Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meaiunincf | Structured version Visualization version GIF version |
Description: Measures are continuous from below (bounded case): if 𝐸 is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
Ref | Expression |
---|---|
meaiunincf.p | ⊢ Ⅎ𝑛𝜑 |
meaiunincf.f | ⊢ Ⅎ𝑛𝐸 |
meaiunincf.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meaiunincf.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
meaiunincf.z | ⊢ 𝑍 = (ℤ≥‘𝑁) |
meaiunincf.e | ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) |
meaiunincf.i | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) |
meaiunincf.x | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) |
meaiunincf.s | ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
Ref | Expression |
---|---|
meaiunincf | ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meaiunincf.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | meaiunincf.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | meaiunincf.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑁) | |
4 | meaiunincf.e | . . 3 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) | |
5 | meaiunincf.p | . . . . . 6 ⊢ Ⅎ𝑛𝜑 | |
6 | nfv 1920 | . . . . . 6 ⊢ Ⅎ𝑛 𝑘 ∈ 𝑍 | |
7 | 5, 6 | nfan 1905 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 ∈ 𝑍) |
8 | meaiunincf.f | . . . . . . 7 ⊢ Ⅎ𝑛𝐸 | |
9 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑛𝑘 | |
10 | 8, 9 | nffv 6678 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘𝑘) |
11 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑛(𝑘 + 1) | |
12 | 8, 11 | nffv 6678 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘(𝑘 + 1)) |
13 | 10, 12 | nfss 3867 | . . . . 5 ⊢ Ⅎ𝑛(𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)) |
14 | 7, 13 | nfim 1902 | . . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
15 | eleq1w 2815 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍)) | |
16 | 15 | anbi2d 632 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍))) |
17 | fveq2 6668 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘)) | |
18 | fvoveq1 7187 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1))) | |
19 | 17, 18 | sseq12d 3908 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))) |
20 | 16, 19 | imbi12d 348 | . . . 4 ⊢ (𝑛 = 𝑘 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))))) |
21 | meaiunincf.i | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) | |
22 | 14, 20, 21 | chvarfv 2241 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
23 | meaiunincf.x | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) | |
24 | breq2 5031 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑀‘(𝐸‘𝑛)) ≤ 𝑦)) | |
25 | 24 | ralbidv 3109 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦)) |
26 | nfv 1920 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) ≤ 𝑦 | |
27 | nfcv 2899 | . . . . . . . . . 10 ⊢ Ⅎ𝑛𝑀 | |
28 | 27, 10 | nffv 6678 | . . . . . . . . 9 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) |
29 | nfcv 2899 | . . . . . . . . 9 ⊢ Ⅎ𝑛 ≤ | |
30 | nfcv 2899 | . . . . . . . . 9 ⊢ Ⅎ𝑛𝑦 | |
31 | 28, 29, 30 | nfbr 5074 | . . . . . . . 8 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) ≤ 𝑦 |
32 | 2fveq3 6673 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑘))) | |
33 | 32 | breq1d 5037 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)) |
34 | 26, 31, 33 | cbvralw 3339 | . . . . . . 7 ⊢ (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦) |
35 | 34 | a1i 11 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)) |
36 | 25, 35 | bitrd 282 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)) |
37 | 36 | cbvrexvw 3349 | . . . 4 ⊢ (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦) |
38 | 23, 37 | sylib 221 | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦) |
39 | meaiunincf.s | . . . 4 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) | |
40 | nfcv 2899 | . . . . 5 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) | |
41 | 40, 28, 32 | cbvmpt 5128 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
42 | 39, 41 | eqtri 2761 | . . 3 ⊢ 𝑆 = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
43 | 1, 2, 3, 4, 22, 38, 42 | meaiuninc 43545 | . 2 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘))) |
44 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑘(𝐸‘𝑛) | |
45 | fveq2 6668 | . . . 4 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛)) | |
46 | 10, 44, 45 | cbviun 4919 | . . 3 ⊢ ∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
47 | 46 | fveq2i 6671 | . 2 ⊢ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
48 | 43, 47 | breqtrdi 5068 | 1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2113 Ⅎwnfc 2879 ∀wral 3053 ∃wrex 3054 ⊆ wss 3841 ∪ ciun 4878 class class class wbr 5027 ↦ cmpt 5107 dom cdm 5519 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 ℝcr 10607 1c1 10609 + caddc 10611 ≤ cle 10747 ℤcz 12055 ℤ≥cuz 12317 ⇝ cli 14924 Meascmea 43513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-disj 4993 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-oadd 8128 df-omul 8129 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-oi 9040 df-card 9434 df-acn 9437 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-xadd 12584 df-ico 12820 df-icc 12821 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-clim 14928 df-sum 15129 df-salg 43376 df-sumge0 43427 df-mea 43514 |
This theorem is referenced by: meaiuninc3v 43548 |
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