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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 1909 | . . . . . 6 ⊢ Ⅎ𝑛 𝑘 ∈ 𝑍 | |
7 | 5, 6 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑘 ∈ 𝑍) |
8 | meaiunincf.f | . . . . . . 7 ⊢ Ⅎ𝑛𝐸 | |
9 | nfcv 2897 | . . . . . . 7 ⊢ Ⅎ𝑛𝑘 | |
10 | 8, 9 | nffv 6895 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘𝑘) |
11 | nfcv 2897 | . . . . . . 7 ⊢ Ⅎ𝑛(𝑘 + 1) | |
12 | 8, 11 | nffv 6895 | . . . . . 6 ⊢ Ⅎ𝑛(𝐸‘(𝑘 + 1)) |
13 | 10, 12 | nfss 3969 | . . . . 5 ⊢ Ⅎ𝑛(𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)) |
14 | 7, 13 | nfim 1891 | . . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
15 | eleq1w 2810 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝑛 ∈ 𝑍 ↔ 𝑘 ∈ 𝑍)) | |
16 | 15 | anbi2d 628 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑘 ∈ 𝑍))) |
17 | fveq2 6885 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘𝑛) = (𝐸‘𝑘)) | |
18 | fvoveq1 7428 | . . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑘 + 1))) | |
19 | 17, 18 | sseq12d 4010 | . . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1)) ↔ (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1)))) |
20 | 16, 19 | imbi12d 344 | . . . 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 2225 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐸‘𝑘) ⊆ (𝐸‘(𝑘 + 1))) |
23 | meaiunincf.x | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) | |
24 | breq2 5145 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ (𝑀‘(𝐸‘𝑛)) ≤ 𝑦)) | |
25 | 24 | ralbidv 3171 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥 ↔ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑦)) |
26 | nfv 1909 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) ≤ 𝑦 | |
27 | nfcv 2897 | . . . . . . . . . 10 ⊢ Ⅎ𝑛𝑀 | |
28 | 27, 10 | nffv 6895 | . . . . . . . . 9 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) |
29 | nfcv 2897 | . . . . . . . . 9 ⊢ Ⅎ𝑛 ≤ | |
30 | nfcv 2897 | . . . . . . . . 9 ⊢ Ⅎ𝑛𝑦 | |
31 | 28, 29, 30 | nfbr 5188 | . . . . . . . 8 ⊢ Ⅎ𝑛(𝑀‘(𝐸‘𝑘)) ≤ 𝑦 |
32 | 2fveq3 6890 | . . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑘))) | |
33 | 32 | breq1d 5151 | . . . . . . . 8 ⊢ (𝑛 = 𝑘 → ((𝑀‘(𝐸‘𝑛)) ≤ 𝑦 ↔ (𝑀‘(𝐸‘𝑘)) ≤ 𝑦)) |
34 | 26, 31, 33 | cbvralw 3297 | . . . . . . 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 3229 | . . . 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 2897 | . . . . 5 ⊢ Ⅎ𝑘(𝑀‘(𝐸‘𝑛)) | |
41 | 40, 28, 32 | cbvmpt 5252 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
42 | 39, 41 | eqtri 2754 | . . 3 ⊢ 𝑆 = (𝑘 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑘))) |
43 | 1, 2, 3, 4, 22, 38, 42 | meaiuninc 45774 | . 2 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘))) |
44 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑘(𝐸‘𝑛) | |
45 | fveq2 6885 | . . . 4 ⊢ (𝑘 = 𝑛 → (𝐸‘𝑘) = (𝐸‘𝑛)) | |
46 | 10, 44, 45 | cbviun 5032 | . . 3 ⊢ ∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘) = ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) |
47 | 46 | fveq2i 6888 | . 2 ⊢ (𝑀‘∪ 𝑘 ∈ 𝑍 (𝐸‘𝑘)) = (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) |
48 | 43, 47 | breqtrdi 5182 | 1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 ∀wral 3055 ∃wrex 3064 ⊆ wss 3943 ∪ ciun 4990 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 1c1 11113 + caddc 11115 ≤ cle 11253 ℤcz 12562 ℤ≥cuz 12826 ⇝ cli 15434 Meascmea 45742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-xadd 13099 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-salg 45602 df-sumge0 45656 df-mea 45743 |
This theorem is referenced by: meaiuninc3v 45777 |
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