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Mirrors > Home > MPE Home > Th. List > mbfpos | Structured version Visualization version GIF version |
Description: The positive part of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014.) |
Ref | Expression |
---|---|
mbfpos.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
mbfpos.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
Ref | Expression |
---|---|
mbfpos | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 10034 | . . . . . . 7 ⊢ 0 ∈ V | |
2 | 1 | fvconst2 6469 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 × {0})‘𝑥) = 0) |
3 | 2 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐴 × {0})‘𝑥) = 0) |
4 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
5 | mbfpos.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
6 | eqid 2622 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
7 | 6 | fvmpt2 6291 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
8 | 4, 5, 7 | syl2anc 693 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
9 | 3, 8 | breq12d 4666 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 0 ≤ 𝐵)) |
10 | 9, 8, 3 | ifbieq12d 4113 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝐴 × {0})‘𝑥)) = if(0 ≤ 𝐵, 𝐵, 0)) |
11 | 10 | mpteq2dva 4744 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝐴 × {0})‘𝑥))) = (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0))) |
12 | 0re 10040 | . . . . 5 ⊢ 0 ∈ ℝ | |
13 | 12 | fconst6 6095 | . . . 4 ⊢ (𝐴 × {0}):𝐴⟶ℝ |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐴 × {0}):𝐴⟶ℝ) |
15 | mbfpos.2 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) | |
16 | 15, 5 | mbfdm2 23405 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ dom vol) |
17 | 0cnd 10033 | . . . 4 ⊢ (𝜑 → 0 ∈ ℂ) | |
18 | mbfconst 23402 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 0 ∈ ℂ) → (𝐴 × {0}) ∈ MblFn) | |
19 | 16, 17, 18 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝐴 × {0}) ∈ MblFn) |
20 | 5, 6 | fmptd 6385 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
21 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑦if(((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝐴 × {0})‘𝑥)) | |
22 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑥((𝐴 × {0})‘𝑦) | |
23 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑥 ≤ | |
24 | nffvmpt1 6199 | . . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) | |
25 | 22, 23, 24 | nfbr 4699 | . . . . 5 ⊢ Ⅎ𝑥((𝐴 × {0})‘𝑦) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) |
26 | 25, 24, 22 | nfif 4115 | . . . 4 ⊢ Ⅎ𝑥if(((𝐴 × {0})‘𝑦) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝐴 × {0})‘𝑦)) |
27 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 × {0})‘𝑥) = ((𝐴 × {0})‘𝑦)) | |
28 | fveq2 6191 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) | |
29 | 27, 28 | breq12d 4666 | . . . . 5 ⊢ (𝑥 = 𝑦 → (((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ ((𝐴 × {0})‘𝑦) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))) |
30 | 29, 28, 27 | ifbieq12d 4113 | . . . 4 ⊢ (𝑥 = 𝑦 → if(((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝐴 × {0})‘𝑥)) = if(((𝐴 × {0})‘𝑦) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝐴 × {0})‘𝑦))) |
31 | 21, 26, 30 | cbvmpt 4749 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ if(((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝐴 × {0})‘𝑥))) = (𝑦 ∈ 𝐴 ↦ if(((𝐴 × {0})‘𝑦) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦), ((𝐴 × {0})‘𝑦))) |
32 | 14, 19, 20, 15, 31 | mbfmax 23416 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(((𝐴 × {0})‘𝑥) ≤ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥), ((𝐴 × {0})‘𝑥))) ∈ MblFn) |
33 | 11, 32 | eqeltrrd 2702 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ifcif 4086 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 ℂcc 9934 ℝcr 9935 0cc0 9936 ≤ cle 10075 volcvol 23232 MblFncmbf 23383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xadd 11947 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-xmet 19739 df-met 19740 df-ovol 23233 df-vol 23234 df-mbf 23388 |
This theorem is referenced by: mbfposb 23420 mbfi1flimlem 23489 itgreval 23563 ibladdlem 23586 iblabslem 23594 mbfposadd 33457 ibladdnclem 33466 iblabsnclem 33473 itgmulc2nclem2 33477 |
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