13261 volcvol 25420 MblFncmbf 25571 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-xadd 13027 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-xmet 21302 df-met 21303 df-ovol 25421 df-vol 25422 df-mbf 25576 |
| This theorem is referenced by: mbfss 25603 mbfresfi 37867 mbfposadd 37868 mbfres2cn 46202 |
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