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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 1918 | . . . . . 6 ⊢ Ⅎ𝑛 𝑘 ∈ 𝑍 | |
7 | 5, 6 | nfan 1903 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 ∈ 𝑍) |
8 | meaiunincf.f | . . . . . . 7 ⊢ Ⅎ𝑛𝐸 | |
9 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑛𝑘 | |
10 | 8, 9 | nffv 6897 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘𝑘) |
11 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑛(𝑘 + 1) | |
12 | 8, 11 | nffv 6897 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘(𝑘 + 1)) |
13 | 10, 12 | nfss 3972 | . . . . 5 ⊢ Ⅎ𝑛(𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)) |
14 | 7, 13 | nfim 1900 | . . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
15 | eleq1w 2817 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍)) | |
16 | 15 | anbi2d 630 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍))) |
17 | fveq2 6887 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘)) | |
18 | fvoveq1 7426 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1))) | |
19 | 17, 18 | sseq12d 4013 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))) |
20 | 16, 19 | imbi12d 345 | . . . 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 2234 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
23 | meaiunincf.x | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) | |
24 | breq2 5150 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑀‘(𝐸‘𝑛)) ≤ 𝑦)) | |
25 | 24 | ralbidv 3178 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦)) |
26 | nfv 1918 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) ≤ 𝑦 | |
27 | nfcv 2904 | . . . . . . . . . 10 ⊢ Ⅎ𝑛𝑀 | |
28 | 27, 10 | nffv 6897 | . . . . . . . . 9 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) |
29 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑛 ≤ | |
30 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑛𝑦 | |
31 | 28, 29, 30 | nfbr 5193 | . . . . . . . 8 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) ≤ 𝑦 |
32 | 2fveq3 6892 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑘))) | |
33 | 32 | breq1d 5156 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)) |
34 | 26, 31, 33 | cbvralw 3304 | . . . . . . 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 279 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)) |
37 | 36 | cbvrexvw 3236 | . . . 4 ⊢ (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦) |
38 | 23, 37 | sylib 217 | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 (𝑀‘(𝐸‘𝑘)) ≤ 𝑦) |
39 | meaiunincf.s | . . . 4 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) | |
40 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) | |
41 | 40, 28, 32 | cbvmpt 5257 | . . . 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 45131 | . 2 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘))) |
44 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑘(𝐸‘𝑛) | |
45 | fveq2 6887 | . . . 4 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛)) | |
46 | 10, 44, 45 | cbviun 5037 | . . 3 ⊢ ∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
47 | 46 | fveq2i 6890 | . 2 ⊢ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
48 | 43, 47 | breqtrdi 5187 | 1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ∀wral 3062 ∃wrex 3071 ⊆ wss 3946 ∪ ciun 4995 class class class wbr 5146 ↦ cmpt 5229 dom cdm 5674 ⟶wf 6535 ‘cfv 6539 (class class class)co 7403 ℝcr 11104 1c1 11106 + caddc 11108 ≤ cle 11244 ℤcz 12553 ℤ≥cuz 12817 ⇝ cli 15423 Meascmea 45099 |
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 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-disj 5112 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-oadd 8464 df-omul 8465 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-oi 9500 df-card 9929 df-acn 9932 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-z 12554 df-uz 12818 df-rp 12970 df-xadd 13088 df-ico 13325 df-icc 13326 df-fz 13480 df-fzo 13623 df-seq 13962 df-exp 14023 df-hash 14286 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15427 df-sum 15628 df-salg 44959 df-sumge0 45013 df-mea 45100 |
This theorem is referenced by: meaiuninc3v 45134 |
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